Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 1-11

Backward Regularity for some Infinite Dimensional
Hypoelliptic Semi-groups
H\’el\‘ene Airault and Paul Malliavin

CONTENTS
1. Regularity of the canonic diﬀusion above Virasoro algebra
2. Infinite dimensionnal non autonomous riemannian metrics
3. Instability of HJM model in Mathematical FinanceWe dedicate this work to Kiyosi It\^o, the “Newton of continuous
Stochastic Dynamic” , one of the most influential scholars of the lastcentury; the topic of this paper underlines in itsef the deep influencethat the 1976 Kyoto Symposium [12] has had on the whole subsequent
carrier of the second author who is also deeply indebted to Kiyosi It\^ofor fifty years of warm personnal relations; his attentive support ﬀom
the beginning to some of our scientific enterprises has been a key step
towards their international recognition.

In classical Stochastic Analysis regularity properties are time independent : the Brownian motion is for all time H\"olderian of orderregular, the tangent space to the Wiener space (i.e. the Cameron-Martin
space) is also time independent. The Stochastic Analysis on Loop groups
have recently confirmed the paradigm that regularity properties are time
independent.
It has been a surprise that regularity exponents for highly non linear
infinite dimensionnal diﬀusion as the canonic diﬀusion above Virasoro
algebra are time dependent $[2],[9]$ . We shall discuss in this paper the
status of tangent space to Virasoro diﬀusion; we shall exhibit a minimal tangent space which is time independent; it is conceivable that the
maximal tangent space is time dependent, fact which will be established
on a toy model. The finite dimensional root of of this phenomen lies indoes not satisfy simple scaling
the fact that hypoelliptic diﬀusion on
relation when the time goes to zero [4], [11].
Stability of interest models in Mathematical Finance are deeply affected by these infinite dimensional eﬀects.
$(\frac{1}{2}-\epsilon)$$R^{d}$

Received April 24, 2003.

2

H. Airault and P. Malliavin

1.

Regularity of the canonical diﬀusion above Virasoro algebra.

, $Diffff(S^{1})$ , has fordiﬀeomorphism of the circle
The group of; we identify a function
thevector fields on
Lie algebra; with this identication the bracket ofto the vector field$[u,v]=\dot{v}u-\dot{u}v$. Complexifying the underlying
vector fields becomes
expression
space
get
following
wethefor this bracket in thereal vector
complex trigonometric basis :$S^{1}$$C^{\infty}$$S^{1}$$C^{\infty}$$diffff(S^{1})$$u(\theta)$$u(\theta)\frac{d}{d\theta}$$[e^{in\theta}, e^{im\theta}]=i(m-n)e^{i(m+n)\theta}$

Given a positive constant $c>0$ , define the bilinear antisymmetric form$\omega_{c}(f, g):=-\frac{c}{12}\int_{S^{1}}(f^{J}+f^{(3)})gd\theta$;then$\omega_{c}([f_{1}, f_{2}], f_{3})+\omega_{c}([f_{2}, f_{3}], f_{1})+\omega_{c}([f_{3}, f_{1}], f_{2})=0$$\omega_{c}(e^{in\theta}, e^{-im\theta})=i\delta_{n}^{m}\frac{c}{6}(n^{3}-n)$Virasoro algebra is defined asbracket :, $n>0$ .

$\mathcal{V}_{c}:=R\oplus diff(S^{1})$with the following$[\xi\kappa+f, \eta\kappa+g]:=\omega_{c}(f, g)\kappa+[f, g]$

Brownian motion on Diﬀ$(S^{1})$

Define the Hilbertian metric$||\emptyset||_{H^{g}}^{2}2=\sum_{n>1}(n^{3}-n)(a_{n}^{2}+b_{n}^{2})$

,

,

.

.

$\frac{3}{2}$by :$\phi(\theta)=\frac{1}{2}a_{0}+\sum_{n=1}^{+\infty}(a_{n}\cos n\theta+b_{n}\sin n\theta)$;

define$e_{n}$:$R^{2}\mapsto diffff(S^{1})$,$e_{n}(\xi)=\frac{1}{\sqrt{n^{3}-n}}(\xi^{1}\cos n\theta+\xi^{2}\sin n\theta)$, $n>1$ .

-valuedLetbe independent copies of Wiener space of the
Brownian motion; define $X=\otimes X_{k}$ and consider the Stratonovich
SDE:$R^{2}$$X_{k}$$dg_{x}(t)=(\sum_{k>1}e_{k}(dx_{k}(t)))og_{x}(t)$,$gx(O)=Identity$

3

Backward regularity for infinite dimensional semi-groups$dg_{x}(rt)=(\sum_{k>1}r^{k}e_{k}(dx_{k}(t)))og_{x}^{r}(t)$,$g_{x}^{r}(0)=Identity$then, $gx(t)$ eDiﬀ $(S^{1})\forall r<1$ .
Theorem. [2], [9].
the group of homemorphism of
Denotemodulus of continuity , then$H^{\beta}(S^{1})$$S^{1}$;, with an H\"olderian$\beta$, $a.s.$ ,$\lim_{r\rightarrow 1}g_{x}^{r}(t):=g_{x}(t)\in H^{\beta(t)}(S^{1})$$\beta(t)=\frac{1-\sqrt{1-e^{-\frac{t}{2}}}}{1+\sqrt{1-e^{-\frac{t}{2}}}}$.

.
The laws
satisfyofRemark. The composition of two homemorphisms of H\"olderian expo: this factnents ,
can have an H\"olderian exponent as worst as.
when
explains the exponential decrease of
It is obvious that the metric used to construct the Brownian mo,, 1. The Lie subagebration degenerates on the vector fields
generated by these three vector fields is isomorphic to $sl(2, R)$ ; the corresponding subgroup of $Diffff(S^{1})$ is the restriction to the circle of thegroup of M\"obius transformations of the unit disk.
is an homogeneous RieIt had be shown [1] thatmannian manifold, that the Hilbert transform on the circle pass to thequotient and defines an integrable almost complex structure for whichbecomes an homogeneous K\"ahler manifold. Denote :and defines the heat
, thenis the Brownian motion on. This section will prove the backward
semi-group on function on
regularity of this heat semi-group.
$\iota/_{t}$$g_{x}(t)$$lJ_{t}*\iota/_{t’}=\iota\nearrow t+t’$$\gamma\gamma^{/}$$\gamma^{/}$$\gamma$$\beta(t)$$\cos\theta$$\rightarrow+\infty$$\sin\theta$$\Gamma$$\mathcal{M}_{1}:=Diffff(S^{1})/\Gamma$$\mathcal{M}_{1}$$\mathcal{M}_{1}$$Diffff(S^{1})\rightarrow$$\pi$$\pi(g_{x}^{-1}(t))$$\mathcal{M}_{1}$$\mathcal{M}_{1}$Background of finite dimensional Stochastic Riemannian Geometry.
Denote by $M$ a Riemannian manifold of dimension ; a frame
is a Euclidean isomorphism of
onto the tangent plane $T_{\pi(r)}(M)$ ; thecollection of all ﬀames on $M$ is a smooth manifold $O(M)$ on which the orthogonal group operates on the right : this is the bundle of orthonormal
frames. The Levi-Civita connection defines on $O(M)$ a parallelism thatis a canonical diﬀerential form of degree 1, with values in. Riemannian geometry is encompassed in the Darbouxlet
Cartan structural equations :
$d$

$r$

$R^{d}$$R^{d}\oplus R^{d}\otimes_{a}R^{d}$$\omega=(\dot{\omega},\dot{\omega})$$<A\wedge B$,$d\dot{\omega}>=\ddot{\omega}(A)\dot{\omega}(B)-\dot{\omega}(B)\dot{\omega}(A)$,H. Airault and P. Malliavin

4$<A\wedge B$,$d\dot{\omega}>=\dot{\omega}(A)\dot{\omega}(B)-\dot{\omega}(B)\dot{\omega}(A)+\Omega(\dot{\omega}(A),\dot{\omega}(B))$

where is the Riemann curvature tensor.
Given anvalued brownian motion
thebyis defined
Stratonovitch SDE,$\Omega$$R^{d}$$<dr_{x},\dot{\omega}>=dx$,$x(\tau)$$<dr_{x},\ddot{\omega}>=0$

the horizontal diﬀusion,$r_{x}(0)=r_{0}$,where $r_{0}\in O(M)$ is fixed. The It\^o parallel transport is the isometry$t_{0-\tau}^{x}$:$T_{\pi(r_{x}(\tau)}(M)\mapsto T_{\pi(r_{0})}(M)$A variation induces
defined bydefined by $t_{0-\tau}^{x}=r_{x}(0)o(r_{x}(\tau))^{-1}$ .

$x\mapsto x+\epsilon\tilde{\zeta}$$\zeta(\tau):=<\frac{dr^{\epsilon}(\tau)}{d\epsilon_{=0}},\dot{\omega}>$,

induces a variation of the path$\rho(\tau):=<\frac{dr^{\epsilon}(\tau)}{d\epsilon_{=0}}$,$\dot{\omega}>$,$(\zeta, \rho)$$r^{\epsilon}(\tau):=r_{x+\epsilon}(\tau)$.

These two variations are linked by the two following key SDE [6], [10],[7], [14], the first being an It\^o SDE, the second a Stratonovitch SDE :(1.1)$d\tilde{\zeta}=d\zeta-\frac{1}{2}Ricci(\zeta)d\tau-\rho dx$,$d\rho=\Omega(\zeta, odx)$.

.
Two parallel transports on
two connections :We follow Bowick-Lahiri [5]. We have on
the Levi-Civita connection
and the connection
induced by theleﬃ invariant Maurer-Cartan form on $Diffff(S^{1})$ ; we introduce a tensorial
operator on
defined by$\mathcal{M}_{1}$$\mathcal{M}_{1}$$\nabla_{X}$$\mathcal{L}_{X}$$T_{0}(\mathcal{M}_{1})$$\phi_{X}=\mathcal{L}_{X}-\nabla_{X}$The operator , extended to the complexification, has the following expression in the complex trigonometric basis :$\phi$(1.2)$\phi_{e^{ir\theta}}(e^{iq\theta})=i(r-q)\ominus(-q-r)$, $r>1$ ,where $\Theta(t):=1_{[0,+\infty[}$ is the Heaviside function. For $s<-1$ we prolon.
by requiring hermitian symmetry :gate
can be expressed in terms
Then the Riemannian curvature ofbyof the operator$\phi_{e^{i\epsilon\theta}}:=(\phi_{e^{-is\theta}})^{*}$$\phi_{*}$$\mathcal{M}_{1}$$\phi_{*}$$\Omega(X, Y)=[\nabla_{X}, \nabla_{Y}]-\nabla[X,Y]=[\phi x, \phi_{Y}]-\phi[X,Y]$,

together withthe last identity results from$=\nabla_{[X,Y]}$ identity coming ﬀom the invariance of the K\"ahlerian metric$[\mathcal{L}_{X}, \mathcal{L}_{Y}]-\mathcal{L}_{[X,Y]}=0$$[\mathcal{L}_{X},\nabla_{Y}]$Backward regularity for infinite dimensional semi-groups

under the left action of Diﬀ[5], and its $trace$ is$(S^{1})$

5. The curvature tensor is of $trace$ class

Ricci $=-\frac{13}{6}\times Identity$(1.3)

Lemma.
,, $k\in[2, q]$ then the
Denotethe space generated bypreserve
operators
and are nilpotent on .
, where,is the matrix associatedDenote,.
toin the real trignometric basis
Theorem
The matrix Stratonovich $SDE$$\cos k\theta$$V_{q}$$\phi_{*}$$\sin k\theta$$V_{q}$$V_{q}$$\xi\in R^{2}$$\eta_{n}(\xi)=\tilde{\phi}_{e_{n}(\xi)}$$\tilde{\phi}_{X}$$(n^{3}-n)^{-\frac{1}{2}}\cos n\theta$$\phi_{X}$(1.4)$M_{t}=\mathcal{U}_{t}o(-\sum_{k>1}\eta_{k}(dx_{k}(t)))$,$(n^{3}-n)^{-\frac{1}{2}}\sin n\theta$$\mathcal{U}_{0}=Identity$

is a unitary matrix.
has a unique solution and
Proof.
The restriction toof this SDE is equivalent to an SDE which isdriven only by $2q$ Brownian motion; this SDE which is solvable by thefinite dimensional theory$\mathcal{U}_{t}$$V_{q}$$o$

Backward regularity. (Minimal tangent space)
Theorem.
Givendefined on

(1.5)

$z$

such that,$||z||_{H^{\frac{3}{2}}}<\infty$

then, for a generic test function$\Phi$$\mathcal{M}_{1}$$|\frac{d}{d\epsilon_{=0}}E((\pi^{*}\Phi)(\exp(\epsilon z)g_{x}(t))))|^{2}$$\leq\frac{13}{6(1-\exp(-\frac{13}{6}t)}||z||^{2}H^{\frac{3}{2}}E(|\pi^{*}\Phi(g_{x}(t))|^{2})$.

Proof.
We follow the strategy that Driver [8] developped in the case of Loopgroups making the change of variables$y_{t}=\int_{0}^{t}\mathcal{U}_{s}dx(s)$;then is a new brownian motion to which we can apply the finite dimensional Riemannian geometry because the curvature operator preservesthe$y_{t}$$V_{q}$$\blacksquare$

6

H. Airault and P. Malliavin

Infinite dimensional non autonomous Riemannian metrics.

2.

Consider a group $G$ of dimension finite or infinite; for instance $G$
could be the group of diﬀeomorphism of a compact manifold, case which
includes the theory of Stochastic Flows.
We consider a left invariant diﬀusion on $G$ ; denote by
its infinitesimal operator where theare left invariant vector
field on $G$ ; denote by
the corresponding gradient : $\nabla\phi*\nabla\psi:=$.
We denote by $p_{T}(dg)$ the law of the process starting from the identity. Given a tangent vector at the identity define the “logarithmic
by the identity
derivative” of$\triangle=\frac{1}{2}\sum_{k\geq 1}\partial_{A_{k}}^{2}$$A_{k}$$+\partial_{A_{0}}$$\nabla$$\triangle(\phi\psi)-\phi\triangle\psi-\psi\triangle\phi$$z$$p_{T}$(2.1)$\frac{d}{d\epsilon_{=0}}E(\Phi(\exp(\epsilon z)g_{x}(T))=E(K_{z,,,T}(g_{x}(T))\Phi(g_{x}(T)))$,

where is a generic test function.
For all $T>0$ define a Hilbertian norm by$\Phi$(2.1)$||z||_{T}^{2}:=E(|K_{z,,,T}(g_{x}(T))|^{2}))$.

Theorem.

If $T<T^{/}$ then
(2.3)$||z||_{T’}\leq||z||_{T}$.

Proof.
For $\eta>0$ define $\Psi(g):=E_{g_{x}(T)=g}(\Phi(g_{x}(T+\eta))$ , then$E(\Phi(\exp(\epsilon z)g_{x}(T+\eta))=E(E^{N_{T}}(\Phi(\exp(\epsilon z)g_{x}(T+\eta))$$=E(\Psi(\exp(\epsilon z)g_{x}(T)))$diﬀerentiating relatively to$\epsilon$;

we obtain$E(K_{z,,,T+\eta}(g_{x}(T+\eta))\Phi(g_{x}(T+\eta)))=E(K_{z,,,T}(g_{x}(T))\Psi(g_{x}(T)))$

letting$\eta\rightarrow 0$

we write $\simeq equalities$ modulo$o(\epsilon)$,; then by It\^o calculus :$K_{z,,,T+\eta}(g_{x}(T+\eta)-K_{z,,,T}(g_{x}(T)\simeq\eta(\frac{\partial K}{\partial T}+\triangle K)+\nabla K*(x(T+\eta)-x(T))$$\Psi(g)-\Phi(g)\simeq\eta\triangle\Phi(g)$,$\Phi(g_{x}(T+\eta))\simeq\Phi(g_{x}(T))+\eta(\triangle\Phi(g_{x}(T)))+\nabla\Phi*(x(T+\eta)-x(T))$

7

Backward regularity for infinite dimensional semi-groups$\frac{1}{\eta}E^{N_{T}}(K_{z,,,T+\eta}(g_{x}(T+\eta))\Phi(g_{x}(T+\eta)))-(K_{z,,,T}(g_{x}(T))\Phi(g_{x}(T)))$$\simeq\Phi(\frac{\partial K}{\partial T}+\triangle K)+K\triangle\Phi+\nabla\Phi*\nabla K$;$\frac{1}{\eta}E(K_{z,,,T+\eta}(g_{x}(T+\eta))\Phi(g_{x}(T+\eta)))-(K_{z,,,T}(g_{x}(T))\Phi(g_{x}(T)))$$\simeq\Phi(\frac{\partial K}{\partial T}+\triangle K)+K\triangle\Phi+\nabla\Phi*\nabla K)-K\triangle\Phi$(2.4)$E(\Phi(\frac{\partial K}{\partial T}+\triangle(K))+\nabla K*\nabla\Phi)=0$.

From the other hand$\frac{\partial}{\partial T}E[(K_{T}(g))^{2})]=E[\triangle(K_{T}^{2})+\frac{\partial K_{T}^{2}}{\partial T}]$$=E[2K_{T}(\frac{\partial K}{\partial T}+\triangle(K_{T}))+\nabla K_{T}*\nabla K_{T})]=-E[\nabla K_{T}*\nabla K_{T}]<0$,

the last equality is obtained by applying (2.4) with $\Phi=K_{T}$generated by vector fields
Consider now the ﬀee Lie algebra,; denote $G$ the infinite dimensional group associated. Denotea -dimensional Brownian motion and define on $G$ the process by thefollowing Stratanovitch SDE$d$$\mathcal{G}$$A_{1}$$\ldots A_{d}$

$x$

$d$

$dg_{x}(t)=g_{x}(t)o\sum_{k=1}^{d}A_{k}dx^{k}(t)$,$g_{x}(0)=Identity$.
denotethe completion of for the norm
Theorem. For $T\neq T’$ , we have
(2.5)
, which means the inequivalence of the corresponding norms.
$\prime\mu_{T}$$||z||_{T}$$\mathcal{G}$$H_{T}\neq\prime\mu_{T’}$

Proof.
We shall use the Ben-Arous expansion [3] ( see Theorem 15)$g_{x}(t)=\exp(\sum_{m=1}^{\infty}\sum_{J\in\sigma_{m}}M_{J}(t)U^{J})$,,where $A^{J}:=[A_{j_{1}}$ , [subset ofsuch that the$A_{j_{2}}$$[1, d]^{m}$$\ldots$, wheredenotes a maximal
are linearly independent in
and$[A_{j_{n-1}}, A_{j_{\mathfrak{n}}}]$$A^{J}$$\sigma_{m}$$\mathcal{G}$8

H. Airault and P. Malliavin

where iterated integrals
mutually orthogonal inhave been constructed by Meyer and are. We decompose$M_{J}$$L^{2}$$z=\sum_{m=1}^{\infty}z_{m}$,$z_{m}=\sum_{J\in\sigma_{m}}c_{J}A^{J}$.

Lemma.
(2.6)$||z||_{T}^{2}=\sum_{m=1}^{\infty}||z_{m}||_{T}^{2}$

By the rescaling of Meyer integrals we have$||z_{m}||_{T}^{2},$$=[\frac{T}{T},]^{m}||z_{m}||_{T}^{2}$

relation which shows the inequivalence of the two norms

3.

Instability of Heath-Jarrow-Merton model of interest rates.

All long terms loans (States bounds, mortgages, companies bounds)
are appearing on a single market, the “ zero coupon default ﬀee bonds
market”. Every day it is possible to buy bonds at any maturity between
1 up to 360 months; for each maturity the market gives a price; allthese prices can be summarized by a single positive function
theinstantaneous forward rate such that the discount price today of a 1
dollar bound paid in five years is equal to$r_{t}(x)$$\exp(-\int_{0}^{60}r_{t}(x)dx)$.

The associated configuration space is $(R^{+})^{360}$ .
The HJM model replace the by the space of continuous positive
functions, $x\in[0,360]$ and propose that “for the risk ﬀee measure”
the interest rate curve dynamic can be described by the following It\^o
SDE, driven by independent Brownian motion $W^{*}(t)$ ,
$C$

$C$

$r_{t}(x)$$q$$dr_{t}(x)=(\frac{\partial r_{t}(x)}{\partial x}+Z_{t}(x))dt+\sum_{k=1}^{q}\phi_{k,,,t}(x)dW^{k}(t)$(3.1)$Z_{t}(x)=\sum_{k=1}^{q}\phi_{k,,,t}(x)\int_{0}^{x}\phi_{k,,,t}(s)ds$.

,

Backward regularity for infinite dimensional semi-groups

9This HJM modell can be mathematically established under the two general assumptions : market where an agent cannot increase his wealthwithout risk (arbitrage free) and market variations ﬀee ﬀom jumps.
A practical fact is that the variance injected in the equation is very
low : $q\leq 4$ . This means that the operator associated with the SDE(3.1) is an hypoellitic operator driven by at most four vectors fields in
Euclidean space of large dimension.
being thedefined as
Consider the Stochastic flow
the Jacobian of thesolution of (3.1) for $r_{W}(t_{0})=r_{0}$ . Denote byflow
which is defined by solving the linearized SDE.
Greeks means the reaction of the market at an infinitesimal pertuof
appearing at time , $W^{*}(s)-W^{*}(t_{0})$ , $s\geq t_{0}$ being
bationfixed :$a$$U_{t-t_{0}}^{W}(r_{0})$$U_{t-t_{0}}^{W}$$J_{t-t_{0}}^{W}$$U_{t-t_{0}}^{W}$$\delta_{0}$$t_{0}$$r_{0}$$\frac{d}{d\epsilon_{\epsilon=0}}U_{t-t_{0}}^{W}(r_{0}+\epsilon\delta_{0})=J_{t-t_{0}}^{W}(\delta_{0}):=\delta^{W}(t)$,is called the Greek propagation.
Every trader can buy or sell european options which is a contractby which the seller obliges himself to pay at maturity $T$ an amount ofmoney equal to $F(r_{T})$ . The option is called digital if the function $F$ isdiscontinuous.
Sensitivities at the option $F$ is defined$\frac{d}{d\epsilon_{\epsilon=0}}E(F(U_{T-t_{0}}^{W}(r_{0}+\epsilon\delta_{0}))=E(<dF, J_{T-t_{0}}^{W}(\delta_{0})>)$.

Sensitivities regularization for digital european options
Denote the vector space of all possible infinitesimal pertubationof the market at time ; consider the Hilbertian norm
definedin (2.2) and denotethe corresponding Hilbert space then$C$$\delta_{0}$$||\delta||_{T,,,t_{0}}$$t_{0}$$C_{t_{0},,,T}$$|\frac{d}{d\epsilon_{\epsilon=0}}E(F(U_{T-t_{O}}^{W}(r_{0}+\epsilon\delta))|\leq||\delta||_{T,,,t_{0}}(E(|F(r_{W}(T)|^{2}))^{\frac{1}{2}}$Compartimentage Principle.
is strictly increas“Generically” the sequence of Hilbert spaces$T$
ing relatively the parameterand strictly decreasing relatively to theparameter .
Hedging
The Clark-Ocone-Karatzas formula(3.2)$C_{T,,,t_{0}}$$t_{0}$$F(r_{W}(T))-E(F(r_{W}(t_{0})))=\sum_{k=1}^{q}\int_{t_{0}}^{T}E^{F_{s}}(D_{s,,,k}(F(r_{W}(T)))dW^{k}(s)$10

H. Airault and P. Malliavin

gives a realization of the option along each trajectory. The corresponding
strategy of replication, consist for the trader to balance at each time
his portfolio according the infinitesimal observed variation of the driving
Brownian $W^{k}(t+\epsilon)-W^{k}(t)$ , multiply by $E^{F_{s}}(D_{s,,,k}(F(r_{W}(T))$ .
The formula (3.2) is a specialization of the general It\^o theorem saying that any random variable of zero expectation is representable by a
Stochastic integral; at this level of generality the integrand is only in$L^{2}([t_{0}, T])$ on each trajectory. As thefinancial replication of the option
is given by this integrand, it is impossible to realize this replication if
this integrand is not at least continuous; otherwise instabilities appear.
$t$

3.3. Theorem [13].
the stopping time such that
Denote$\Theta$$ J_{t_{0}-t}^{W}(\Phi_{k}(r_{W}(t)))\in C_{T,,,t}\forall t\leq\Theta$,$\forall k\in[1, q]$;then $E^{F_{\Theta}}(F(r_{W}(T)))$ is replicable by a stable Clark-Ocone-Karatzas formula.
Proof.
$E^{F_{s}}(D_{s,,,k}(F(r_{W}(T)))=E(<dF, J_{T-s}^{W}(\Phi_{k}(r_{W}(s))>)$

Consequence : Traders must try to sale digital options before the stopping time .
$\Theta$

BIBLIOGRAPHY
[1]
[2]

[3]

[4]

[5]

[6]
[7]

Airault(H.), Malliavin(R) and Thalmaier(A), Support of Virasoro unitarizing measures : C. R. Acad.Sci. Paris Ser. I 335 (2002) 621-626.
Airault(H.) and Ren(J.), Modulus of continuity of the canonic Brownian
motion “on the group of diﬀeomorphism of the circle” : J. Fund.
Analysis 196 (2002) 325-426.
Ben-Arous (G.), Flots et s\’eries de Taylor stochastiques : Prob. Theory
Rel. Fields 81 (1989) 29-77.
Ben-Arous (G.) and L\’eandre (R.), D\’ecroissance exponentielle du noyau
de la chaleur sur la diagonale: Prob. Th. Rel. Fields 90 (1991) 175-202
and 377-402.
Bowick(M.J.) and Lahiri(A.), The Ricci curvature of $Diff(S^{1})/SL(2, R)$ :J.Math. Phys. 29(1988) 1979-1981.
Bismut(J.M.), Large deviations and the Malliavin Calculus: Birkhk\"auser,
Prog. Math. 45 (1984) 216 pp.
Cruzeiro(A.B.) and Malliavin(P.), Renormalized diﬀerential geometry on
path spaces : Structural equation, curvature : J. Fund. Analysis 139(1996) 119-181.

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[8]

[9]
[10]
[11]
[12]

[13]
[14]

11

Driver (B), Integration by parts and quasi-invariance for heat kemel
measure on Loop groups : J.Fund.Analysis 149 (1997) 470-547 and
155 (1998) 297-301.
Fang(S.), Canonical brownian motion on the diﬀeomorphism group of
the circle : J. Fund. Analysis 196 (2002) 162-179.
Fang(S.) and Malliavin (P.), Stochastic Analysis on the path space of a
Riemannian manifold : J. Fund. Analysis 118 (1993) 249-274.
Kusuoka(S.) and Stroock(D.), Precise asymptotics of certain Wiener
functionals : J. Ebnct. Analysis 99 (1991) 1-74.
Malliavin(R), Stochastic Calculus of Variations and hypoelliptic operators : In Proc. Int. Symp. on SDE Kyoto 1976, K. It\^o ed., John Wiley
1978, 195-263.
Malliavin(P.) and Thalmaier(A.), Stochastic Calculus of Variations inMathematical Finance : Springer 2003, 120 pp.
Stroock(D.), An introduction to the Analysis of Paths on a Riemannian
manifold : American Math. Society 2000, pp. 269.

H. A.
INSSET, Universit\’e de Picardie, 48 rue Raspail, 02100 Saint-Quentin, France;hairault@insset.u-picardie.fr
P. M.
10 rue Saint Louis en VIsle, 75004 Paris, France;sli@ccr.iussieu.fr

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 13-29

Invariant Measures for a Stochastic Porous
Medium Equation
Giuseppe Da Prato and Michael R\"ockner
Abstract.
We prove the existence of (. finitesimally) invariant measures
for a stochastic version of the porous medium equation (of exponent.
$m=3)$ with Dirichlet Laplacian on an open set in$\mathbb{R}^{d}$\S 1. Introduction
The porous medium equation(1.1)$\frac{\partial X}{\partial t}=\triangle(X^{m})$,$m\in \mathbb{N}$,has been studied extensively. We refer
on a bounded open set
to [1] for both the mathematical treatment and the physical backgroundand also to [2, Section 4.3] for the general theory of equations of such
type.
In this paper we are interested in a stochastic version of (1.1).
Throughout this paper we assume$D\subset \mathbb{R}^{d}$(H1)$m=3$ .

We believe our approach can be extended for other odd values of $m$ , but
this would require a technically much more complicated proof. To avoid
the latter and to explain the main idea we restrict to the above case.
We consider Dirichlet boundary conditions for the Laplacian . So,
the stochastic partial diﬀerential equation we would like to analyze forsuitable initial conditions is the following:$\triangle$(1.2)$dX(t)=\triangle(X^{3}(t))dt+\sqrt{C}dW(t)$ ,$t$$\geq 0$.

Received April 15, 2003.
2000 Mathematics Subject Classification AMS : $76S05,35J25,37L40$ .

G. Da Prato and M. R\"ockner

14

replaced
As in [3], where similar equations were studied (but withof linear growth, satisfying, in particular, $\beta’\geq c>$by some :, it turns out that the appropriate state space is $H^{-1}(D)$ , i.e. the dualof the Sobolev space $H_{0}^{1}:=H_{0}^{1}(D)$ . Below we shall use the standard$L^{2}(D)$ dualization
and $H=H^{-1}(D)$ induced by
betweenthe embeddings$x\rightarrow x^{3}$$\beta$$\mathbb{R}\rightarrow \mathbb{R}$$0)$$\langle\cdot, \cdot\rangle$$H_{0}^{1}(D)$$H_{0}^{1}(D)\subset L^{2}(D)^{/}=L^{2}(D)\subset H^{-1}(D)=H$

without further notice. Then for $x\in H$$|x|_{H}^{2}=\int_{D}(-\triangle)^{-1}x(\xi)x(\xi)d\xi$

and for the dualof $H$ we have $H^{/}=H_{0}^{1}$ .
is a cylindrical Brownian motion in $H$ and $C$ is a positive
definite bounded operator on $H$ of $trace$ class. To be more concrete
below we assume:$H^{/}$$(W_{t})_{t\geq 0}$

There exists(H2)$\{e_{k}|k\in \mathbb{N}\}$, such that for the eigenbasis, $k\in[0, +\infty)$ ,of (with Dirichlet boundary conditions) we have$k\in \mathbb{N}$$\lambda_{k}$$\triangle$$Ce_{k}=\sqrt{\lambda_{k}}e_{k}$For$\alpha_{k}:=\sup_{\xi\in D}|e_{k}(\xi)|^{2}$(H3)$ K:=\sum_{k=1}^{\infty}\alpha_{k}\lambda_{k}<+\infty$,$k\in \mathbb{N}$for$dl$ $k\in \mathbb{N}$.

, we have.

Our aim in this paper is to construct invariant measures for (1.2). Existence of solutions to (1.2) will be studied in another paper. To formulate
what is meant by “invariant measure” without refering to a solution of(1.2) we need to consider the generator, also called Kolmogorov operator,
corresponding to (1.2).
Applying It\^o’s formula (on a heuristic level) to (1.2) one finds what
, should be,
the corresponding Kolmogorov operator, let us call it
namely$N_{0}$(1.3)$N_{0}\varphi(x)=\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}D^{2}\varphi(e_{k}, e_{k})+D\varphi(x)(\triangle(x^{3}))$,$x\in H$,,denote the first and second Pr\’echet derivatives of :. So, we take.
at least
In order to make sense of (1.3) one needs thatis,for “relevant” $x\in H$ . Here one clearly sees the diﬃculties sincewhere$ D\varphi$$H\rightarrow \mathbb{R}$$ D^{2}\varphi$$\varphi$$\varphi\in C_{b}^{2}(H)$$\triangle(x^{3})\in H$$x^{3}$

Invariant measures for a stochastic porous medium equation15of course, not defined for any Schwartz distribution in $H=H^{-1}$ , not to. An invariant measure for (1.2) is
mention that it will not be in
now defined “infinitesimally” (cf. [4]), without having a solution to (1.2),
as the solution to the equation$H_{0}^{1}(D)$(1.4)$N_{0}^{*}\mu=0$

with the property thatmakes sense and$\mu$

is supported by those $x\in H$ for which. (1.4) is a short form for$x^{3}$$\triangle(x^{3})\in H$(1.5)$N_{0}\varphi\in L^{1}(H, \mu)$

and $\int_{H}N_{0}\varphi d\mu=0$ for all.

$\varphi\in C_{b}^{2}(H)$

Any invariant measure for any solution of (1.2) in the classical sense will
satisp (1.4).
In \S 2 we construct a solution
to (1.4) and prove the necessary, $M$,
support properties of , more precisely, that for all $M$$\mu$$\in \mathbb{N}$$\mu$$\mu(\{x\in L^{2}(D)|x^{M}\in H_{0}^{1}\})=1$$\geq 2$,

. We rely onin (1.3) is $\mu-a.e$ . well defined for all
results in [3] which we apply to suitable approximations, i.e. the function
is replaced byso that$\varphi\in C_{b}^{2}(H)$$N_{0}$$x\mapsto x^{3}$$\beta_{\Xi}(x):=\frac{x^{3}}{1+\epsilon x^{2}}+\in x$,$\in\in(0,1]$,

to which the results in [3] apply.

\S 2.

Existence of an infinitesimal invariant measureThroughout this section $(H1)-(H3)$ are still in force. So, we first.
consider the following approximations for the Kolmogorov operator$x\inL^{2}(D)$$\in\in(0,1]$
,
that
we
for
such
For
define$N_{0}$$\varphi\in C_{b}^{2}(H)$$(2.1)$$\beta_{\epsilon}(x)\in H_{0}^{1}$$N_{\Xi}\varphi(x):=\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}D^{2}\varphi(x)(e_{k}, e_{k})+D\varphi(x)(\triangle\beta_{\Xi}(x))$

,

where
(2.2)

$\beta_{\Xi}(r):=\frac{r^{3}}{1+\in r^{2}}+\in r$,$r\in \mathbb{R}$.

We note that is Lipschitz and recall the following result ﬀom [3] whichis crucial for our further analysis, see [3, Theorems 3.1, 3.9, Remark 3.1].
$\beta_{\Xi}$16

G. Da Prato and M. R\"ockner

Theorem 2.1. $Let\in\in(0,1]$ . Then there exists a probability meaon $H$ such thatsure$\mu_{\Xi}$

,

(2.3)

$\mu_{\Xi}(H_{0}^{1})=1$(2.4)$\int_{H}|x|_{H_{0}^{1}}^{2}\mu_{\epsilon}(dx)<+\infty$(2.5)$\int_{H}|\beta_{\Xi}|_{H_{0}^{1}}^{2}d\mu_{\epsilon}=\int_{H}|\triangle\beta_{\epsilon}|_{H}^{2}d\mu_{\Xi}<+\infty$

,

and
(2.6)

for all$\int_{H}N_{\Xi}\varphi d\mu_{\Xi}=0$$\varphi\in C_{b}^{2}(H)$.

Remark 2.2. (i). In [3] only$\mu_{\Xi}(\{x\in L^{2}(D)|\beta_{\in}(x)\in H_{0}^{1}\})=1$

was proved. But since $\beta_{\Xi}(0)=0$ ,(2.7)$\beta_{\Xi}(\mathbb{R})=\mathbb{R}$, andfor all$\beta_{\epsilon}^{J}(r)=r^{2}\frac{3+\epsilon r^{2}}{(1+\epsilon r^{2})^{2}}+\in\geq\in$

it follows that the inverse
is equivalent to$\beta_{\epsilon}(x)\in H_{0}^{1}$$\beta_{\epsilon}^{-1}$of$x\in H_{0}^{1}$$r\in \mathbb{R}$,is Lipschitz with $\beta_{\Xi}^{-1}(0)=0$ , soand (2.4) follows ﬀom (2.5), since$\beta_{\Xi}$$|\nabla x|=|\nabla\beta_{\in}^{-1}(\beta_{\Xi}(x))|\leq\in-1|\nabla\beta_{\Xi}(x)|$.

We thank V. Barbu for pointing this out to us.
(ii) By Theorem 2.1 we have thatis well defined for$x\in H$ .
$N_{\epsilon}\varphi(x)$For$N\in \mathbb{N}$$\mu_{\epsilon}-a.e$.

we define$P_{N}x=\sum_{k=1}^{N}\langle x, e_{k}\rangle_{k}e_{k}$,$x\in H$.

Note that, sinceis the eigenbasis of the Laplacian we havethat the respective restrictionis also an orthogonal projection on$L^{2}(D)$ andand on both spaces
also converges strongly tothe identity.
$\in\in(0,1]$ , is the following:
The first new result on$\{e_{k}|k\in \mathbb{N}\}$$P_{N}$$(P_{N})_{N\in N}$$H_{0}^{1}$$\mu_{\Xi},$

Invariant measures for a stochastic porous medium equation

17

Proposition 2.3. $\{\mu_{\in}, \in\in(0,1]\}$ is tight on H. For any weaklimit point$\mu$$\int_{H}|x|_{L^{2}(D)}^{2}\mu(dx)\leq\int_{D}1d\xi+\frac{1}{2}RC$.

In particular, $\mu(L^{2}(D))=1$ .
, $\chi_{7L}(x)=x$ on $[-n, n]$ , $\chi_{n}(x)=$let
Proof. For$(n+1)sign$ , for $x\in \mathbb{R}\backslash [-(n+2), n+2]$ ,and. Define for ,$n\in \mathbb{N}$$\chi_{n}\in C^{\infty}(\mathbb{R})$$0\leq\chi_{n}’\leq 1$$x$$+\infty$$n$$N\in \mathbb{N}$$\varphi_{N,n}(x):=\frac{1}{2}\chi_{n}(|P_{n}x|_{H}^{2})$Then$\varphi_{N,r\iota}\in C_{b}^{2}(H)$$N_{\epsilon}\varphi_{N,n}(x)$$=$$\sup_{n\in N}|\chi_{n}^{\prime/}|<$.

and for $x\in H$$\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}[2\chi_{n}^{/\prime}(|P_{n}x|_{H}^{2})\langle P_{N}x, e_{k}\rangle_{H}^{2}+\chi_{n}^{J}(|P_{n}x|_{H}^{2})]$$+\chi_{n}^{J}(|P_{n}x|_{H}^{2})\langle P_{N}x, \triangle\beta_{\Xi}(x)\rangle_{H}$Hence integrating with respect to$\mu_{\Xi}$.

, by (2.6) we find$\int_{H}\chi_{7\iota}^{/}(|P_{n}x|_{H}^{2})\langle P_{N}x, \beta_{\in}(x)\rangle_{L^{2}(D)}\mu_{\Xi}(dx)$$=\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}\int_{H}[2\chi_{n}^{J/}(|P_{n}x|_{H}^{2})\langle P_{N}x, e_{k}\rangle_{H}^{2}+\chi_{n}^{/}(|P_{7\iota}x|_{H}^{2})]\mu_{\epsilon}(dx)$$\leq\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}+\sup_{k\in N}\lambda_{k}\int_{H}|\chi_{n}^{/\prime}(|P_{n}x|_{H}^{2})||P_{N}x|_{H}^{2}\mu_{\Xi}(dx)$For all$n\in \mathbb{N}$.

the integrand in the left hand side is bounded by$1_{\{|P_{n}x|_{H}^{2}\leq n+2\}}|P_{N}x|_{H}|\beta_{\Xi}(x)|_{H_{0}^{1}}$,and similar bounds for the integrand in the right hand side hold. Therefore, (2.5) and Lebesgue’s dominated convergence theorem allow us to18

G. Da Prato and M. R\"ocknertake$ N\rightarrow\infty$

and obtain$\int_{H}\chi_{n}^{J}(|x|_{H}^{2})\langle x, \beta_{\Xi}(x)\rangle_{L^{2}(D)}\mu_{\Xi}(dx)$$\leq\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}+\sup_{k\in N}\lambda_{k}\int_{H}|\chi_{n}^{//}(|x|_{H}^{2})||x|_{H}^{2}\mu_{\Xi}(dx)$$\leq\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}+\sup_{k\in N}\lambda_{k}\int_{\{|x|_{H}^{2}\geq n\}}|x|_{H}^{2}\mu_{\Xi}(dx)$Hence taking
arrive at$ n\rightarrow\infty$.

.

by (2.4) and using the definition (2.2) of$\int_{H}\int_{D}(\frac{x^{4}(\xi)}{1+\epsilon x^{2}(\xi)}+\in x^{2}(\xi))d\xi\mu_{\in}(dx)\leq\frac{1}{2}EC$$\beta_{\Xi}$we.

$Since\in\in(0,1]$ , this implies(2.8)$\int_{H}|x|_{L^{2}(D)}^{2}\mu_{\Xi}(dx)\leq\int_{D}(1+\frac{x^{4}(\xi)}{1+x^{2}(\xi)})d\xi\mu_{\in}(dx)$$\leq\int_{D}1d\xi+\frac{1}{2}$

Tr $C$ .

Since $L^{2}(D)\subset H$ is compact, this implies that $\{\mu_{\Xi}|\in\in(0,1]\}$ is tight on$H$ . Since the map
is lwer semicontinuous and nonnegative
on $H$ all assertions follow.
$x\rightarrow|x|_{L^{2}(D)}^{2}$$\square $Later we need better support properties of . Therefore, our next
aim is to prove the following:$\mu$

Theorem 2.4. Let $(H1)-(H3)$ hold. Then:, $M$, there exists a constant $C_{M}=C_{M}(D, K)$(i) For all$>0$ such that
$M\in \mathbb{N}$$\geq 2$$\sup_{\in\in(0,1]}\int_{H}\int_{D}x^{2(M-1)}(\xi)|\nabla x(\xi)|^{2}d\xi\mu_{\epsilon}(dx)\leq C_{M}$(ii)For all
2. 3$M\in \mathbb{N}$, $M\geq 2$ and any limit point$\int_{H}\int_{D}|\nabla(x^{M})(\xi)|^{2}d\xi\mu(dx)\leq C_{M}$

$\mu$

.

.

as in Proposition

Invariant measures for a stochastic porous medium equation

19

In particular, setting$H_{0,M}^{1}:=\{x\in L^{2}(D)|x^{M}\in H_{0}^{1}\}$

we have$\mu(H_{0,M}^{1})=1$

for all $M\geq 2$ .

In order to prove Theorem 2.4 we need some preparation, i.e. more$\in\in(0,1]$ . This can be deduced ﬀom
precise information about theis an infinitesimally invariant measure(2.6), i.e. ﬀom the fact that
. So, we fix $\in\in(0,1]$ and for the rest of this section we assume
forthat $(H1)-(H3)$ hold.
, $M$:
We need to apply (2.6) with replaced by, given by$\mu_{\epsilon},$$\mu_{\epsilon}$$N_{\Xi}$$\varphi_{M}$$\varphi$$L^{2}(D)\rightarrow[0, \infty]$$\in \mathbb{N}$$\varphi_{M}(x):=\int_{D}x^{2M}(\xi)d\xi$,$x\in L^{2}(D)$.

Clearly, such functions are not inso we have to construct proper$\delta\in(0,1]$
approximations. So, define for$C_{b}^{2}(H)$(2.9)

Then for(2.10)$f_{M,\delta}(r):=\frac{r^{2M}}{1+\delta r^{2}}$,$r\in \mathbb{R}$.

$r\in \mathbb{R}$$f_{M,\delta}^{J}(r)=(1+\delta r^{2})^{-2}[2Mr^{2M-1}+2\delta(M-1)r^{2M+1}]$

and(2.11)$f_{M,\delta}^{//}(r)=2(1+\delta r^{2})^{-3}[M(2M-1)r^{2M-2}+\delta(4M^{2}-6M-1)r^{2M}$$+\delta^{2}(M-1)(2M-3)r^{2M+2}]$ .

We have chosen this approximation since below (cf. Lemma 2.7) it will
. More precisely we haveis nonnegative if $M$
be crucial that$f_{M,\delta}^{/\prime}$$\geq 2$$0\leq f_{M,\delta}(r)\leq\frac{1}{\delta}r^{2M-2}$(2.12)$0\leq f_{M,\delta}^{/}(r)\leq\frac{2M}{\delta}|r|^{2M-3}$$0\leq f_{M,\delta}^{//}(r)\leq 16M^{2}|r|^{2M-4}\inf\{r^{2},1/\delta\}$.

20

G. Da Prato and M. R\"ockner

Remark 2.5. The following will be used below: if xthat for M

is such$\in H_{0}^{1}$$\in \mathbb{N}$(2.13)$\int_{H}x^{2(M-1)}(\xi)|\nabla x(\xi)|^{2}d\xi<\infty$,

thenand $x^{M-1}\nabla x=\frac{1}{M}\nabla x^{M}$ , or using the notation introduced in Theorem 2.4(ii) equivalently $x\in H_{0,M}^{1}$ . The proof is standard
by approximation. So, we omit it. We also note that by Poincar\’e’s inequality, $H_{0,M}^{1}\subset L^{2M}(D)$ . More precisely, there exists $C(D)\in(0, \infty)$
such that$x^{M}\in H_{0}^{1}$(2.14)$ C(D)\int_{D}x^{2M}(\xi)d\xi\leq\int_{D}|\nabla x^{M}(\xi)|^{2}d\xi$$=M\int_{D}x^{2(M-1)}(\xi)|\nabla x^{M}(\xi)|^{2}d\xi$

for all$x$,

as above.

The following lemma is a consequence of (2.6) and crucial for our$\in\in(0,1]$ and their limit points.
analysis of$\mu_{\Xi},$

Lemma 2.6. Let(2.15)$M\in \mathbb{N}$, $\delta\in(0,1]$ . Assume that$\int_{H}\int_{D}x^{2(M-2)}(\xi)|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)<\infty$

if $M$$\geq 3$.

Then$\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}\int_{H}\int_{D}f_{M,\delta}^{\prime/}(x(\xi))e_{k}^{2}(\xi)d\xi\mu_{\in}(dx)$(2.16)$=\int_{H}\int_{D}f_{M,\delta}^{J/}(x(\xi))\beta_{\epsilon}^{/}(x(\xi))|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)$.

Proof. We first note that (2.15) holds for $M=2$ by (2.3). For$\kappa\in(0,1]$ we define$f_{M,\delta,\kappa}(r):=f_{M,\delta}(r)e^{-\frac{1}{2}\kappa r^{2}}$and$f_{1,\delta,\kappa}:=f_{1,\delta}$,$r\in \mathbb{R}$. Then (2.11) implies that$\varphi_{M,\delta,\kappa}(x):=\int_{D}f_{M,\delta,\kappa}(x(\xi))d\xi$

if $M$$\geq 2$$f_{M,\delta,\kappa}\in C_{b}^{2}(\mathbb{R})$,$x\in L^{2}(D)$.

. DefineInvariant measures for a stochastic porous medium equation

Then it is easy to check thatand that for all , $z\in L^{2}(D)$$\varphi M,\delta,\kappa$

21

is G\^ateaux diﬀerentiable on $L^{2}(D)$$y$(2.17)$\varphi_{M,\delta,\kappa}’(x)(y)=\int_{D}f_{M,\delta,\kappa}^{/}(x(\xi))y(\xi)d\xi$(2.17)$\varphi_{M,\delta,\kappa}^{/\prime}(x)(y, z)=\int_{D}f_{M,\delta,\kappa}^{\prime/}(x(\xi))y(\xi)z(\xi)d\xi$

,

.

Hence
$\varphi_{M,\delta,\kappa}\circ P_{N}\in C_{b}^{2}(H)$and for all$x\in H_{0}^{1}$(hence$\beta_{\in}(x)\in H_{0}^{1}$$=$$N_{\Xi}(\varphi_{M,\delta,\kappa}\circ P_{N})(x)$)$\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}\int_{D}f_{M,\delta,\kappa}^{\prime/}(P_{N}x(\xi))e_{k}^{2}(\xi)d\xi$$+\int_{D}f_{M,\delta,\kappa}’(P_{N}x(\xi))P_{N}(\triangle\beta_{\in}(x))(\xi)d\xi$

Since$P_{N}\triangle=\triangle P_{N}$.

, integrating by parts we obtain$N_{\Xi}(\varphi_{M,\delta,\kappa}\circ P_{N})(x)$$=$$\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}\int_{D}f_{M,\delta,\kappa}^{;/}(P_{N}x(\xi))e_{k}^{2}(\xi)d\xi$$-\int_{D}f_{M,\delta,\kappa}’’(P_{N}x(\xi))\langle\nabla(P_{N}x)(\xi), \nabla(P_{N}\beta_{\Xi}(x))(\xi)\rangle_{\mathbb{R}^{d}}d\xi$

Since(H3) that$(P_{N})_{N\in N}$

converges strongly to the identity in$\lim_{N\rightarrow\infty}N_{\Xi}(\varphi_{M,\delta,\kappa}\circ P_{N})(x)$$=$$H_{0}^{1}$, we conclude by$\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}\int_{D}f_{M,\delta,\kappa}^{//}(x(\xi))e_{k}^{2}(\xi)d\xi$$-\int_{D}f_{M,\delta,\kappa}^{\prime/}(x(\xi))\beta_{\in}^{/}(x)(\xi)|\nabla x(\xi)|^{2}d\xi$

Since
in

$\beta_{\in}$.

is Lipschitz, by (2.3)-(2.5) and (H3) this convergence also holds. Hence (2.6) implies that$L^{1}(H, \mu_{\Xi})$$\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}\int_{H}\int_{D}f_{M,\delta,\kappa}^{//}(x(\xi))e_{k}^{2}(\xi)d\xi\mu_{\Xi}(dx)$(2.19)$=\int_{H}\int_{D}f_{M,\delta,\kappa}^{//}(x(\xi))\beta_{\Xi}^{/}(x)(\xi)|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)$

.

.

22

G. Da Prato and M. R\"ockner

So, for $M=1$ the assertion is proved. If $M\geq 2$ , an elementary calculation shows that by (2.12) there exists a constant $C(M, \delta)>0$ (onlydepending on $M$ and ) such that$\delta$(2.20)$|f_{M,\delta,\kappa}’’(r)|\leq C(M, \delta)r^{2(M-2)}$,$r\in \mathbb{R}$.

Hence by (H3), Remark 2.5 and assumption (2.15) we can apply
Lebesgue’s dominated convergence theorem to (2.19) and letting
we obtain the assertion.
$\kappa\rightarrow 0$$\square $Lemma 2.7. Let(i) We have$M\in \mathbb{N}$and assume that (2.15) holds if $M$$\geq 3$$\frac{K}{2}\int_{H}\int_{D}x^{2(M-1)}(\xi)d\xi\mu_{\in}(dx)$(2.21)$\geq\int_{H}\int_{D}x^{2(M-1)}(\xi)(\frac{x^{2}(\xi)}{1+x^{2}(\xi)}+\in)|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)$.

(ii) If $M\geq 2$ , we have$\frac{K}{2}\int_{H}\int_{D}(x^{2(M-1)}(\xi)+x^{2(M-2)}(\xi))d\xi\mu_{\in}(dx)$(2.22)$\geq\int_{H}\int_{D}x^{2(M-1)}(\xi)|\nabla x(\xi)|^{2}d\xi\mu_{\in}(dx)$$=\frac{1}{M^{2}}\int_{H}\int_{D}|\nabla x^{M}(\xi)|^{2}d\xi\mu_{\Xi}(dx)$.

(ii)$\int_{H}\int_{D}|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)\leq\frac{K}{2\in}$Proof,.

(i) By (H3) the leﬃ hand side of (2.16) is dominated by$\frac{K}{2}\int_{H}\int_{D}f_{M,\delta}^{\prime/}(x(\xi))d\xi\mu_{\in}(dx)$.

If $M\geq 2$ , by assumption (2.15) and Remark 2.5 we know that$\int_{H}\int_{D}x^{2(M-1)}(\xi)d\xi\mu_{\Xi}(dx)<\infty$.

23

Invariant measures for a stochastic porous medium equation

which trivially also holds for $M=1$ . So, by (2.11), (2.12) and Lebesgue’s
dominated convergence theorem we obtain that for $M\geq 2$$\frac{K}{2}\int_{H}\int_{D}2M(2M-1)x^{2(M-1)}(\xi)d\xi\mu_{\Xi}(dx)$$\geq\lim_{\delta\rightarrow}\inf_{0}\int_{H}\int_{D}f_{M,\delta(x(\xi))\beta_{\Xi}’(x(\xi))|\nabla x(\xi)|^{2}d\xi\mu_{\Xi}(dx)}^{JJ}$

Since$f_{M,\delta}^{//}\geq 0$

for $M$$\geq 2$.

and$\beta_{g}^{/}(r)\geq\frac{r^{2}}{1+r^{2}}+\in\geq 0$

for all$r\in \mathbb{R}$,

we can apply Fatou’s lemma to prove the assertion. If $M$ $=1$ we conclude in the same way by (2.3) and Lebesgue’s dominated convergence
for all $\delta\in(0,1]$ .
theorem which applies since is bounded and(ii) Since (2.15) holds for $M$ $=2$ , by H\"older’s inequality (2.15) holdswith $M-1$ replacing $M$ , since by assumption it holds for $M$ . So, theinequality in (i) also holds with $M$ $-2$ replacing $M$ $-1$ . Estimating$\in on$ the right hand sides from below by 0 and adding both resulting
inequalities we obtain the inequality in (2.22). The equality in (2.22)
follows by Remark 2.5.
(iii) The assertion follows from (2.21) setting $M=1$ .
$\beta_{\Xi}^{/}$$|f_{1,\delta}^{\prime/}|\leq 6$$\square $By an induction argument we shall now prove that the integrals in(2.22) are all finite and at the same time prove the bounds claimed in
Theorem 2.4.
Proof of Theorem 2.4. (i). If $M=2$ , then the leﬃ hand side of(2.22) is finite by (2.8) and moreover (2.22) applies, so that by (2.8) wehave(2.23)$\int_{H}\int_{D}x^{2}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\Xi}(dx)\leq\frac{K}{2}(\frac{1}{2}rbC+2\int_{D}1d\xi)<\infty$

Suppose the left hand side of (2.22) is finite for(2.15) holds. Then (2.22) holds and by Remark 2.5$M\in \mathbb{N}$$\infty>\int_{H}\int_{D}x^{2(M-1)}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\Xi}(dx)$(2.24)$=\frac{1}{M^{2}}\int_{H}\int_{D}|\nabla(x^{M}(\xi))|^{2}d\xi\mu_{\Xi}(dx)$$\geq\frac{C(D)^{2}}{M^{2}}\int_{H}\int_{D}x^{2M}(\xi)d\xi\mu_{\Xi}(dx)$

.

, $M$

$\geq 2$

.

, and

24

G. Da Prato and M. R\"ockner

Hence (2.15) holds with $M-1$ replacing $M$ $-2$ and the leﬃ hand sideof (2.22) is finite for $M+1$ replacing $M$ , hence by induction for all. Furthermore, for all $M$ first applying (2.22) and then applying(2.24) first with $M$ $-1$ replacing $M$ and then with $M-2$ replacing $M$
respectively we obtain$M\in \mathbb{N}$$\int_{H}\int_{D}x^{2(M-1)}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\Xi}(dx)$(2.25)$\leq\frac{K}{2}[(\frac{M-1}{C(D)})^{2}\int_{H}\int_{D}x^{2(M-2)}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\epsilon}(dx)$$+\int_{H}\int_{D}x^{2(M-2)}(\xi)d\xi\mu_{\Xi}(dx)]$$\leq\frac{K}{2C(D)^{2}}[(M-1)^{2}\int_{H}\int_{D}x^{2(M-2)}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\Xi}(dx)$(2.26)$+(M-2)^{2}\int_{H}\int_{D}x^{2(M-3)}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\in}(dx)]$.

If $M=3$ we cannot use (2.26) since for the second summand we have, but from (2.25) we obtain by (2.23)
no bound which is independentand (2.8) that$of\in$$\int_{H}\int_{D}x^{4}(\xi)|\nabla(x(\xi))|^{2}d\xi\mu_{\Xi}(dx)$$\leq\frac{K}{2}[(\frac{2}{C(D)})^{2}\frac{K}{2}(\frac{1}{2}EC+2\int_{D}1d\xi)+\frac{1}{2}EC+\int_{D}1d\xi]$.

Now assertion (i) follows ﬀom (2.26) by induction.
To prove (ii) we start with the following
Claim: For all$M\in \mathbb{N}$(2.27)$\Theta_{M}(x):=1_{H_{0,M}^{1}}(x)\int_{D}|\nabla x^{M}(\xi)|^{2}d\xi+\infty\cdot 1_{H\backslash H_{0,M}^{1}}(x)$is a lower semicontinuous function on $H$ .

,$x\in H$

Invariant measures for a stochastic porous medium equation

25

, the
Since is a weak limit point of $\{\mu_{g}|\in\in(0,1]\}$ andclaim immediately implies assertion (ii).
,, suchTo prove the claim let $\alpha>0$ and. By Poincar\’e’s inequality
isin $H$ asthatasalso weakly in $L^{2}(D)$ ,a bounded set in $L^{2M}(D)$ . So$x\inL^{2}(D)$, there. Sinceis bounded inin particularandsuch thatasexists a subsequence
weakly in
and$\Theta_{M}\geq 0$$\mu$$n\in \mathbb{N}$$x_{n}\in\{\Theta_{M}\leq\alpha\}$$\{x_{n}|n\in \mathbb{N}\}$$ n\rightarrow\infty$$x_{n}\rightarrow x$$x_{n}\rightarrow x$$ n\rightarrow\infty$$\{x_{n}^{M}|n\in \mathbb{N}\}$$H_{0}^{1}$$y\in H_{0}^{1}$$(x_{n_{k}}^{M})_{k\in N}$$ k\rightarrow\infty$$x_{n_{k}}^{M}\rightarrow y$$H_{0}^{1}$$\int_{D}|\nabla y(\xi)|^{2}d\xi\leq\alpha$.

is compact,inasSince the embedding$L^{2}(D)$ . Selecting another subsequence if necessary, this convergence is$d\xi-a.e.$ , hence, $d\xi-a.e$ .
as$H_{0}^{1}\subset L^{2}(D)$$x_{n_{k}}^{M}\rightarrow y$$ k\rightarrow\infty$$ k\rightarrow\infty$$x_{n_{k}}\rightarrow y^{\frac{1}{M}}$

Since (selecting another subsequence if necessary) we also know that theCesaro mean ofhas as an accumulation point in the topology.
of $d\xi-a.e$ . convergence, we must have $x^{M}=y$ , so
As a consequence of the previous proof we obtain:$(x_{n_{k}})_{k\in N}$$x$$x\in\{\Theta_{M}\leq\alpha\}$Corollary 2.8. Let$M\in \mathbb{N}$. Then$\Theta_{M}$$\square $has compact level sets in $H$ .

is lower
Proof. We already know ﬀom the previous proof that
semicontinuous. The relative compactness of their level sets is, however,clear by Poincar\’e’s inequality since $L^{2M}(D)\subset H$ is compact.
$O-_{M}$$\square $Since for$M\in \mathbb{N}$(2.28)$|\triangle x^{M}|_{H}=\int_{D}|\nabla x^{M}(\xi)|^{2}d\xi$,, we can define the Kolmogorov operator in (1.3) rigorously. So, forso$\triangle x^{M}\in H$

for$x\in H_{0,3}^{1}$(2.29)

and $x\in H_{0,M}^{1}$$\varphi\in C_{b}^{2}(H)$$N_{0}\varphi(x):=\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}D^{2}\varphi(x)(e_{k}, e_{k})+D\varphi(x)(\triangle x^{3})$.

We note that by Theorem 2.4(ii) and (2.28), $N_{0}\varphi\in L^{2}(H, \mu)$ for any
weak limit point of $\{\mu_{\Xi}|\in\in(0,1]\}$ on $H$ . Now we can prove our main
result, namely that any such is an infinitesimally invariant measureforin the sense of [4], i.e. satisfies (1.4).
$\mu$

$\mu$

$N_{0}$

G. Da Prato and M. R\"ockner

26

Theorem 2.9. Assume that $(H1)-(H3)$ hold. Let
sition 2.3. Then$\int_{H}N_{0}\varphi d\mu=0$

Proof. Let$\varphi\in C_{b}^{2}(H)$. For

for all$N\in \mathbb{N}$$\varphi\in C_{b}^{2}(H)$define$\mu$

be as in Propo-.

$\varphi_{N}:=\varphi\circ P_{N}$. Then for$x\in H_{0,3}^{1}$$N_{0}\varphi_{N}(x)$$=$$\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}D^{2}\varphi(P_{N}x)(P_{N}e_{k}, P_{N}e_{k})+D\varphi_{N}(x)(\triangle x^{3})$$=$$\frac{1}{2}\sum_{k=1}^{N}\lambda_{k}D^{2}\varphi(P_{N}x)(e_{k}, e_{k})+D\varphi(P_{N}x)(P_{N}(\triangle x^{3}))$.

If we can prove that
(2.30)$\int_{H}N_{0}\varphi_{N}d\mu=0$

for all$N\in \mathbb{N}$,the same is true for by Lebesgue’s dominated convergence theorem.
So, fix. Then by (2.6)$\varphi$$N\in \mathbb{N}$$\int_{H}N_{0}\varphi_{N}d\mu$$=$$\lim_{\in\rightarrow 0}\int_{H}\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}D^{2}\varphi_{N}(x)(e_{k}, e_{k})\mu_{\Xi}(dx)$$+\int_{H}D\varphi_{N}(x)(\triangle x^{3})\mu(dx)$$=$-$\lim_{\zeta\rightarrow 0}\int_{H}D\varphi_{N}(x)(\triangle\beta_{\in}(x))\mu_{\Xi}(dx)$(2.30)$+\int_{H}D\varphi_{N}(x)(\triangle x^{3})\mu(dx)$$=$$\lim_{\in\rightarrow 0}\sum_{i=1}^{N}\int_{H}[D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle x^{3}\rangle_{H}\mu(dx)$$-D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle\beta_{\in}(x)\rangle_{H}\mu_{\in}(dx)]$.

27

Invariant measures for a stochastic porous medium equation

For $i\in\{1, \ldots, N\}$ fixed we have$|\int_{H}D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle x^{3}\rangle_{H}\mu(dx)$$-\int_{H}D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle\beta_{\Xi}(x)\rangle_{H}\mu_{\in}(dx)|$(2.32)$\leq|\int_{H}D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle x^{3}\rangle_{H}(\mu-\mu_{\in})(dx)|$.

$+|\int_{H}D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle(x^{3}-\beta_{\Xi}(x))\rangle_{H}\mu_{\in}(dx)|$

The right hand side’s second summand is bounded by(2.33)$|e_{i}|_{L^{2}(D)}\sup_{x\in H}|D\varphi(x)|_{H_{0}^{1}}\int_{H}(\int_{D}|x^{3}(\xi)-\beta_{\Xi}(x(\xi))|^{2}d\xi)^{1/2}\mu_{\Xi}(dx)$.

We have$|r^{3}-\beta_{\Xi}(r)|=|\frac{\epsilon r^{5}}{1+\in r^{2}}-\in r|\leq\in(|r|^{5}+|r|)$,$r\in \mathbb{R}$.

So, the term in (2.33) is dominated by$\in|e_{i}|_{L^{2}(D)}\sup_{x\in H}|D\varphi(x)|_{H_{O}^{1}}\int_{H}(||x|^{5}|_{L^{2}(D)}+|x|_{L^{2}(D)})\mu_{\Xi}(dx)$which by Theorem 2.4(i), Remark 2.5 and Poincar\’e’s inequality con.
verges to 0
Now we estimate the first summand in the right hand side of (2.32).
So, we define$as\in\rightarrow 0$$f(x):=D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle x^{3}\rangle_{H}$.

, it follows by the proof of theThen since
lower semicontinuity
that is continuous on the level sets(withdefined as in (2.27)). Furthermore, since$\langle e_{i}, \triangle(x^{3})\rangle_{H}=\langle e_{i}, x^{3}\rangle_{L^{2}(D)}$$of\ominus_{3}$$of\ominus_{3}$$f$$\Theta_{3}$$|f(x)|\leq\sup_{x\in H}|D\varphi(x)|_{H_{0}^{1}}|x^{3}|_{L^{2}(D)}$it follows that$\lim_{R\rightarrow\infty}\sup_{\{\ominus_{3}\geq R\}}\frac{|f(x)|}{1+\Theta_{3}(x)}=0$.

,

G. Da Prato and M. R\"ockner28

has compact level
Furthermore, by Corollary 2.8 the function,
sets. Hence by [8, Theorem 5.1 (ii)], there exist $f_{n}\in C_{b}(H)$ ,such that$1+\Theta_{3}$$n\in \mathbb{N}$(2.34)$\lim_{n\rightarrow\infty}\sup_{x\in H}\frac{|f(x)-f_{n}((x)|}{1+\Theta_{3}x)}=0$.

But$|\int_{H}D\varphi(P_{N}x)(e_{i})\langle e_{i}, \triangle x^{3}\rangle_{H}(\mu-\mu_{\in})(dx)|$$\leq\int_{H}|f(x)-f_{n}(x)|(\mu+\mu_{\epsilon})(dx)+|\int_{H}f_{n}(x)(\mu-\mu_{\Xi})(dx)|$

For fixed the second summand tends to 0 as
dominated by$n$$\in\rightarrow 0$.

and the first is$\sup_{x\in H}\frac{|f(x)-f_{n}((x)|}{1+\ominus_{3}x)}\sup_{\in>0}\int_{H}(1+\ominus_{3})d(\mu+\mu_{\Xi})$

,

. So,
which in turn by Theorem 2.4 and (2.34) tends to zero as. Hence the rightalso the first summand in (2.32) tends to zero
hand side of (2.31) is zero and (2.30) follows which completes the proof.
$ n\rightarrow\infty$$as\in\rightarrow 0$$\square $Aknowledgement. The first author would like to thank the University of Bielefeld for its kind hospitality and financial support. This
Kolwork was also supported by the research program “Equazioni
mogorov” from the Italian “Ministero della Ricerca Scientifica Tecnologica”.
The second named author would like to thank the Scuola Normale
Pisa for the hospitality and the financial support during
Superiorea very pleasant stay in Pisa when most of this work was done. Financial support of the BiBoS-Research Centre and the DFG-Forschergruppe“Spectral Analysis, Asymptotic Distributions, and Stochastic Dynamics” is also gratefully acknowledged.
$di$

$e$

$di$

References[1] D. G. Aronson, The porous medium equation, in Lect. Notes Math. Vol.
1224, (A. Fasano and al. editors), Springer, Berlin, 1-46, 1986.
[2] V. Barbu, Analysis and control of nonlinear infinite dimensional systems,
Academic Press, San Diego, 1993.

Invariant measures for a stochastic porous medium equation

29[3] V. Barbu and G. Da Prato, The two phase stochastic Stefan problem,
Probab. Theory Relat. Fields, 124, 544-560, 2002.
[4] V. Bogachev and M. R\"ockner, Elliptic equations for measures on infinite dimensional spaces and applications, Probab. Theory Relat. Fields,
120, 445-496, 2001.
[5] G. Da Prato and M. R\"ockner, Singular dissipative stochastic equations inEilbert spaces, Probab. Theory Relat. Fields, 124, 2, 261-303, 2002.
[6] A. Eberle, Uniqueness and non-uniqueness of singular diﬀusion operators, Lecture Notes in Mathematics 1718, Berlin, Springer-Verlag,
1999.
-analysis of finite and infinite dimensional diﬀusions,[7] M. R\"ockner,
Lecture Notes in Mathematics, 1715, G. Da Prato (editor), SpringerVerlag, 65-116, 1999.
[8] M. R\"ockner and Z. Sobol, Kolmogorov equations in infinite dimensions:well-posedness, regularity of solutions, and applications to generalized
stochastic Burgers equations, Preprint 2003.
[9] W. Stannat, The theory of generalized Dirichlet forms and its applications
in Analysis and Stochastics, Memoirs AMS, 678, 1999.
[10] D.W. Stroock and S.R.S. Varadhan, Multidimensional Diﬀusion Processes, Springer-Verlag, 1979.
$L^{p}$

Giuseppe Da Prato
Scuola Normale Superiore,
Piazza dei Cavalieri 7, 56126 Pisa,
Italy
Michael R\"ockner
Fakult\"at f\"ur Mathematik, Universit\"at Bielefeld,
Postfach 100131, D- 33501 Bielefeld,
GermanyAdvanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics

pp. 31-47

Equivariant Diﬀusions on Principal Bundles
K. David Elworthy, Yves Le Jan and Xue-Mei Li: $P\rightarrow M$ be a smooth principal bundle with structure group $G$ .
This means that there is aright multiplication $P\times G\rightarrow P$ , $u\mapsto u$ .
say, of the Lie group $G$ such that identifies the space of orbits of $G$ with the
manifold $M$ and is locally trivial in the sense that each point of $M$ has anopen neighbourhood $U$ with a diﬀeomorphism

Let

$\pi$

$C^{\infty}$

$g$

$\pi$

$\pi$

$\tau_{U}$: $\pi^{-1}(U)$$U\times G$$U$over $U$ , which is equivariant with respect to the right action of $G$ , i.e. if$(\pi(b), k)$ then $\tau_{u}(b\cdot g)=(\pi(b), kg)$ . Assume for simplicity that $M$ is compact.
, $x\in M$ are diﬀeomorphic to $G$ and their
Set $n=dimM$ . The fibres,tangent spaces $VT_{u}P(=kerT_{u}\pi)$ , $u\in P$ , are the ’vertical’ tangent spaces
to $P$ . A connection on $P$ , (or on ) assigns a complementary ’horizontal’
subspace $HT_{u}P$ to VTUP in $T_{u}P$ for each , giving a smooth horizontal subbundle $HTP$ of the tangent bundle $TP$ to $P$ . Given such a connection it is a
curve: : $[0, T]\rightarrow M$ and $u_{0}\in\pi^{-1}(\sigma(0))$
classical result that for any
which is a lift of , i.e.
there is a unique horizontal :$\tau_{u}(b)=$$\pi^{-1}(x)$

$\pi$

$u$

$C^{1}$$\tilde{\sigma}$$\sigma$$[0, T]\rightarrow P$$\sigma$$\pi(\tilde{\sigma}(t))=$.
and hasIn his startling ICM article [8] It\^o showed how this construction could be
extended to give horizontal lifts of the sample paths of diﬀusion processes. In
fact he was particularly concemed with the case when $M$ is given a Riemann, $x\in M$ , the diﬀusion is Brownian motion on $M$ , and $P$ is theian metricorthonormal frame bundle : $OM\rightarrow M$ . Recall that each $u\in OM$ with
$u\in\pi^{-1}(x)$ can be considered as an isomery
,
:
and a
$\sigma(t)$$\tilde{\sigma}(0)=u_{0}$$\langle, \rangle_{x}$

$\pi$

$u$

$\mathbb{R}^{n}\rightarrow T_{x}M$$\langle, \rangle_{x}$Received March 12, 2003.
Research supported by EPSRC grants $GR/NOO845$ and $GR/S40404/01$ , NSF
grant DMS 0072387, and EU grant ERBF MRX $CT$ 960075 A.

32

K. D. Elworthy, Y. Le Jan and X.-M. Li

horizontal lift$\tilde{\sigma}$

determines parallel translation of tangent vectors along$//t\equiv//(\sigma)_{t}$:$\sigma$$T_{\sigma(\cdot)}M\rightarrow T_{\sigma(t)}M$$v\mapsto\tilde{\sigma}(t)(\tilde{\sigma}(0))^{-1}v$.

The resulting parallel translation along Brownian paths extends also to parallel translation of forms and elements $of\wedge^{p}TM$ . This enabled It\^o to use hisconstruction to obtain a semi-group acting on diﬀerential forms$ P_{t}\phi=E(//t-1)_{*}(\emptyset)=E\phi(//t^{-)}\cdot$

As he pointed out this is not the semi-group generated by the Hodge-Kodaira
Laplacian, . To obtain that generated by the Hodge-Kodaira Laplacian, ,some modification had to be made since the latter contains zero order terms,
the so called Weitzenbock curvature terms. The resulting probabilistic expression for the heat semi-groups on forms has played a major role in subsequentdevelopment.
In [5] we go in the opposite direction starting with a diﬀusion with smooth
generator on $P$ , which is $G$ -invariant and so projects to a diﬀusion generator$A$ on $M$ . We assume the symbol
has constant rank so determining a sub$E$$A$bundleof $TM$ , (so $E=TM$ if is elliptic). We show that this set-up
induces a ’semi-connection’ on $P$ over $E$ (a connection if $E=TM$ ) withand arespect to which can be decomposed into a horizontal component
vertical part
. Moreover any vertical diﬀusion operator such as
induces
only zero order operators on sections of associated vector bundles.
There are two particularly interesting examples. The first when $\pi:GLM\rightarrow$$M$ is the full linear frame bundle and we are given a stochastic flow {:$t<\infty\}$ on $M$ , generator $A$ , inducing the diﬀusion $\{u_{t} : 0\leq t<\infty\}$ on
GLM by$\triangle$$\triangle$$B$$\sigma_{A}$$A^{H}$$\mathcal{B}$$\mathcal{B}^{V}$$\mathcal{B}^{V}$$\xi_{t}$$u_{t}=T\xi_{t}(u_{0})$$ 0\leq$.

Here we can determine the connection on GLM in terms of the LeJan-Watanabe
connection of the flow [12], [1], as defined in [6], [7], in particular giving conditions when it is a Levi-Civita connection. The zero order operators arisingfrom the vertical components can be identified with generalized Weitzenbock
curvature terms.
The second example slightly extends the above framework by letting :$P\rightarrow M$
be the evaluation map on the diﬀeomorphism group DiﬀﬀM of $M$
in $M$ . The group $G$ corresponds
given by $\pi(h):=h(x_{0})$ for a fixed point:
to the group of diﬀeomorphisms fixing . Again we take a flow {$M$ , $t\geq 0\}$ on $M$ , but now the process on DiﬀﬀM is just the right invariant
process determined by $\{\xi_{t} : 0\leq t<\infty\}$ . In this case the horizontal lift tothe diﬀeomorphism group of the diﬀusion $\{\xi_{t}(x_{0}) : 0\leq t<\infty\}$ on $M$ is$\pi$$x_{0}$$x_{0}$$\xi_{t}(x)$$ x\in$33

Equivariant diﬀusions on principal bundles

obtained by ’removal of redundant noise’, . [7] while the vertical process isa flow of diﬀeomorphisms preserving , driven by the redundant noise.
Here we report briefly on some of the main results to appear in [5] and givedetails of a more probabilistic version Theorem 2.5 below: a skew productdecomposition which, although it has a statement not explicitly mentioning
connections, relates to It\^o’s pioneering work on the existence of horizontal
lifts. The derivative flow example and a simplified version of the stochastic
flow example are described in \S 3.
The decomposition and lifting apply in much more generality than withthe full structure of a principal bundle, for example to certain skew products
and invariant processes on foliated manifolds. This will be reported on later.
Earlier work on such decompositions includes [4] [13].
$c.f$$x_{0}$\S 1. Construction
A. If $A$ is a second order diﬀerential operator on a manifold $X$ , denote: $T^{*}X\rightarrow TX$ its symbol determined byby$\sigma^{A}$$df(\sigma^{A}(dg))=\frac{1}{2}A(fg)-\frac{1}{2}A(f)g-\frac{1}{2}fA(g)$,forfunctions , . The operator is said to be semi-elliptic if $df$ $(\sigma^{A}(df))\geq 0$
for each $f\in C^{2}(X)$ , and elliptic if the inequality holds strictly. Ellipticity isequivalent tobeing onto. It is called a diﬀusion operator if it is semi-elliptic
and annihilates constants, and is smooth if it sends smooth functions to smoothfunctions.
$C^{2}$$f$$g$$\sigma^{A}$

Consider a smooth map : $N\rightarrow M$ between smooth manifolds $M$ and$N$ .
By a lift of a diﬀusion operator $A$ on $M$ over we mean a diﬀusion
operator on $N$ such that
$p$

$p$

$B$

(1)

$\mathcal{B}(fo p)=(Af)\circ p$

for allfunctions$B$
and is a lift of $A$ .
$C^{2}$$f$

on $M$ . Suppose $A$ is a smooth diﬀusion operator on $M$

be respectively the symbols forLemma 1.1. Letandfollowing diagram is commutative for all $u\in p^{-1}(x)$ , $x\in M$ :$\sigma^{B}$$\sigma^{A}$$\sigma_{u}^{B}$$T_{u}N$$T_{u}^{*}N$$\downarrow Tp$$(Tp)^{*}|$$T_{x}M$$T_{x}^{*}M$$\sigma_{x}^{A}$.

$B$

and A. The

34

K. D. Elworthy, Y. Le Jan and X.-M. Li

B. Semi-connections on principal bundles. Let M be a smooth finite dimensional manifold and $P(M, G)$ a principal fibre bundle over $M$ withthe projection andstructure group $G$ a Lie group. Denote by :the right translation by .
$\pi$

$P\rightarrow M$$R_{a}$$a$Definition 1.2. Let $E$ be a sub-bundle ofTM and :a principal$P\rightarrow M$$E$$G$ -bundle.
:Ansemi-connection on
is a smooth sub-bundle$H^{E}TP$$TP$of such that(i)maps the fibres $H^{E}T_{u}P$ bijectively onto
for all $u\in P$ .
(ii) $H^{E}TP$ is $G$ -invariant.
$P\rightarrow M$

$\pi$

$\pi$

$ T_{u}\pi$$E_{\pi(u)}$Notes.
(1) Such a semi-connection determines and is determined by, a smooth horizontal lift:$u\in P$,:$h_{u}$$E_{\pi(u)}\rightarrow T_{u}P$such that(i) $T_{u}\pi\circ h_{u}(v)=v$ , for all $v\in E_{x}\subset T_{x}M$ ;(ii) $h_{u\cdot a}=T_{u}R_{a}\circ h_{u}$ .

The horizontal subspace $H^{E}T_{u}P$ at$u$is then the image at

of

$u$

$h_{u}$, and theis a projection onto $H^{E}T_{u}P$ .
(2) Let $F=P\times V/\sim be$ an associated vector bundle to $P$ with fibre $V$ . Anelement of $F$ is an equivalence class $[(u, e)]$ such that $(ug, g^{-1}e)\sim(u, e)$ .
Set $\overline{u}(e)=[(u, e)]$ . An $E$ semi-connection on $P$ gives a covariant derivative
on $F$ . Let $Z$ be a section of $F$ and $w\in E_{x}\subset TXM$ , the covariant derivativeis defined, as usual for connections, bycomposition$h_{u}\circ T_{u}P$$\nabla_{w}Z\in F_{x}$$\nabla_{w}Z=u(d\tilde{Z}(h_{u}(w)), u\in\pi^{-1}(x)=F_{x}$

.

:
Here
is $\tilde{Z}(u)=\tilde{u}^{-1}Z(\pi(u))$ considering as an isomorphism:. This agrees with the ‘semi-connections on $E$ defined in
Elworthy-LeJan-Li [7] when $P$ is taken to be the linear frame bundle of $TM$
and $F=TM$ . As described there, any semi-connection can be completed toa genuine connection, but not canonically.
$\tilde{Z}$$P\rightarrow V$$\overline{u}$’$\overline{u}$$V\rightarrow F_{\pi(u)}$Consider on $P$ a diﬀusion generator $B$ , which is equivariant, i.e.
$Bf\circ R_{a}=B(f\circ Ra)$ ,$\forall f$, $g\in C^{2}(P, R)$ ,$a$$\in G$.

The operator $B$ induces an operator $A$ on the base manifold $M$ by setting(2)$Af(x)=B$ $(f\circ\pi)(u)$ ,$u\in\pi^{-1}(x)$, $f\in C^{2}(M)$ ,which is well defined since$B$$(f\circ\pi)(u\cdot a)=B((f\circ\pi))(u)$ .

35

Equivariant diﬀusions on principal bundles. Assume the dimension
Let $E_{x}:=Image(\sigma_{x}^{A})\subset T_{x}M$ , the image ofof $E_{x}=p$ , independent of . Set $E=\bigcup_{x}E_{x}$ . Then : $E\rightarrow M$ is a subbundle of $TM$ .
$\sigma_{x}^{A}$

$x$

$\pi$

gives rise to
Theorem 1.3. Assume
has constant rank Then$P$
semi-connection on the principal bundle whose horizontal map is given by$\sigma^{A}$$\sigma^{B}$

$a$

(3)

$h_{u}(v)=\sigma^{B}((T_{u}\pi)^{*}\alpha)$

where$\alpha\in T_{\pi(u)}^{*}M$

Proof. To provesatisfies$\sigma_{x}^{A}(\alpha)=v$.

is well defined we only need to show
. Now
for every 1-form on $P$ and for every in
by Lemma 1.1 that$0$$\psi(\sigma^{B}(T_{u}\pi^{*}(\alpha)))=$$h_{u}$$\psi$$\sigma^{A}\alpha=0$$ker\sigma_{x}^{A}$$\alpha$$0=\alpha\sigma^{A}(\alpha)=(T\pi)^{*}(\alpha)\sigma^{B}((T\pi)^{*}(\alpha))$implies.

Thus $T\pi^{*}(\alpha)\sigma^{B}(T\pi^{*}(\alpha))=0$ . On the other hand we may consider,
as a bilinear form on $T^{*}P$ and then for all$\sigma^{B}$$\beta\in T_{u}^{*}P$$\sigma^{B}(\beta+t(T\pi)^{*}(\alpha), \beta+t(T\pi)^{*}(\alpha))$$=\sigma^{B}(\beta, \beta)+2t\sigma^{B}(\beta, (T\pi)^{*}(\alpha))+t^{2}\sigma^{B}((T\pi)^{*}\alpha, (T\pi)^{*}\alpha)$$=\sigma^{B}(\beta, \beta)+2t\sigma^{B}(\beta, (T\pi)^{*}(\alpha))$Suppose$\sigma^{B}(\beta, (T\pi)^{*}(\alpha))\neq 0$.

. We can then choose$t$such that$\sigma^{B}(\beta+t(T\pi)^{*}(\alpha), \beta+t(T\pi)^{*}(\alpha))<0$,

which contradicts the semi-ellipticity of $B$ .
We must verify (i) $T_{u}\pi oh_{u}(v)=v$ , $v\in E_{x}\subset T_{x}M$ and (ii)
$T_{u}R_{a}o$. The first is immediate by Lemma 1.1 and for the second use the$T\pioTR_{a}=T\pi$ for all $a\in G$ and the equivariance of.
fact that$h_{u\cdot a}=$$h_{u}$$\sigma^{B}$$\blacksquare$\S 2. Horizontal lifts of diﬀusion operators and decompositions of equivariant operators
A. Denote by
the space of smooth diﬀerential -forms on a manifold $M$ . To each diﬀusion operator $A$ we shall associate a unique operator .
The horizontal lift of $A$ can be defined to be the unique operator such that theareassociated operator vanishes on vertical 1-forms and such that andintertwined by the lift map
acting on 1-forms.
$C^{\infty}\Omega^{p}$$p$$\delta^{A}$$\overline{\delta}$$\overline{\delta}$$\delta^{A}$$\pi^{*}$

Proposition 2.1. For each smooth diﬀusion operator $A$ there is a unique:smooth diﬀerential operator
such that$\delta^{A}$(1)$C^{\infty}(\Omega^{1})\rightarrow C^{\infty}\Omega^{0}$$\delta^{A}(f\phi)=df\sigma^{A}(\phi)_{x}+f\cdot\delta^{A}(\phi)$

36

K. D. Elworthy, Y. Le Jan and X.-M. Li(2).

$\delta^{A}(df)=A(f)$For example if $A$ has H\"ormander representation$A=\frac{1}{2}\sum_{j=1}^{m}\mathcal{L}_{X^{j}}\mathcal{L}_{X^{j}}+\mathcal{L}_{A}$for some$C^{1}$

vector fields$X^{i}$, $A$ then$\delta^{A}=\frac{1}{2}\sum_{j=1}^{m}\mathcal{L}_{X^{j}}\iota_{X^{j}}+\iota_{A}$

whereforms.

$\iota_{A}$denotes the interior product of the vector field $A$ acting on diﬀerential

Definition 2.2. Let be a
sub-bundle of $TN$ for some smooth man$B$
ifold N. A diﬀusion operator on $N$ is said to be along iffor;
all 1-foms which vanish on
it is said to be strongly cohesive if
hasconstant rank and is along the image of .
$S$$C^{\infty}$$\delta^{B}\phi=0$$S$$\sigma^{B}$$S$$\phi$$\sigma^{B}$$B$

To be along implies that any H\"ormander form representation of $B$ uses
only vector fields which are sections of .
$S$

$S$

Definition 2.3. When a diﬀusion operator on $P$ is along the verticalfoliation $VTP$ of the : $P\rightarrow M$ we say $B$ is vertical, and when the bundle
has a semi-connection and $B$ is along the horizontal distribution we say is
$B$

$\pi$

$B$

horizontal.
If : $P\rightarrow M$ has an $E$ semi-connection and $A$ is a smooth diﬀusion, i.e.
operator along $E$ it is easy to see that $A$ has a unique horizontal lift$P$$A$
a smooth diﬀusion operator
on which is horizontal and is a lift of in
the sense of (1). By uniqueness it is equivariant.
$\pi$$A^{H}$$A^{H}$$G$

B. The action of $G$ on $P$ induces a homomorphism of the Lie algebra 9 of,with the algebra of right invariant vector fields on $P$ : if$\alpha\in g$$A^{\alpha}(u)=\frac{d}{dt}|_{t=0}u\exp(t\alpha)$and,$A^{\alpha}$

is called the fundamental vector field corresponding to . Take a basis,of and denote the corresponding fundamental vector fields by$\alpha$$A_{k}$$A_{1}$$\ldots$,$g$.
We can now give one of the main results from [5]:$\{A_{i}^{*}\}$

37

Equivariant diﬀusions on principal bundles

Theorem 2.4. Let be an equivariant operator on $P$ with A the induced
operator on the base manifold. Assume $A$ is strongly cohesive. Then there isunique semi-connection on $P$ over $E$ for which has a decomposition$\mathcal{B}$

$a$

$B$

$B$

$=A^{H}+B^{V}$ ,has the expresis vertical. Furthemore, whereand are smoothfunctions on $P$ ,, andgiven byfor any connection 1-form$P$
on which vanishes on the horizontal subspaces of this semi-connection.
where
sion

$A^{H}$

is horizontal and$B^{V}$$\mathcal{B}^{V}$$\alpha^{ij}$$\beta^{k}$$\sum\alpha^{ij}\mathcal{L}_{A_{i}^{*}}\mathcal{L}_{A_{j}^{*}}+\sum\beta^{k}\mathcal{L}_{A_{k}^{*}}$$\beta^{\ell}=\delta^{B}(\tilde{\omega}^{\ell})$$\alpha^{k\ell}=\tilde{\omega}^{k}(\sigma^{B}(\tilde{\omega}^{\ell}))$$\tilde{\omega}$

here. The semi-connection
We shall only prove the first part of Theorem
is the one given by Theorem 1.3, and we define
to be the horizontal lift of$A$ . The proof thatis vertical is simplified by using the fact
$P$$V$functionsthat a diﬀusion operator on is vertical if and only if for all$P$$M$
on and on$A^{H}$$\mathcal{B}^{V}:=\mathcal{B}-A^{H}$$C^{2}$$f_{2}$$f_{1}$$D(f_{1}(f_{2}o\pi))=(f_{2}o\pi)D(f_{1})$ .

(4)

Set

$\tilde{f}_{2}=f_{2}\circ\pi$. Note$(B -A^{H})(f_{1}\tilde{f}_{2})=\overline{f}_{2}(B-A^{H})f_{1}+f_{1}(B-A^{H})\tilde{f}_{2}+2(df_{1})\sigma^{B-A^{H}}(d\tilde{f}_{2})$.

Therefore to show $(B -A^{H})$ is vertical we only need to prove$f_{1}(B-A^{H})\overline{f}_{2}+2(df_{1})\sigma^{B-A^{H}}(d\tilde{f}_{2})=0$

Recall Lemma 1.1 and use the natural extension of
to seethe fact that by (3)to$\sigma^{A}$.

$\sigma^{A}$:$E^{*}\rightarrow E$

and$h\circ\sigma_{x}^{A}=\sigma^{B}(T_{u}\pi)^{*}$$\sigma^{A^{H}}(d\tilde{f}_{2})$and so$=$$(h\circ\sigma^{A}h^{*})(df_{2}\circ T\pi)=h\circ\sigma^{A}df_{2}$$=$$\sigma^{B}(df_{2}\circ T\pi)=\sigma^{B}(d\tilde{f}_{2})$$\sigma^{(B-A^{H})}(d\tilde{f}_{2})=0$,. Also by equation (1)$(B -A^{H})\overline{f}_{2}=Af_{2}-A^{H}\tilde{f}_{2}=0$This shows that
Define$\alpha$:$\mathcal{B}-A^{H}$

is vertical.

$P\rightarrow g\otimes g$

and$\beta$:$P\rightarrow \mathfrak{g}$.
$\blacksquare$by$\alpha(u)=\sum\alpha^{ij}(u)A_{i}\otimes A_{j}$$\beta(u)=\sum\beta^{k}(u)A_{k}$.

38

K. D. Elworthy, Y. Le Jan and X.-M. Li

It is easy to see that
depends only on , and the expression is independentof the choice of basis of . From the invariance of we obtain$B^{V}$$\alpha$$\beta$$\mathcal{B}$$g$$\alpha(ug)$$=$$(ad(g)\otimes ad(g))\alpha(u)$$\beta(ug)$$=$$ad(g)\beta(u)$,for all $u\in P$ and $g\in G$ .

C. Theorem 2.4 has a more directly probabilistic version. For this let: $P\rightarrow M$ be as before and for $ 0\leq l<r<\infty$ let $C(l, r;P)$ be the spaceof continuous paths :with its usual Borel -algebra. For such
write $l_{y}=l$ and $r_{y}=r$ . Let $C(*, *;P)$ be the union of such spaces. It has
the standard additive structure under concatenation: if and are two paths
with $r_{y}=l_{y’}$ and $y(r_{y})=y’(l_{y’})$ let $y+y’$ be the corresponding element in$C(l_{y}, r_{y’} ; P)$ . The basic-algebra of $C(*, *, P)$ is defined to be the pull backby of the usual Borel -algebra on $C(*, *;M)$ .
Consider the lawsof the process running frombetween times and , associated to a smooth diﬀusion operator on $P$ .
Assume for simplicity that the diﬀusion has no explosion. Thus
is a kernel from $P$ to $C(l, r;P)$ . The right action
by in $G$ extends to$G$, ofgive a right action, also written
on $C(*, *, P)$ . Equivariance of isequivalent to$\pi$$[l, r]\rightarrow P$$y$$\sigma$

$y’$

$y$

$\sigma$$\pi$$\sigma$$\{\mathbb{P}_{ti^{r}}^{l} :

$l$

$a$

0\leq l<r, a\in P\}$

$B$

$r$

$\{\mathbb{P}^{l}(i^{r}, a\in P\}$

$R_{g}$$g$$B$$R_{g}$$\mathbb{P}_{ag}^{l,r}=(R_{g})_{*}\mathbb{P}_{a}^{l,r}$

for all $0\leq l\leq r$ and $a\in P$ . If sodepends only on $\pi(a)$ , , andgives the law of the induced diﬀusion $A$ on $M$ . We say that such a diﬀusion$B$
is basic if for all $a\in P$ and $ 0\leq l<r<\infty$ the basic -algebra on$C(l, r;P)$ contains all Borel sets up to
negligible sets, i.e. for all $a\in P$$B$
and Borel subsets of $C(l, r;P)$ there exists a Borel subset $A$ of $C(l, r, M)$$l$$\pi_{*}(\mathbb{P}_{ti^{r})}^{l}$$r$$\sigma$$\mathbb{P}_{a}^{l,r}$$s.t$. $\mathbb{P}_{a}(\pi^{-1}(A)\triangle B)=0$ .

For paths in $G$ it is more convenient to consider the space $C_{id}(l, r;G)$ ofwith $\sigma(l)=id$ for
continuous :the identity element. The
corresponding space $C_{id}(*, *, G)$ has a multiplication$\sigma$$‘ id’$$[l, r]\rightarrow G$$C_{id}(s, t;G)\times C_{id}(t, u).G)\rightarrow Cid(s, u;G)$$(g, g’)\mapsto g\times g’$

where $(g\times g’)(r)=g(r)$ for $r\in[s, t]$ and $(g\times g’)(r)=g(t)g’(r)$ for$r\in[t, u]$ .
Given probability measures , on $C_{id}(s, t;G)$ and $C_{id}(t, u;G)$ respecof withtively this determines a convolution
which is a probability$C_{id}(s,u;G)$.
measure on$\mathbb{Q}$$\mathbb{Q}’$$\mathbb{Q}*\mathbb{Q}^{J}$$\mathbb{Q}$$\mathbb{Q}^{/}$

39

Equivariant diﬀusions on principal bundles

Theorem 2.5. Given the laws $\{\mathbb{P}_{ci^{r}}^{l} : a\in P, 0\leq l<r<\infty\}$ of an equivariant diﬀusion as above with A strongly cohesive there exist probability,
kernelsfrom $P$ to $C(l, r;P)$ , $ 0\leq l<r<\infty$ and. from $C(l, r, P)$ to $C_{id}(l, r;G)$ such thatdefined(i)
is equivariant, basic and detemining a strongly
cohesive generator.
$B$$\{\mathbb{P}_{a}^{H,l,r} :a\in P\}$$\mathbb{Q}_{y}^{l,r}$$\mathbb{P}^{l,r}a.s$$\{\mathbb{P}_{a}^{H,l,r} :(ii)$y\mapsto \mathbb{Q}_{y}^{l,r}$

a\in P\}$satisfies,$\mathbb{Q}_{y+y}^{l_{y},r_{y’}},=\mathbb{Q}_{y^{y’ y}}^{lr}*\mathbb{Q}_{y}^{l_{y’},r_{y’}}$almost all , with $r_{y}=l_{y’}$ .
For $U$ a Borel subset of $C(l, r, P)$ ,

for
(iii)

$\mathbb{P}^{l_{y},r_{y}}\otimes \mathbb{P}^{l_{y’},r_{y’}}$

$y’$

$y$

$\mathbb{P}_{ci^{r}}^{l}(U)=\int\int\chi U(y.

\cdot g)\mathbb{Q}_{y}^{l,r}(dg)\mathbb{P}_{a}^{H,l,r}(dy)$The kernels
are uniquely detemined as are the$a.s$ .
depends onin for all in P. Furthemore
projection $\pi(y)$ and its initial point .
$\mathbb{P}_{a}^{H,l,r}$$\{\mathbb{Q}_{y}^{l,r} :

$a$

$y$

.

$\mathbb{Q}_{y}^{l,r}$$y$y\in \mathbb{R}\}$,$\mathbb{P}_{a}^{H,l,r}$

only through its$y_{l}$and let $\{b_{t} : l \leq r\leq t\}$ be a process with law
Theorem 2.4 we can assume that is given by an s.d.e. of the form

Proof Fix in
$a$

$P$

$\mathbb{P}_{a}^{l,r}$

. By

$b$

(5)

$db_{t}=\tilde{X}(b_{t})\circ dB_{t}+\tilde{X}^{0}(b_{t})dt+A(b_{t})\circ d\beta_{t}+V(b_{t})dt$,:
is the horizontal lift of some $X$ :
where
$A$
$M$
on , while :is the horizontal lift of a vector field. Here B. and . are
and the vector field $V$ are vertical and determineandrespectively, some , and we are
independent Brownian motions on$B$$P$using the semi-connection on induced by as in Theorem 1.3.
Let
satisfy$\tilde{X}$$M\times \mathbb{R}^{p}\rightarrow E$$P\times \mathbb{R}^{p}\rightarrow TP$$P\times \mathbb{R}^{1}\rightarrow TP$$X^{0}$$\tilde{X}^{0}$$B^{V}$$\mathbb{R}^{p}$$\beta$$\mathbb{R}^{q}$$q$$\{\tilde{x}_{t} :l \leq t\leq r\}$$d\tilde{x}_{t}=\tilde{X}(\tilde{x}_{t})\circ dB_{t}+\tilde{X}^{0}(\tilde{x}_{t})dt$(6)$\tilde{x}_{l}=a$is the horizontal lift of $\{\pi(b_{t}) : l \leq t\leq r\}$ . Then there is a uniquecontinuous process $\{g_{t} : l \leq t\leq r\}$ in $G$ with $g_{l}=id$ such thatso$\tilde{x}$.

$\tilde{x}_{t}g_{t}=b_{t}$

$\pi$

.

We have to analyse $\{g_{t} : l \leq t\leq r\}$ . Using local trivialisations of:
we see it is a semi-martingale. As in [9], Proposition 3.1 on page$P\rightarrow M$

69,$db_{t}=TR_{9t}(\circ d\tilde{x}_{t})+A^{g_{t}^{-1}od_{9t}}(b_{t})$40

K. D. Elworthy, Y. Le Jan and X.-M. Li

giving$\overline{\omega}(\circ db_{t})=\tilde{\omega}(A^{g_{t}^{-1}od_{9t}}(b_{t}))=g_{t}^{-1}\circ dg_{t}$for any smooth connection formThus$\tilde{\omega}$:$P\rightarrow g$

on $P$ which vanishes on $H^{E}TP$ .

$dg_{t}=TL_{9t}\tilde{\omega}(A(\overline{x}_{t}g_{t})\circ d\beta_{t}+V(\overline{x}_{t}g_{t})dt)$(7)$g_{l}=id$

For $y\in C(l, r:P)$ let,$l$$\{g_{t}^{y} :$\leq t\leq r$

l \leq t\leq r\}$.

be the solution of$d^{y}g_{t}=TL_{g_{t}^{y}}\tilde{\omega}(A(y_{t}g_{t}^{y})\circ d\beta_{t}+V(y_{t}g_{t}^{y})dt)$(8)$g_{l}^{y}=id$(where the Stratonovich equation is interpreted with $‘ dy_{t}d\beta_{t}=0’$ ). Since$\beta$and B. and hence and are independent we see
almost surely. Fora discussion of some technicalities conceming skew products, see [16].
For . in $C(*, *;P)$ let $\{h(y)_{t} : l_{y}\leq t\leq r_{y}\}$ be the horizontal lift of$\pi(y).$ , starting at. This exists for almost all as can be seen either by theextension of It\^o’s result to general principal bundles, e.g. using (6), or by theexistence of measurable sections using the fact thatis basic. Define
to be the law of above andto be that of. Clearly conditions (i) is$g=g^{\overline{x}}$$\tilde{x}$$\beta$$y$$y$$y_{l_{y}}$$A^{H}$$\tilde{x}$$\mathbb{P}_{a}^{H,l,r}$$g^{h(y)}$$Q_{y}^{l,r}$

satisfied.
To check (ii) take

$y$

and

$y’$

with $r_{y}=l_{y’}$ . Then$h(y+y^{/})=h(y)+h(y’)(g_{r_{y}}^{h(y)})^{-1}$ ,writing $y=h(y)g^{h(y)}$ and $y’=h(y’)g^{h(y^{l})}$ . For$r_{y}\leq t\leq r_{y’}$$(y+y’)_{t}=h(y^{J})_{t}(g_{r_{y}}^{h(y)})^{-1}g_{t}^{h(y+y’)}$this shows.

But $(y+y’)_{t}=y_{t}’=h(y’)_{t}g_{t}^{h(y^{J})}$ and so we havefor, giving $g^{h(y+y^{J})}=g^{h(y)}\times g^{h(y’)}$ almost surely. This proves (ii).
For uniqueness suppose we have another set of probability measures denoted
and
which satisfy (i), (ii), (iii). Since
is equivari$A$$M$
we can apply the preceding argument to it in place ofant and induces on. However since it is basic the term involving in the stochastic diﬀerential equation (6) must vanish. Since it is also strongly cohesive the vertical. On the other hand in thepart $V$ must vanish also and we havethe law of . is determined by those of . and . but
decompositionis the conditional law ofgiven
and so is uniquely determined
as described.
$g_{t}^{h(y+y^{J})}=g_{r_{y}}^{h(y)}g_{t}^{h(y^{l})}$$t\geq r_{y}$$\tilde{\mathbb{Q}}_{y}^{l,r}$$\tilde{P}_{a}^{H,l,r}$$\{\tilde{\mathbb{P}}_{a}^{H,l,r}\}_{a}$$\{\mathbb{P}_{ti^{r}}^{l}\}_{a}$$\beta$$\overline{\mathbb{P}}_{a}^{H,l,r}=\mathbb{P}_{a}^{H,l,r}$$b_{t}=\overline{x}_{t}g_{t}^{\overline{x}_{t}}$$\mathbb{Q}_{y}^{l,r}$$b$$g^{\overline{x}}$$g^{\overline{x}}$$\tilde{x}$$\tilde{x}=y$$\blacksquare$

41

Equivariant diﬀusions on principal bundles

is associated to the time dependent generator which at $g\in G$
forandand $t\in[l, r]$ isis associated to.
as defined in Theorem 2.4 while
In fact$\mathbb{Q}_{y}^{l,r}$$\sum\alpha^{ij}(h(y)_{t}g)\mathcal{L}_{A_{i}}\mathcal{L}_{A_{j}}+\sum\beta^{k}(h(y)_{t}g)\mathcal{L}_{A_{k}}$$\alpha^{ij}$$\beta^{k}$$A^{H}$$\mathbb{P}^{H,l,r}$\S 3. Stochastic flows and derivative flows
A. Derivative flows. Let $A$ on $M$ be given in H\"ormander form
$A=\frac{1}{2}\sum_{j=1}^{m}\mathcal{L}_{X^{j}}\mathcal{L}_{X^{j}}+\mathcal{L}_{A}$,, $A$ . As before let $E_{x}=span\{X^{1}(x)$ ,,for some vector fields$E\subset
TM$$X^{m}(x)\}$ and assume $dimE_{x}$ is constant,, say, giving a sub-bundle.
$X^{m}(x)$,oftrivialmap
thebundle,determine a vector bundle$X^{1}$$\ldots X^{m}$$\ldots$$p$$\ldots$$TheX^{1}(x)\underline{\mathbb{R}}^{m}$$X$:$\underline{\mathbb{R}}^{m}\rightarrow TM$.
with $\sigma^{A}=X(x)X(x)^{*}$ . We can, and will, consider $X$ as a map $X$ :$E$
on (theAs such it determines (a) a Riemannian metric$E$same as that determined by ) and (b) a metric connection on uniquely
defined by the requirement that for each in $M$ ,$\underline{\mathbb{R}}^{m}\rightarrow E$$\{\langle, \rangle_{x} :x\in M\}$$\dot{\nabla}$$\sigma^{A}$$x$$\dot{\nabla}_{tJ}X(e)=0$

for all $v\in T_{x}M$ whenever is orthogonal to the kemel of $T_{x}M$ . Then for any
diﬀerentiable section $U$ of $E$ ,$e$$\nabla\vee vU=Y(x)d(Y(U(\cdot)))(v)$ ,(9)where $Y$ is the$\mathbb{R}^{m}$$v\in T_{x}M$,valued 1-form on $M$ given by$\langle Y_{x}(v), e\rangle_{\mathbb{R}^{m}}=\langle X(x)(e), v\rangle_{x}$,$e\in \mathbb{R}^{m}$, $v\in E_{x}$ , $x\in M$

e.g. [7] where it is referred to as the LeJan-Watanabe connection in this context. By a theorem of Narasimhan and Ramanan [14] all metric connections
on $E$ arise this way, see [15], [7].
, the stochastic diﬀerenFor $\{B_{t} : 0\leq t<\infty\}$ a Brownian motion on
tial equation$\mathbb{R}^{m}$(10)$dx_{t}=X(x_{t})\circ dB_{t}+A(x_{t})dt$

determines a Markov process with diﬀerential generator $A$ . Over each solution$\{x_{t} : 0\leq t<\rho\}$ , whereis the explosion time, there is a ’derivative’ pro$\{v_{t} : o\leq t<\rho\}$
$TM$
cess
in
which we can write as $\{T\xi_{t}(v_{0}) : 0\leq t<\rho\}$$\rho$

42

with

K. D. Elworthy, Y. Le Jan and X.-M. Li:

linear. This would be the derivative of the flow$\{\xi_{t} : 0\leq t<\rho\}$ of the stochastic diﬀerential equation when the stochastic differential equation is strongly complete. In general it is given by a stochastic
diﬀerential equation on the tangent bundle $TM$ , or equivalently by a covariant
equation along $\{x_{t} : 0\leq t<\rho\}$ :$T\xi_{t}$$T_{x_{0}}M\rightarrow T_{x_{C}}M$$Dv_{t}=\nabla X(v_{t})\circ dB_{t}+VA(vt)dt$

with respect to any torsion free connection. Take $P$ to be the linear frame bun.
dle $GL(M)$ of $M$ , treating $u\in GL(M)$ as an isomorphism$\{u_{t}:0\leq
t<\rho\}$$u_{0}\in
GLM$
$GLM$
For
we obtain a processonby$u:\mathbb{R}^{n}\rightarrow T_{\pi(u)}M$$u_{t}=T\xi_{t}ou_{0}$.

Let $B$ be its diﬀerential generator. Clearly it is equivariant and a lift of $A$ .
A proof of the following in the context of stochastic flows, is given later.
For $w\in E_{x}$ , set(11)$Z^{w}(y)=X(y)Y(x)(w)$ .

Theorem 3.1. The semi-connection
induced by $B$ is the adjoint connection of the LeJan-Watanabe connection detemined by $X$ , as defined by(9), [7]. Consequently $\nabla_{w}V=L_{Z^{w}}V$ for any vector field $V$ and $w\in E$ also
vanishes if $w\in E_{x}$ .
$\nabla$$\dot{\nabla}$$\nabla_{V(x)}Z^{w}$

In the case of the derivative flow theexpression: for $u\in GLM$ ,$\alpha$,$\beta$of Theorem 2.4 have an explicit$\alpha(u)=\frac{1}{2}\sum(u^{-1}(-)\dot{\nabla}_{u(-)}X^{p})\otimes(u^{-1}(-)\dot{\nabla}_{u(-)}X^{p})$

(12)

$\{$

$\beta(u)=-\frac{1}{2}\sum u^{-1}\dot{\nabla}_{\dot{\nabla}_{u(-)}X^{p}}X^{p}-\frac{1}{2}u^{-1}Ric\# u(-)$

Here is the curvature tensor for
defined by$\check{R}$$\dot{\nabla}$

and$\dot{R}ic^{\neq}(v)=\sum_{j=1}^{p}\dot{R}(v, e^{j})e^{j}$$\dot{R}ic^{\#}$.

: $TM\rightarrow E$ the Ricci curvature, $v\in T_{x}M$ .

Equivariant operators on $GLM$ determine operators on associated bundles, such $as\wedge^{q}TM$ . If the original operator was vertical this turns out to bea zero order operator (as is shown in [5] for general principal bundles) and in
the case $of\wedge^{q}TM$ these operators are the generalized Weitzenbock curvature
operators described in [7]. In particular for diﬀerential 1-forms the operator
is $\phi\mapsto\phi(Ric^{\neq}-)$ . To see this, as an illustrative example, given a 1-form$\phi$

43

Equivariant diﬀusions on principal bundles

define:$\tilde{\phi}$$GLM\rightarrow L(\mathbb{R}^{n}; \mathbb{R})$$L_{A_{j}^{*}}(\tilde{\phi})(u)$by$\tilde{\phi}(u)=\phi_{\pi u}u$so$\tilde{\phi}(ug)=\phi_{\pi u}(ug-)$$=$$\frac{d}{dt}\tilde{\phi}(u\cdot e^{A_{j}t})|_{t=0}$$=$$\frac{d}{dt}\phi_{\pi u}(u\cdot e^{A_{j}t})|_{t=0}$$=$$\phi_{\pi u}(uA_{j}-)=\tilde{\phi}(u)(A_{j}-)$. Then.

Iterating we have$\mathcal{B}^{V}(\tilde{\phi})(u)$$=$$\sum_{i,j}\alpha^{i,j}(u)\phi_{\pi u}(uA_{j}A_{i}-)+\sum_{k}\beta^{k}(u)\phi_{\pi u}(uA_{k}-)$1$=$$-\overline{2}\phi(u)(u^{-1}Ric^{\#}(u-))$as required, by using the map $gl(n)\otimes gl(n)\rightarrow gl(n)$ , $S\otimes T\mapsto S\circ T$ , andequation (12).
B. Stochastic flows. In fact Theorem 3.1 can be understood in the moregeneral context of stochastic flows as diﬀusions on the diﬀeomorphism groups.
For this assume that $M$ is compact and for $r\in\{1, 2, \ldots\}$ and $s>r+$
manifold of diﬀeomorphisms of $M$ of$dim(M)/2$ let $D^{s}=D^{s}M$ be the, (for example see Ebin-Marsden [2] or Elworthy [3].) AlterSobolev classwe
could
take the spaceofdiﬀeomorphisms with diﬀerennatively
in $M$ and let :betiate structure as in [11]. Fix a base point
into a principal bundle over $M$ with group theevaluation at . This makes
manifoldof-diﬀeomorphisms with $\theta(x_{0})=x_{0}$ , acting on the right).
, for $r=0,1$ , 2,
by composition (although the action of $D^{s+r}$ is only
Let $\{\xi_{t}^{s} : 0\leq s\leq t<\infty\}$ be the flow of (10) starting at time . Write
for . The more general case allowing for infinite dimensional noise ison $C([s.t];M)$
given in [5]. We define probability measures(These correspond to thebe letting
be the law ofdiﬀusion process on
associated to the right-invariant stochastic diﬀerential
satisfied by $\{\xi_{t} : 0\leq t<\infty\}$ as in [3].) These are equivariant
equation on
and project by to the laws given by the stochastic diﬀerential equation on $M$ .
Assuming that these give a strongly cohesive diﬀusion on $M$ we are essentiallyin the situation of Theorem 2.5.
Let $K(x)$ :be the orthogonal projection onto the kemel of$X(x)$ , each $x\in M$ . set $K^{\perp}(x)=id-K(x)$ . Consider the-valuedprocess
given by (or as the flow of)$C^{\infty}$$H^{s}$$C^{\infty}$$D^{\infty}$$\pi$$x_{0}$$D^{s}$$x_{0}$$H^{s}$$D_{x_{0}}^{s}$$D^{s}\rightarrow M$$\theta$$C^{r}$$\ldots$$s$$\xi_{t}$$\xi_{t}^{0}$$\{\mathbb{P}_{\theta}^{s,t} :$\{\xi_{r}^{s}\circ\theta :$\mathbb{P}_{\theta}^{s,t}$\theta\in D^{s}\}$s\leq r\leq t\}$$D^{s}$$D^{s}$$\pi$$\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$$D^{\infty}$$\{\theta_{t} :0\leq t<\infty\}$(13) $d\theta_{t}(x)=X(\theta_{t}(x))K^{\perp}(\theta_{t}(x_{0}))\circ dB_{t}+X(\theta_{t}(x))Y(\theta_{t}(x_{0}))A(\theta_{t}(x_{0}))$

44

K. D. Elworthy, Y. Le Jan and X.-M. Li

for given$\theta_{0}$(14)$dg_{t}$in$D^{\infty}$

and, define a$D_{x_{0}}^{\infty}$-valued process $\{g_{t} : 0\leq t<\infty\}$ by$T\theta_{t}^{-1}\{X(\theta_{t}g_{t}-)K(\theta_{t}x_{0})\circ dB_{t}$$=$$+A(\theta_{t}g_{t}-)dt-X(\theta_{t}g_{t}-)Y(\theta_{t}x_{0})A(\theta_{t}x_{0})dt\}$$g_{0}$

Set

$id$

$=$

$x_{t}^{\theta}=\xi_{t}(\theta_{0}(x_{0}))$.

. Note that$\pi(\theta_{t})=\theta_{t}(x_{0})=x_{t}^{\theta}$

since$X(\theta_{t}(x_{0}))K^{\perp}(\theta_{t}(x_{0}))=X(\theta_{t}(x_{0}))$and$X(\theta_{t}(x_{0}))Y(\theta_{t}(x_{0}))A(\theta_{t}(x_{0}))=A(\theta_{t}(x_{0}))$.

. It can be considered toThus $\{\theta_{t} : 0\leq t<\infty\}$ is alift of, i.e. by
be driven by the ’relevant noise’, (from the point of view of
the Brownian motion . given by$\{x_{t}^{\theta}, 0\leq t<\infty\}$$\xi(\theta_{0}(x_{0}))$$\overline{B}$$\tilde{B}_{t}=\int_{0}^{t}\overline{//}(x^{\theta})_{s}^{-1}K^{\perp}(x_{s}^{\theta})dB_{s}$with respect to theis parallel translation along, sodetermined by $K$ andconnection on the trivial bundle
that
:

where

$\{\overline{//}(x.\theta ), 0\leq s<\infty\}$$x^{\theta}$$K^{\perp}$$M\times \mathbb{R}^{m}\rightarrow M$$\overline{//}(x^{\theta})_{s}$$\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$foris orthogonal and maps the kemel of $X(\theta(x_{0}))$ onto the kernel of$ 0\leq s<\infty$ , see $[7](chapter3)$ .
Correspondingly there is the ’redundant noise’, the Brownian motion { :$0\leq t<\infty\}$ given by$X(x_{s}^{\theta})$$\beta_{t}$$\beta_{t}=\int_{0}^{t}//(x^{\theta})_{s}^{-1}K(x_{s}^{\theta})dB_{s}\sim$.

Then, as shown in $[7](chapter3)$ ,
(i)
(ii)
(iii)

same filtration as. and . are independent$\tilde{B}$. has the$\{x_{s}^{\theta} :0\leq s<\infty\}$$\tilde{B}$$\beta$$dB_{t}=\overline{//}ld\beta_{t}+//\sim td\tilde{B}t$

We wish to see how$g$.

is driven by$\beta$. For this observe$\int_{0}^{t}K(x_{s}^{\theta})\circ dB_{s}=\int_{0}^{t}K(x_{s}^{\theta})dB_{s}+\int_{0}^{t}\Lambda(x_{s}^{\theta})ds$

for$\Lambda$:$M\rightarrow \mathbb{R}$given by the Stratonovich correction term. By (iii)$\int_{0}^{t}K(x_{s}^{\theta})dB_{s}=\int_{0}^{t}//_{s}d\beta_{s}-=\int_{0}^{t}\overline{/}/_{s}\circ d\beta_{s}$45Equivariant diﬀusions on principal bundles

since//-. is independent of by (i) and (ii). Thus equation (14) for . can be$\beta$$g$written as$T\theta_{t}^{-1}\{X(\theta_{t}g_{t}-)//(\theta-.(x_{0}))_{t}\circ d\beta_{t}+X(\theta_{t}g_{t}-)\Lambda(\theta_{t}(x_{0}))dt$$=$$dg_{t}$$+A(\theta_{t}g_{t}-)dt-X(\theta_{t}g_{t}-)Y(\theta_{t}x_{0})A(\theta_{t}x_{0})dt\}$

and if we define$dg_{t}y$$Ty_{t}-1\{X(y_{t}g_{t}-)\overline{//}(y.(x_{0}))_{t}\circ d\beta_{t}+X(y_{t}g_{t}-)\Lambda(y_{t}(x_{0}))dt$$=$$+A(y_{t}g_{t}-)dt-X(y_{t}g_{t}-)Y(y_{t}x_{0})A(y_{t}x_{0})dt\}$
$id$

$=$

$90$

for any continuous :that .
.
It\^o’sformula on
By
$y$

$g$

$[0, \infty)\rightarrow D^{\infty}$, we see, by the independence of$\beta$and$\theta$$=g^{\theta}$$\cdot$$D^{s}$, for $x\in M$ ,$d(\theta_{t}g_{t}(x_{0}))=(\circ d\theta_{t})(g_{t}(x))+T\theta_{t}(\circ dg_{t}\theta(x))$.

Now$T\theta_{t}(\circ dg_{t}\theta(x))$$=$$\{X(\theta_{t}g_{t}(x))K(\theta_{t}x_{0})\circ dB_{t}$$+A(\theta_{t}g_{t}(x))dt-X(\theta_{t}g_{t}(x))Y(\theta_{t}x_{0})A(\theta_{t}x_{0})dt\}$

and so by (13) we see that
we have
Taking$\theta_{t}g_{t}=\xi_{t}\circ\theta_{0}$, $a.s$ .

$\theta_{0}=id$

Proposition 3.2. The flow$\xi$

. has the decomposition$\xi_{t}=\theta_{tg_{t}}^{\theta}$for$\theta$and$g^{\theta}$.

$\equiv g$,$ 0\leq t<\infty$. given by (13) and (14) above. For almost allwith $\sigma(0)=x_{0}$ and bounded measurable$\sigma$:$[0, \infty)\rightarrow M$$F:C(0, \infty;D^{\infty})\rightarrow \mathbb{R}$$E\{F(\xi)|\xi.(x_{0})=\sigma\}=E\{F(\tilde{\sigma}g^{\overline{\sigma}}.)\}$

where$\overline{\sigma}$:$[0, \infty)\rightarrow D^{\infty}$

is the horizontal lift of with$\sigma$$\tilde{\sigma}(O)=id$.

To define the ’horizontal lift’ above we can use the fact, from (i) above,that . has the same filtration as
and so furnishes a lifting map.
over $E$ , fromIn terms of the semi-connection induced on :above, by uniqueness or directly, we see the horizontal lift$\theta$$\xi.(x_{0})$$\pi$$h_{\theta}$:$E_{\theta(x_{0})}\rightarrow T_{\theta}D^{s}$$h_{\theta}(v)$:$M\rightarrow TM$$D^{s}\rightarrow M$

46

K. D. Elworthy, Y. Le Jan and X.-M. Li

is given by $h_{\theta}(v)=X(\theta(x))Y(\theta(x_{0}))v$ and the horizontal lift . from, all , is given byandacurve on $M$ with$\overline{\sigma}$$\overline{\sigma}_{0}$$C^{1}$$\overline{\sigma}_{0}(x_{0})=\sigma_{0}$$\sigma$$\dot{\sigma}(t)\in E_{\sigma(t)}$

of

$t$

$\frac{d}{dt}\tilde{\sigma}_{t}=X(\tilde{\sigma}_{t}-)Y(\sigma_{t})\dot{\sigma}_{t}$for$\overline{\sigma}_{0}=id$. The lift is the solution flow of the diﬀerential equation$\dot{y}_{t}=Z^{\sigma}(y_{t})$on $M$ .
For each frame
bundles$u$:$\mathbb{R}^{n}\rightarrow T_{x_{0}}M$(15)$ D^{s}\theta$

there is a homomorphism of principal$\mapsto T_{x_{0}}\theta\circ u\rightarrow GL(M)$.

. (If the latter satisfies
This sends
to the derivative process
the strongly cohesive condition we could apply our analysis to this submersion$D^{s}\rightarrow GLM$
and get another decomposition of )$\{\xi_{t} :t\geq 0\}$$T_{x}\xi_{t}\circ u$$\xi..$

Results in Kobayashi-Nomizu [9] (Proposition 6.1 on page 79) apply tothe homomorphism $D^{s}\rightarrow GL(M)$ of (15). This gives a relationship between
on $GLM$ dethe curvature and holonomy groups of the semi-connection
termined by the derivative flow and those of the connection induced by thediﬀusion on. It also shows that the horizontal lift$\{x_{t}
:
t\geq
0\}$
for
the flow
to $GL\{M$ ) is justthrough of$\theta_{0}=id$, i.e. the solution flow of the stochastic diﬀerential
given by (13) with$\hat{\nabla}$$D^{s}\rightarrow\pi M$$u$$\{\overline{x}_{t} :$T_{x_{0}}\theta_{t}\circ u$$\{\theta_{t} :t\geq 0\}$t\geq 0\}$equation$dy_{t}=Z^{odx_{t}}(y_{t})$.

is the adjoint of the LeJanFrom this and Lemma 1.3.4 of [7] we see that
Watanabe connection determined by the flow, so proving Theorem 3.1 above.
However the present construction applies with GLM replaced by any natural
bundle over $M$ (e.g. jet bundles, see Kolar-Michor-Slovak [10]), to give semiconnections on these bundles.
$\hat{\nabla}$

References
L. Accardi, A. Mohari, and Centro V. Volterra. On the structure of classical andquantum flows. J. Fund. Anal, 135(2):421-455, 1996.
[2] D. G. Ebin and J. Marsden. Groups of diﬀeomorphisms and the motion of an
incompressible fluid. Ann. Math., pages 102-163, 1970.
[1]

Equivariant diﬀusions on principal bundles

47

K. D. Elworthy. Stochastic Diﬀerential Equations on Manifolds. LMS Lecture
Notes Series 70, Cambridge University Press, 1982.
[4] K. D. Elworthy and W. S. Kendall. Factorization of harmonic maps and Brownian
motions. In From local times to global geometry, control and physics (Coventry,
1984/85), Piman Res. Notes Math. Ser., 150, pages 75-83. Longman Sci. Tech.,
Harlow, 1986.
[5] K. D. Elworthy, Y LeJan, and Xue-Mei Li. In preparation.
[6] K. D. Elworthy, Y. LeJan, and Xue-Mei Li. Conceming the geometry of stochastic
diﬀerential equations and stochastic flows. In ’New Trends in stochastic Analysis’, Proc. Taniguchi Symposium, Sept. 1994, Charingworth, ed. K. D. Elworthy
and S. Kusuoka, I. Shigekawa. World Scientific Press, 1996.
[7] K. D. Elworthy, Y LeJan, and Xue-Mei Li. On the geometry ofdiﬀuision operatorsand stochastic flows, Lecture Notes in Mathematics 1720. Springer, 1999.
[8] K. It\^o. The Brownian motion and tensor fields on Riemannian manifold. In Proc.
Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leﬄer, Djursholm, pages 536-539, 1963.
[9] S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry, Vol. I. Interscience Publishers, 1969.
[10] I. Kolar, P. W. Michor, and J. Slovak. Natural operations in diﬀerential geometry.
Springer-Verlag, Berlin, 1993.
[11] A. Kriegl and P. W. Michor. The convenient setting of global analysis. Mathematical Surveys and Monographs. 53. American Mathematical Society, 1997.
[12] Y LeJan and S. Watanabe. Stochastic flows of diﬀeomorphisms. In Stochastic
analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32,, pages 307332. North-Holland, Amsterdam, 1984.
[13] M. Liao. Factorization of diﬀusions on fibre bundles. Trans. Amer. Math. Soc,
[3]

311(2):813-827, (1989).
[14] M. S. Narasimhan and S. Ramanan. Existence of universal connections. American

J. Math., S3, 1961.
[15] D. Quillen. Superconnection character forms and the Cayley transform. Topology,27(2):211-238, 1988.
[16] J. C. Taylor. Skew products, regular conditional probabilities and stochastic diﬀerential equation: a technical remark. In S\’eminaire de probabilit\’es, XXVI, Lecture
Notes in Mathematics 1526, ed. J. Az\’ema, P. A. Meyer and M. Yor, pages 113-

126, 1992.

K. D. Elworthy
Mathematics Institute, Warwick University,
Coventry CV47AL, UK
Y. Le JanD\’epartment de Math\’ematique, Universit\’e Paris Sud,
91405 Orsay, France

Xue-Mei Li
Department of Computing and Mathematics, The Nottingham Trent University,
Nottingham NG7IAS, UK. e-mail address. xuemei.li@ntu.ac.uk

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 49-74

Monge-Kantorovitch Measure Transportation,
Monge-Amp\‘ere Equation and the It\^o Calculus
Denis Feyel and Ali S\"uleyman \"Ust\"unelDedicated to Professor Kiyosi It\^o for his 88th birthday
Abstract.

Let $(W, \mu, H)$ be an abstract Wiener space assume two , $i=$
, 2 probabilities on $(W, B(W))$ . Assume that the Wasserstein disandwith respect to the Cameron-Martin norm
tance between$l\nearrow i$$1$$\nu_{2}$$\nu_{1}$$d_{H}(1/_{1}, \nu_{2})=\{\inf_{\beta}\int_{W\times W}|x-y|_{H}^{2}d\beta(x, y)\}^{1/2}$

is finite, where the infimum is taken on the set of probability measuresandand that
on $W\times W$ whose first and second marginals are
has regular disintegration along a sequence of finite dimensional
projections. Then there exists a unique (cyclically monotone) map$T$ $=I_{W}+\xi$ , with
: $W\rightarrow H$ , such that $T$ maps
toanduniquethesolutionoftheMongeisthe measure, then $T$ is stochasticallyKantorovitch problem. Besides, if
invertible, i.e., there exists: $W\rightarrow W$ such that $SoT=I_{W}$$\nu_{1}a.s$ . and $ToS=I_{W}\nu_{2}a.s$ .
If $I/_{1}=\mu$ , then there exists a 1, such thatconvex functionin the Gaussian Sobolev space. These results imply that the quasi-invariant transformationsof the Wiener space with finite Wasserstein distance ﬀom can be
written as the composition of a transport map $T$ and a rotation,i.e., a measure preserving map. We give also 1-convex sub-solutions
using by calculating the Gaussian jacobian. Finally the full solutionsof the Monge-Amp\‘ere equation on $W$ are given with the help of the
It\^o calculus.
$\beta$$\nu_{2}$$\iota/_{1}$$\iota/_{1}$$\xi$$\gamma=$$\nu_{2}$$\nu_{1}$$(I \times T)\nu_{1}$$\nu_{2}<<\mu^{1)}$$S$$\phi$$I\supset_{2,1}$$\xi=\nabla\phi$

$\mu$

\S 1.

Introduction

In 1781, Gaspard Monge has published his celebrated memoire about
the most economical way of earth-moving [20]. The configurations of1)

Received March 6, 2003.
In fact this hypothesis is too strong, cf. Theorem 4.1.

D. Feyel and A. S.

50$\dot{U}$st\"unel

excavated earth and remblai were modelized as two measures of equal
mass, say and , that Monge had supposed absolutely continuous withrespect to the volume measure. Later Amp\‘ere has studied an analogous
question about the electricity current in a media with varying conductivity. In modern language of measure theory we can express the problem
in the following terms: let $W$ be a Polish space on which are given two
positive measures and , of finite, equal mass. Let $c(x, y)$ be a costfunction on $W\times W$ , which is, usually, assumed to be positive. Does
there exist a map $T$ : $W\rightarrow W$ such that $ T\rho=\nu$ and $T$ minimizes theintegral$\nu$$\rho$$\nu$$\rho$$\int_{W}c(x, T(x))d\rho(x)$between all such maps? The problem has been further studied by Appell$[1, 2]$ and by Kantorovitch [16].
Kantorovitch has succeeded to transform this highly nonlinear problem of Monge into a linear problem byreplacing the search for $T$ with the search of a measure on $W\times W$
with marginals and such that the integral$\gamma$$\nu$$\rho$$\int_{W\times W}c(x, y)d\gamma(x, y)$is the minimum of all the integrals$\int_{W\times W}c(x, y)d\beta(x, y)$whereruns in the set of probability measures on $W\times W$ whose
marginals are and . Since then the problem addressed above is called
the Monge problem and the quest of the optimal measure is called theMonge-Kantorovitch problem.
In this paper we survey and complete the recent results (cf. [12, 13])
about the Monge-Kantorovitch and the Monge problem in the frameof an abstract Wiener space with a (infinitesimal) cost function on$W\times W$ , which is singular with respect to the natural Fr\’echet topologyof this space. Let us explain all this more rigourously: let $W$ be a
separable Fr\’echet space with its Borel sigma algebra $B(W)$ and assume
that there is a separable Hilbert space $H$ which is injected densely andcontinuously into $W$ , thus the topology of $H$ is, in general, stronger than
the topology induced by $W$ . The cost function $c:W\times W\rightarrow R_{+}\cup\{\infty\}$
is defined as$c(x, y)=|x-y|_{H}^{2}$ ,$\beta$$\rho$$\nu$$c$

we suppose that $ c(x, y)=\infty$ if $x-y$ does not belong to $H$ . Clearly,
this choice of the function is not arbitrary, in fact it is closely related
$c$

51

Monge-Amp\‘ere equation and the It\^o calculus

to It\^o Calculus, hence also to the problems originating ﬀom Physics,
quantum chemistry, large deviations, etc. Since for all the interestingmeasures on $W$ , the Cameron-Martin space is a negligeable set, the costdenote the set offunction will be infinity very ﬀequently. Let$W\timesW$with given marginals and . It is
probability measures on
a convex, compact set under the weak topology $\sigma(\Sigma, C_{b}(W\times W))$ . As
explained above, the problem of Monge consists of finding a measurable
map $T$ : $W\rightarrow W$ , called the optimal transport of to , i.e., $T\rho=\nu^{2)}$which minimizes the total cost$\Sigma(\rho, \nu)$$\nu$$\rho$$\rho$$U\rightarrow\int_{W}|x-U(x)|_{H}^{2}d\rho(x)$

$\nu$

,

between all the maps $U$ : $W\rightarrow W$ such that $ U\rho=\nu$ . The MongeKantorovitch problem will consist of finding a measure on $W\times W$ , which, defined byminimizes the function$\theta\rightarrow J(\theta)$(1.1)$J(\theta)=\int_{W\times W}|x-y|_{H}^{2}d\theta(x, y)$where runs in
Wasserstein metric$\theta$$H$$\Sigma(\rho, \nu)$,. Note thatis the square ofwith respect to the Cameron-Martin space$\inf\{J(\theta) :\theta\in\Sigma(\rho, \nu)\}$$d_{H}(\rho, \nu)$.

Any solution of the Monge-Kantorovitch problem will give a solution to the Monge problem provided that its support is included in thegraph of a map. Hence our work consists of realizing this program. Although in the finite dimensional case this problem is well-studied in thepath-breaking papers of Brenier [4] and McCann $[18, 19]$ , cf. also $[25, 26]$ ,the things do not come up easily in our setting and the diﬃculty is dueto the fact that the cost function is not continuous with respect to the
Fr\’echet topology of $W$ , for instance the weak convergence of the probability measures does not imply the convergence of the integrals of thecost function. In other words the function $|x-y|_{H}^{2}$ takes the value plus
infinity “very often”. On the other hand the results we obtain seem tohave important applications to several problems of stochastic analysis
that we shall explain while enumerating the contents of the paper.
In Section 2, are given the basic results of functional analysis on
the Wiener space (cf., for instance [10, 28]) and the related probabilistic
theory of convex functions developed in [11]. Section 3 deals with thederivation of some inequalities which control the Wasserstein distance.
$\gamma$2)$ T\rho$.

We denote the push-forward of$\rho$

by $T$ , i.e., the image of$\rho$

under $T$ , by

D. Feyel and A. S.

52$\dot{U}$st\"unel

In particular, with the help of the Girsanov theorem and the It\^o calculus, we give a very simple proof of an inequality, initially discovered
by Talagrand ([24]), some simple applications are $ako$ illustrated. The
facility with which one obtains these results gives an idea about the efficiency of the infinite dimensional techniques, namely the It\^o calculus
for the Monge-Kantorovitch like problems.
In Section 4 we give the full statement for the existence and theuniqueness of solution of the Monge problem and the uniqueness of thesolution of the Monge-Kantorovitch problem under the hypothesis that
the Wasserstein distance is finite. We have avoided to give the corresponding proofs which are quite technical (cf. [13]), however all theapplications are provided with proofs or explanations in such a way thatthe reader can have an idea about how to do it.
Section 5 studies the Monge-Amp\‘ere equation for the measures which
are absolutely continuous with respect to the Wiener measure. First we
define the Alexandrov versions of the Ornstein-Uhlenbeck operator andthe second order Sobolev derivatives for 1-convex Wiener maps. With
the help of these, we write the corresponding Jacobian using the modified Carleman-Fredholm determinant which is natural in the infinite
dimensional case (cf., [29]). Here we have a major diﬃculty which originates from the pathology of the Radon-Nikodym derivatives of the vector
measures with respect to a scalar measure: in fact even if the second
order Sobolev derivative of a Wiener function is a vector measure withvalues in the space of Hilbert-Schmidt operators, its absolutely continuous part has no reason to be Hilbert-Schmidt. Hence the CarlemanFredholm determinant may not exist, however due to the 1-convexity,
the detereminants of the approximating sequence are all with values in
the interval $[0, 1]$ . Consequently we can construct the subsolutions withthe help of the Fatou lemma.
Last but not the least, in section 6, we remark that all these difficulties can be overcome thanks to the natural renormalization of the
It\^o stochastic calculus. In fact using the It\^o representation theorem andthe Wiener space analysis extended to the distributions [27], we can givethe explicit solution of the Monge-Amp\‘ere equation. This is a remarkable result in the sense that such techniques do not exist in the finite
dimensional case.

\S 2.

Preliminaries and notations

Let $W$ be a separable Fr\’echet space equipped with a Gaussian
measure of zero mean whose support is the whole space. The corresponding Cameron-Martin space is denoted by $H$ . Recall that the
$\mu$

53

Monge-Amp\‘ere equation and the It\^o calculus

injection $Hc\rightarrow W$ is compact and its adjoint is the natural injection. The triple $(W, \mu, H)$ is called an abstract Wiener
space. Recall that $W=H$ if and only if $W$ is finite dimensional. A subspace $F$ of $H$ is called regular if the corresponding orthogonal projection
has a continuous extension to $W$ , denoted again by the same letter. It is
well-known that there exists an increasing sequence of regular subspaces$(F_{n}, n\geq 1)$ , called total, such thatis dense in $H$ and in $W$ . Let, then for any $f\in L^{p}(\mu)$ ,be the -algebra generated by
the martingale sequence $(E[f|\sigma(\pi_{F_{n}})], n \geq 1)$ converges to (strongly. Observe that the function $f_{n}=E[f|\sigma(\pi_{F_{n}})]$ can
if $ p<\infty$ ) in
be identified with a function on the finite dimensional abstract Wiener, where.
space
Since the translations of with the elements of $H$ induce measuresequivalent to , the G\^ateaux derivative in $H$ direction of the random-spaces and this closure will bevariables is a closable operator on
denoted by cf., for example $[10, 28]$ . The corresponding Sobolev spaces(the equivalence classes) of the real random variables will be denoted as$D_{p,k}$ , where $k\in N$ is the order of diﬀerentiability and $p>1$ is the orderof integrability. If the random variables are with values in some separable
Hilbert space, say , then we shall define similarly the corresponding
Sobolev spaces and they are denoted as $I)_{p,k}(\Phi)$ , $p>1$ , $k\in N$ . Since: $D_{p,k}\rightarrow D_{p,k-1}(H)$ isacontinuous and linear operator its adjoint isa well-defined operator which we represent by . In the case of classicalWiener space, i.e., when $W=C(1R_{+}, 1R^{d})$ , then coincides with the It\^ointegral of the Lebesgue density of the adapted elements of $FJ_{p,k}(H)$ (cf.
[28] .
, we note by
For any $t\geq 0$ and measurable :$W^{\star}L\rightarrow H^{\star}\subset L^{2}(\mu)$$\bigcup_{n}F_{n}$$\sigma(\pi_{F_{n}})^{3)}$$\sigma$$\pi_{F_{n}}$$f$$L^{p}(\mu)$$(F_{n}, \mu_{n}, F_{n})$$\mu_{7\iota}=\pi_{n}\mu$

$\mu$

$\mu$

$L^{p}(\mu)$$\nabla$$\Phi$$\nabla$$\delta$$\delta$$)$$f$$W\rightarrow \mathbb{R}_{+}$$P_{t}f(x)=\int_{W}f(e^{-t}x+\sqrt{1-e^{-2t}}y)\mu(dy)$ ,is a hypercontractive semigroup on
it is well-known that$L^{p}(\mu),p>1$ , which is called the Ornstein-Uhlenbeck semigroup (cf.
$[10, 28])$ . Its infinitesimal generator is denoted by
and we call theOrnstein-Uhlenbeck operator (sometimes called the number operator by
the physicists). The norms defined by$(P_{t}, t\in \mathbb{R}_{+})$$-\mathcal{L}$(2.2)$\mathcal{L}$$||\phi||_{p,k}=||(I+\mathcal{L})^{k/2}\phi||_{L^{p}(\mu)}$are equivalent to the norms defined by the iterates of the Sobolev derivative . This observation permits us to identify the duals of the space$\nabla$3) For the notational simplicity, in the sequel we shall denote it by$\pi F_{n}$.

D. Feyel and A. S.

54$\dot{U}$st\"unel$D_{p,k}(\Phi);p>1$ , $k\in N$ by $D_{q,-k}(\Phi^{/})$ ,with $q^{-1}=1-p^{-1}$ , where thelatter space is defined by replacing in (2.2) by $-k$ , this gives us the
distribution spaces on the Wiener space $W$ (in fact we can take as,
any real number). An easy calculation shows that, formally,
and this permits us to extend the divergence and the derivative operators to the distributions as linear, continuous operators. In fact
: $D_{q,k}(H\otimes\Phi)\rightarrow D_{q,k-1}(\Phi)$ and: $D_{q,k}(\Phi)\rightarrow D_{q,k-1}(H\otimes\Phi)$
continuously, for any $q>1$ and, wheredenotes the completed Hilbert-Schmidt tensor product (cf., for instance [28]).
Let us recall some facts ﬀom the convex analysis. Let $K$ be a Hilbert
space, a subset
of $K\times K$ is called cyclically monotone if any finite
subset $\{(x_{1}, y_{1}), \ldots, (x_{N}, y_{N})\}$ of satisfies the following algebraic condition:
$k$

$k$

$\delta o\nabla=\mathcal{L}$$\nabla$$\delta$$k\in \mathbb{R}$$ H\otimes\Phi$

$S$

$S$

$\langle y_{1}, x_{2}-x_{1}\rangle+\langle y_{2}, x_{3}-x_{2}\rangle+\cdots+\langle y_{N-1}, x_{N}-x_{N-1}\rangle+\langle y_{N}, x_{1}-x_{N}\rangle\leq 0$

denotes the inner product of K.
wherecyclically monotone if and only if

It turns out that$\langle\cdot, \cdot\rangle$$\sum_{i=1}^{N}(y_{i}, x_{\sigma(i)}-x_{i})\leq 0$

$S$

,

is

,

:
for any permutation of $\{1, \ldots, N\}$ and for any finite subset {$i=1$ ,
, $N\}$ of . Note that is cyclically monotone if and only if any
translate of it is cyclically monotone. By a theorem of Rockafellar, any
cyclically monotone set is contained in the graph of the subdiﬀerentialof a convex function in the sense of convex analysis ([22]) and even if
the function may not be unique its subdiﬀerential is unique.
Let now $(W, \mu, H)$ be an abstract Wiener space; a measurable function:
is called 1-convex if the map$(x_{i}, y_{i})$$\sigma$$S$$S$$\ldots$$f$$W\rightarrow 1R\cup\{\infty\}$$h\rightarrow f(x+h)+\frac{1}{2}|h|_{H}^{2}=F(x, h)$is convex on the Cameron-Martin space $H$ with values in. Notethat this notion is compatible with the -equivalence classes of randomvariables thanks to the Cameron-Martin theorem. It is proven in [11]that this definition is equivalent the following condition: Let $(\pi_{n}, n \geq 1)$
be a sequence of regular, finite dimensional, orthogonal projections of$H$ , increasing to the identity map. Denote also by
its continuous
extension to $W$ and define. For $x\in W$ , let $x_{n}=\pi_{n}x$ and. Then is 1-convex if and only if$L^{0}(\mu)$$\mu$$I_{H}$$\pi_{n}^{\perp}=I_{W}-\pi_{n}$$x_{n}^{\perp}=\pi_{n}^{\perp}x$$f$$x_{n}\rightarrow\frac{1}{2}|x_{n}|_{H}^{2}+f(x_{n}+x_{n}^{\perp})$$\pi_{n}$Monge-Amp\‘ere equation and the It\^o calculusis$\pi_{n}^{\perp}\mu$

\S 3.

55

-almost surely convex.

Some Inequalities

be two probabilities on $(W, B(W))$ .
andDefinition 3.1. Let
We say that a probability on $(W\times W, B(W\times W))$ is a solution of theMonge-Kantorovitch problem associated to the couple
if the fifirstmarginal of is , the second one is and if$\xi$$\eta$$\gamma$$(\xi, \eta)$$\xi$$\eta$$\gamma$$J(\gamma)=\int_{W\times W}|x-y|_{H}^{2}d\gamma(x, y)$$=\inf\{\int_{W\times W}|x-y|_{H}^{2}d\beta(x, y)$:$\beta\in\Sigma(\xi, \eta)\}$

,

where
denotes the set of all the probability measures on $W\times W$
and . We shallwhose fifirst and second marginals are respectively
denote the Wasserstein distance between and , which is the positive.
square-root of this infifimum, with$\Sigma(\xi, \eta)$

$\xi$

$\xi$

$\eta$$\eta$$d_{H}(\xi, \eta)$

Remark: By the weak compacteness of probability measures on$W\times W$ and the lower semi-continuity of the strictly convex cost function, the infimum in the definition is attained even if the functional isidentically infinity.
The following result, whose proof is outlined below (cf. also[12, 13]) is anextension of an inequality due to Talagrand [24] and it gives a suﬃcientcondition for the Wasserstein distance to be finite:$J$

Theorem 3.1. Let $L\in L\log L(\mu)$ be a positive random variablewith $E[L]=1$ and let be the measure $ d\nu=Ld\mu$ . We then have
$\nu$

(3.3)

$d_{H}^{2}(\nu, \mu)\leq 2E[L\log L]$.

Proof: Let us remark first that we can take $W$ as the classicalWiener space $W=C_{0}([0,1])$ and, using the stopping techniques of themartingale theory, we may assume that is upper and lower boundedalmost surely. Then a classical result of the It\^o calculus implies thatcan be represented as an exponential martingale
$L$

$L$

$L_{t}=\exp\{-\int_{0}^{t}\dot{u}_{\tau}dW_{\tau}-\frac{1}{2}\int_{0}^{t}|\dot{u}_{\tau}|^{2}d\tau\}$,with $L=L_{1}$ . Let us define: $W\rightarrow H$ as
and$U$ : $W\rightarrow W$ as
implies.
The
theoremthat
Girsanov$U(x)=x+u(x)$$u$$ u(t, x)=\int_{0}^{t}\dot{u}_{\tau}d\tau$D. Feyel and A. S.

56$\gamma$st\"unel

is a Browian motion under , hence $\beta=(U\times I)\nu\in\Sigma(\mu, \nu)$ .
be any optimal measure, then$x\rightarrow U(x)$Let$\dot{U}$$\nu$$J(\gamma)$$=$$d_{H}^{2}(\nu, \mu)\leq\int_{W\times W}|x-y|_{H}^{2}d\beta(x, y)$$=$$E[|u|_{H}^{2}L]$$=$$2E[L\log L]$ ,where the last equality follows also ﬀom the Girsanov theorem and the
It\^o stochastic calculus.

$\square $Combining Theorem 3.1 with the triangle inequality for the Wasserstein distance gives:
Corollary 3.1. Assume that $\nu_{i}(i=1,2)$ have Radon-Nikodym denwhich are insities $L_{i}(i=1,2)$ with respect to the Wiener measure$L\log$ L. Then.
$\mu$

$ d_{H}(\nu_{1}, \nu_{2})<\infty$

Let us give a simple application of the above result in the lines of [17]:Corollary 3.2. Assume that $A\in B(W)$ is any set of positive
Wiener measure. Defifine the $H$ -gauge function of $A$ as$q_{A}(x)=\inf(|h|_{H} :h\in(A-x)\cap H)$ .

Then we have$E[q_{A}^{2}]\leq 2\log\frac{1}{\mu(A)}$,in other words$\mu(A)\leq\exp\{-\frac{E[q_{A}^{2}]}{2}\}$, if
Similarly if $A$ and $B$ are $H$ -separated,, where $A_{\in}=\{x\in W : q_{A}(x)\leq\in\}$ , then$i.e.$.
$ A_{\Xi}\cap B=\emptyset$, for some$\epsilon>0$$\mu(A_{\in}^{c})\leq\frac{1}{\mu(A)}e^{-\epsilon^{2}/4}$

and consequently$\mu(A)\mu(B)\leq\exp(-\frac{5^{2}}{4})$isRemark 3.1. We already know that, from the 0–1-law,$|q_{A}(x+h)-q_{A}(x)|\leq|h|_{H}$, hencealmost surely fifinite, besides it satisfifies$ E[\exp\lambda q_{A}2]<\infty$$B.8$$\lambda<1/2$(cf. the Appendix
of [29]). In fact
for any
all these assertions can also be proved with the technique used below.
$q_{A}$

57

Monge-Amp\‘ere equation and the It\^o calculus

Proof: Let$\nu_{A}$be the measure defined by$ d\nu_{A}=\frac{1}{\mu(A)}1_{A}d\mu$.

Letbe the solution of the Monge-Kantorovitch problem, it is easy to
see that the support ofis included in $W\times A$ , hence$\gamma_{A}$$\gamma_{A}$$|x-y|_{H}\geq\inf\{|x-z|_{H} :z\in A\}=q_{A}(x)$ ,-almost surely. This implies in particular that
finite. It follows now from the inequality (3.3)$q_{A}$$\gamma_{A}$$E[q_{A}^{2}]\leq-2\log\mu(A)$is almost surely,

hence the proof of the first inequality follows. For the second let $B=$
and letbe the solution of the Monge-Kantorovitch problem. Then we have ﬀom the Corollary 3.1,,
corresponding to$A_{\Xi}^{c}$$\gamma AB$$\nu_{A}$$\nu_{B}$$d_{H}^{2}(\nu_{A}, \nu_{B})\leq-4\log\mu(A)\mu(B)$.

is in $A\times B$ , hence
Besides the support of the measuresurely $|x-y|_{H}\geq\epsilon$ and the proof follows.
$\gamma AB$\S 4.

$\gamma AB$

almost$\square $Construction of the transport map

In this section we call optimal every probability $measure^{4)}\gamma$ on$W\times W$ such that $ J(\gamma)<\infty$ and that $J(\gamma)\leq J(\theta)$ for every otherprobability having the same marginals as those of . We recall that a
finite dimensional subspace $F$ of $W$ is called regular if the corresponding
projection is continuous. Similarly a finite dimensional projection of $H$
is called regular if it has a continuous extension to $W$ .
The proof of the next theorem, for which we refer the reader to [13],can be done by choosing a proper disintegration of any optimal measurein such a way that the elements of this disintegration are the solutions offinite dimensional Monge-Kantorovitch problems. The latter is provenwith the help of the section-selection theorem [6].
$\theta$$\gamma$

Theorem 4.1 (General case). Suppose thatability measures on $W$ such that
$ d_{H}(\rho, \nu)<\infty$

4)$\rho$

and$\nu$

art two prob-.

In fact the results of this section are essentially true for the bounded,
positive measures.

D. Feyel and A. S.

58$\dot{U}$st\"unel

Let $(\pi_{n}, n\geq 1)$ be a total increasing sequence of regular projections (of$H$ , converging to the identity map,of $H$). Suppose that, for any
vanish-almostthe regular conditional probabilities
with Hausdorﬀ dimension $n-1$ .
surely on the subsets ofThen there exists a unique solution of the Monge-Kantorovitch problem,
and is supported by the graph of a Borel map
denoted by$T$
which is the solution of the Monge problem. $T$ : $W\rightarrow W$ is of theform $ T=I_{W}+\xi$ , where $\xi\in H$ almost surely. Besides we have$n$$\rho(\cdot|\pi_{n}^{\perp}=x^{\perp})$$\geq 1$$\pi_{n}^{\perp}\rho$$(\pi_{n}^{\perp})^{-1}(W)$$\gamma\in\Sigma(\rho, \nu)$$d_{H}^{2}(\rho, \nu)$$\gamma$$=$$\int_{W\times W}|T(x)-x|_{H}^{2}d\gamma(x, y)$$=$$\int_{W}|T(x)-x|_{H}^{2}d\rho(x)$,, the mapand for-almost almost all, in the sense thatmonotone on$\pi_{n}^{\perp}\rho$$u\rightarrow\xi(u+x_{n}^{\perp})$$x_{n}^{\perp}$

is cyclically$(\pi_{n}^{\perp})^{-1}\{x_{n}^{\perp}\}$$\sum_{i=1}^{N}$($u_{i}+\xi(x_{n}^{\perp}+u_{i})$, $u_{i+1}-u_{i})_{H}\leq 0$-almost surely, for any cyclic sequence $\{u_{1}, \ldots, u_{N}, u_{N+1}=u_{1}\}$-almost surely,
from $\pi_{n}(W)$ . Finally, if, for any $n\geq 1$ ,,also vanishes on the $n-1$ -Hausdorﬀ dimensional subsets of$W\rightarrow W$$T$then is invertible, , there exists :of the form $ S=I_{W}+\eta$$\eta\inH$such thatsatisfifies a similar cyclic monotononicity property as
and that$\pi_{n}^{\perp}\rho$$\nu(\cdot|\pi_{n}^{\perp}=$$\pi_{n}^{\perp}\nu$$(\pi_{n}^{\perp})^{-1}(W)$$y^{\perp})$

$S$

$i.e$

$\xi$

1

$=$

$\gamma\{(x, y)\in W\times W : ToS(y)=y\}$$=$$\gamma\{(x, y)\in W\times W : SoT(x)=x\}$.

In particular we have$d_{H}^{2}(\rho, \nu)$$=$$\int_{W\times W}|S(y)-y|_{H}^{2}d\gamma(x, y)$$=$$\int_{W}|S(y)-y|_{H}^{2}d\nu(y)$.

Remark 4.1. In particular, for all the measures which $aoe$ absolutely continuous with respect to the Wiener measure , the second, the measure
hypothesis is satisfified,vanishes on thesets of Hausdorﬀ dimension $n-1$ .
$\rho$$\mu$$i.e.$$\rho(\cdot|\pi_{n}^{\perp}=x_{n}^{\perp})$

The case where one of the measures is the Wiener measure and theother is absolutely continuous with respect to is the most important
$\mu$

59

Monge-Amp\‘ere equation and the It\^o calculus

one for the applications. Consequently we give the related results separately in the following theorem where the tools of the Malliavin calculus
give more information about the maps and of Theorem 4.1:$\xi$$\eta$

Theorem 4.2 (Gaussian case). Let be the measure $ d\nu=Ld\mu$ ,is a positive random variable, with $E[L]=1$ . Assume thatwhere
$ d_{H}(\mu, \nu)<\infty$ (for instance $L\in L\log L$ ).
Then there exists 1-convex
function $\phi\in D_{2,1}$ , unique upto a constant, such that the map $T=$is the unique solution of the original problem of Monge. Moreover, its graph supports the unique solution of the Monge-Kantorovitch
problem . Consequently
$\nu$

$L$

$a$

$ I_{W}+\nabla\phi$$\gamma$$(I_{W}\times T)\mu=\gamma$

In particular $T$ maps to and $T$ is almost surely invertible,
such that $ T^{-1}\nu=\mu$ and thatexists some
$\mu$

$\nu$

$i.e.$, there$T^{-1}$

1$=$$\mu\{x:T^{-1}\circ T(x)=x\}$$=$$\nu\{y\in W : ToT^{-1}(y)=y\}$ .

is closable with respect
Remark 4.2. Assume that the operator
are equivalent,. In particular, if andto , then we havethen we have$\nabla$$\eta=\nabla\psi$

$\nu$

$\nu$

$ T^{-1}=I_{W}+\nabla\psi$

where is$\psi$

is

$a$

$\mu$

,

1-convex function.

Remark 4.3. Let $(e_{n}, n\in N)$ be a complete, orthonormal in $H$ ,denote by
the sigma algebra generated byand let $L_{n}=$$E[L|V_{n}]$ .
If $\phi_{n}\in D_{2,1}$ is the function constructed in Theorem ,, then, using the inequality (3.3) we can prove that
corresponding to$(\phi_{n},n\inN)$ converges toin $D_{2,1}$ .
the sequence$\{\delta e_{1}, \ldots, \delta e_{n}\}$$V_{n}$$4\cdot 2$$L_{n}$$\phi$

Remark 4.4. Assume that $L\in L_{+}^{1}(\mu)$ , with $E[L]=1$ and let$(D_{k}, k\in N)$ be a measurable partition of $W$ such that on eachis,
. It follows from Theorem
bounded. Defifine $ d\nu=Ld\mu$ and$d_{H}(\mu,\nu_{k})<\infty$
3.1, thatbe the map constructed in Theorem. Let then$T_{k}\mu=\nu_{k}$$n(dk)$
4-2 satisfying. Defifine
as the probability distribution on$N$ given by $n(\{k\})=\nu(D_{k})$ , $k\in N$ . Then we have$D_{k}$$L$$\nu_{k}=\nu(\cdot|D_{k})$$T_{k}$$\int_{W}f(y)d\nu(y)=\int_{W\times N}f(T_{k}(x))\mu(dx)n(dk)$ .

A similar result is given in [9], the diﬀerence with that of above lies in
the fact that we have a more precise information about the probability
space on which $T$ is defifined.

D. Feyel and A. S.

60$\dot{U}$st\"unelLet us give some applications of the above theorem to the factorization of the absolutely continuous transformations of the Wiener measure.
Assume that $V=I_{W}+v$ : $W\rightarrow W$ be an absolutely continuoustransformation and let $L\in L_{+}^{1}(\mu)$ be the Radon-Nikodym derivative ofwith respect to . Let $ T=I_{W}+\nabla\phi$ be the transport map suchthat $ T\mu=L.\mu$ . Then it is easy to see that the map $s=T^{-1}oV$ is arotation, i.e., $ s\mu=\mu$ (cf. [29]) and it can be represented as $ s=I_{W}+\alpha$ .
In particular we have
$ V\mu$

$\mu$

(4.4)

$\alpha+\nabla\phi\circ s=v$.

is almost surely
Since is a 1-convex map, we have:.
be another rotation withconvex (cf. [11]). Let,
By the 1-convexity of we have$h\rightarrow\frac{1}{2}|h|_{H}^{2}+\phi(x+h)$$\phi$$s^{/}=I_{W}+\alpha^{/}$$\alpha^{J}$$W\rightarrow H$$\phi$$\frac{1}{2}|\alpha’|_{H}^{2}+\phi\circ s^{J}\geq\frac{1}{2}|\alpha|_{H}^{2}+\phi\circ s+$($\alpha+\nabla\phi\circ s$,$\alpha^{/}-\alpha$) ,-almost surely. Taking the expectation of both sides, using the factthat and preserve the Wiener measure and the identity (4.4), we
obtain$\mu$$s$$s^{J}$$\mu$$E[\frac{1}{2}|\alpha|_{H}^{2}-(v, \alpha)_{H}]\leq E[\frac{1}{2}|\alpha’|_{H}^{2}-(v, \alpha^{/})_{H}]$

Hence we have proven the existence part of the following
Proposition 4.1. Let
denote the subset of $L^{2}(\mu, H)$ whose ele,ments are defifined by the property that $x\rightarrow x+\eta(x)$ is a rotation,is the unique element of
it preserves the Wiener measure. Then
which minimizes the functional$\mathcal{R}_{2}$$i.e.$$\mathcal{R}_{2}$$\alpha$$\eta\rightarrow M_{v}(\eta)=E[\frac{1}{2}|\eta|_{H}^{2}-(v, \eta)_{H}]$Proof: The only claim to prove is the uniqueness and it follows
easily from Theorem 4.2.

$\square $The following theorem, whose proof is rather easy, gives a better
understanding of the structure of absolutely continuous transformationsof the Wiener measure:

Theorem 4.3. Assume that $U$ : $W\rightarrow W$ be a measurable map and$L\in L\log L$ a positive random variable with $E[L]=1$ . Assume that the, that one has
measure $\nu=L$ . is a Girsanov measure for $U$ ,$i.e.$$\mu$$E[foUL]=E[f]$ ,Monge-Amp\‘ere equation and the It\^o calculus

61

Then there exists a unique map $ T=I_{W}+\nabla\phi$with $\phi\in D_{2,1}$ is -convex, and a measure preserving transformation$R$ : $W\rightarrow W$ such that $ U\circ T=R\mu$ -almost surely and $U=R\circ T^{-1}$-almost surely.

for any $f\in C_{b}(W)$ .
$l$

$\nu$

Another version of Theorem 4.3 can be announced as follows:
Theorem 4.4. Assume that $Z$ : $W\rightarrow W$ is a measurable map suchthat $ Z\mu\ll\mu$ , with $ d_{H}(Z\mu, \mu)<\infty$ . Then $Z$ can be decomposed as$Z=T\circ s$ ,where $T$ is the unique transport map of the Monge-Kantorovitch problem
and is a rotation.
for$\Sigma(\mu, Z\mu)$$s$

Although the following result is a translation of the results of thissection, it is interesting ﬀom the point of view of stochastic diﬀerential
equations:,
Theorem 4.5. Let $(W, \mu, H)$ be the standard Wiener space on, $W=C(1R_{+}, R^{d})$ . Assume that there exists a probability $ P<<\mu$which is the weak solution of the stochastic diﬀerential equation$R^{d}$$i.e.$$dy_{t}=dW_{t}+b(t, y)dt$ ,such that $ d_{H}(P, \mu)<\infty$ . Then there exists a process $(T_{t}, t\in 1R_{+})$
which is a pathwise solution of some (anticipative) stochastic diﬀerential
equation whose law is equal to $P$ .
Proof: Let $T$ be the transport map constructed in Theorem 4.2
such that
corresponding to $ dP/d\mu$ . Then it has an inverse$T(x)=x\}=1$ . Let be the 1-convex function such that $ T=I_{W}+\nabla\phi$
in $L^{2}(R_{+}, ds)$ .
and denote by $(D_{s}\phi, s\in R_{+})$ the representation ofDefine $T_{t}(x)$ as the trajectory $T(x)$ evaluated at $t\in 1R_{+}$ . Then it is easy
to see that $(T_{t}, t\in R_{+})$ satisfies the stochastic diﬀerential equation$T^{-1}$$\mu\{T^{-1}\circ$$\phi$$\nabla\phi$$T_{t}(x)=W_{t}(x)+\int_{0}^{t}l(s, T(x))ds$, $t\in 1R_{+}$ ,where $W_{t}(x)=x(t)$ and $l(s, x)=D_{s}\phi\circ T^{-1}(x)$ if $x\in T(W)$ and zerootherwise.

$\square $62

D. Feyel and A. S.

\S 5.

The Monge-Amp\‘ere equation$\dot{U}$st\"unel

Assume that $W=R^{n}$ and take a density $ L\in L\log$ L. Let $\phi\in D_{2,1}$
be the 1-convex function such that $ T=I+\nabla\phi$ maps to . . Let$ S=I+\nabla\psi$ be its inverse with $\psi\in D_{2,1}$ . Let now
be the second
Alexandrov derivative of , i.e., the Radon-Nikodym derivative of theabsolutely continuous part of the vector measurewith respect to
the Gaussian measureon ] . Since is 1-convex, it follows thatin the sense of the distributions, consequently$-I_{R^{n}}\mu$ -almost surely. Define also the Alexandrov versionofas
the Radon-Nikodym derivative of the absolutely continuous part of the. Since we are in finite dimensional situation, we have
distributionasthe explicit expression for
$L$

$\mu$

$\mu$

$\nabla_{a}^{2}\phi$$\phi$$\nabla^{2}\phi$$R^{n}$$\phi$$\mu$$\nabla^{2}\phi\geq-I_{R^{n}}$$\nabla_{a}^{2}\phi\geq$$\mathcal{L}_{a}\phi$$\mathcal{L}\phi$$\mathcal{L}\phi$$\mathcal{L}_{a}\phi$$\mathcal{L}_{a}\phi(x)=(\nabla\phi(x), x)_{R^{n}}-trace(\nabla_{a}^{2}\phi)$Let$\Lambda$

be the Gaussian Jacobian$\Lambda=\det_{2}(I_{R^{n}}+\nabla_{a}^{2}\phi)\exp\{-\mathcal{L}_{a}\phi-\frac{1}{2}|\nabla\phi|_{R^{n}}^{2}\}$.

Remark 5.1. In this expression as well as in the sequel, the notation
denotes the modified Carleman-Fredholm determinant of
the operator $I_{H}+A$ on a Hilbert space $H$ . If $A$ is an operator of finite
rank, then it is defined as$\det_{2}(I_{H}+A)$$\det_{2}(I_{H}+A)=\prod_{i=1}^{n}(1+l_{i})e^{-l_{i}}$,where $(l_{i}, i\leq n)$ denotes the eigenvalues of $A$ counted with respect totheir multiplicity. In fact this determinant has an analytic extension to
the space of Hilbert-Schmidt operators on a separable Hilbert space, cf.
[7] and Appendix A.2 of [29]. As explained in [29], the modified determinant exists for the Hilbert-Schmidt operators while the ordinary determinant does not, since the latter requires the existence of the $trace$ of$A$ .
Hence the modified Carleman-Fredholm determinant is particularly
useful when one studies the absolute continuity properties of the imageof a Gaussian measure under non-linear transformations in the settingof infinite dimensional Banach spaces (cf., [29] for further information).
It follows from the change of variables formula given in Corollary
4.3 of [19], that, for any $f\in C_{b}(R^{n})$ ,$E[f\circ T\Lambda]=E[f1_{\partial\Phi(M)}]$,63

Monge-Amp\‘ere equation and the It\^o calculus

where $M$ is the set of non-degeneracy of$ I_{1R^{n}}+\nabla_{a}^{2}\phi$,$\Phi(x)=\frac{1}{2}|x|^{2}+\phi(x)$anddenotes the subdiﬀerential of the convex function . Let us
note that, in case $L>0$ almost surely, $T$ has a global inverse , i.e.,$ S\circ T=T\circ S=I_{1R^{n}}\mu$ -almost surely and $\mu(\partial\Phi(M))=\mu(S^{-1}(M))$ .
Assume now that $\Lambda>0$ almost surely, i.e., that $\mu(M)=1$ . Then, for
any $f\in C_{b}(1R^{n})$ , we have$\Phi$$\partial\Phi$$S$$E[f\circ T]$$=$$E[f\circ T\frac{\Lambda}{\Lambda oT^{-1}\circ T}]$$=$$E[f\frac{1}{\Lambda\circ T^{-1}}1_{\partial\Phi(M)}]$$=$$E[fL]$ ,wheredenotes the left inverse of $T$ whose existence is guaranteedby Theorem 4.2. Since $T(x)\in\partial\Phi(M)$ almost surely, it follows from theabove calculations,$T^{-1}$$\frac{1}{\Lambda}=L\circ T$

almost surely. Take now any $t\in[0,1)$ , the map
is strictly convex and a simple calculation implies that the map$
T_{t}=I+t\nabla\phi$
is $(1-t)$ -monotone (cf. [29], Chapter 6), consequently
ping
the Legendre
it has a left inverse denoted by . Let us denote bytransformation of :$x\rightarrow\frac{1}{2}|x|_{H}^{2}+t\phi(x)=$$\Phi_{t}(x)$$\Psi_{t}$$S_{t}$$\Phi_{t}$$\Psi_{t}(y)=\sup_{x\in R^{n}}\{(x, y)-\Phi_{t}(x)\}$.

A simple calculation shows that$\Psi_{t}(y)$$=$$\sup_{x}[(1-t)\{(x, y)-\frac{|x|^{2}}{2}\}+t\{(x, y)-\frac{|x|^{2}}{2}-\phi(x)\}]$$\leq$$(1-t)\frac{|y|^{2}}{2}+t\Psi_{1}(y)$

.

Since
is the Legendre transformation of $\Phi_{1}(x)=|x|^{2}/2+\phi(x)$ andsince $L\in L\log L$ , it is finite on a convex set of full measure, hence it
is finite everywhere. Consequently $\Psi_{t}(y)<\infty$ for any $y\in R^{n}$ . Sincea finite, convex function is almost everywhere diﬀerentiable,
existseverywhereequal
isalmoston
totheeverywhere
almoston and it.
, whereleﬃ inverseis the set of non-degeneracy of$\Psi_{1}$$\nabla\Psi_{t}$$T_{t}(M_{t})$$T_{t}^{-1}$$M_{t}$$ I_{R^{n}}+t\nabla_{a}^{2}\phi$D. Feyel and A. S.

64$\dot{U}$st\"unelis Lipschitz
Note that $\mu(M_{t})=1$ . The strict convexity implies thatbe the Gaussian Jacobian. Let nowwith a Lipschitz constant$T_{t}^{-1}$$\Lambda_{t}$$\frac{1}{1-t}$.

$\Lambda_{t}=\det_{2}(I_{R^{n}}+t\nabla_{a}^{2}\phi)\exp\{-t\mathcal{L}_{a}\phi-\frac{t^{2}}{2}|\nabla\phi|_{R^{n}}^{2}\}$,
almost surely,
Since the domain of is the whole spacehence, as we have explained above, it follows ﬀom the change of variables
is absolutely continuous with respect to andformula of [19] that
that,
$R^{n}$$\phi$$\Lambda_{t}>0$$ T_{t}\mu$$\mu$$\frac{1}{\Lambda_{t}}=L_{t}\circ T_{t}$-almost surely.
Let us come back to the infinite dimensional case: we first give an inequality which may be useful.
$\mu$

Theorem 5.1. Assume that $(W, \mu, H)$ is an abstract Wiener space,assume that $K$ , $L\in L_{+}^{1}(\mu)$ with $K>0$ almost surely and denote by$T$ : $W\rightarrow W$ thetransfer map $ T=I_{W}+\nabla\phi$ , which maps the measure$ Kd\mu$ to the measure $ Ld\mu$ .
Then the following inequality holds:(5.5)$\frac{1}{2}E[|\nabla\phi|_{H}^{2}]\leq E[-\log K+\log LoT]$

Proof: Let us define
we have$k$$\int_{W}f(y)L(y)d\mu(y)$.

as $k=K\circ T^{-1}$ , then for any $f\in C_{b}(W)$ ,$=$$\int_{W}f\circ T(x)K(x)d\mu(x)$$=$$\int_{W}f\circ T(x)k\circ T(x)d\mu(x)$

,

hence
$ T\mu=\frac{L}{k}.\mu$.

It then follows from the inequality (3.3) that$\frac{1}{2}E[|\nabla\phi|_{H}^{2}]$$\leq$$E[\frac{L}{k}\log\frac{L}{k}]$$=$$E[\log\frac{L\circ T}{k\circ T}]$$=$$E[-\log K+\log L\circ T]$ .
$\square $Monge-Amp\‘ere equation and the It\^o calculus

65

we have another inequalandIn case $K$ andare given as(cf.
[21] for the finite dimensional
ity containing the Fisher informationcase):$e^{-V}$$e^{-U}$$L$

Theorem 5.2. Assume that $U$, $V\in D_{2,1}$ are such that $E[e^{-U}]=$$E[e^{-V}]=1$ and that $ E[e^{-U}|\nabla U|_{H}^{2}]+E[e^{-V}|\nabla V|_{H}^{2}]<\infty$ . Defifine $d\rho=$
and $ d\nu=e^{-V}d\mu$ . Then we have$ e^{-U}d\mu$(5.6)$d_{H}(\rho, \nu)\leq E_{\rho}[|\nabla U|_{H}^{2}]^{1/2}+E_{\nu}[|\nabla V|_{H}^{2}]^{1/2}$

The equation (5.6) can be refifined in the following way: let$\hslash\pm(U, V)=E_{\rho}[|\nabla U|_{H}^{2}]^{1/2}\pm\{E_{\rho}[|\nabla U|_{H}^{2}]+2(E_{\rho}(U)-E_{\nu}(V))\}^{1/2}$

and let$\kappa\pm(V, U)=E_{\nu}[|\nabla V|_{H}^{2}]^{1/2}\pm\{E_{\nu}[|\nabla V|_{H}^{2}]+2(E_{\nu}(V)-E_{\rho}(U))\}^{1/2}$

We then have
(5.7)$\max\{\kappa_{-}(U, V), \kappa_{-}(V, U)\}\leq d_{H}(\rho, \nu)\leq\min\{\kappa_{+}(U, V), \kappa_{+}(V, U)\}$.

Proof: By taking the conditional expectations with respect to thesigma algebras generated by a finite number of elements of the first
Wiener chaos and using the Jensen inequality, we can reduce the problem
to the finite dimensional case. Let now $ T=I+\nabla\phi$ be the transport
map sending to , let also $ S=I+\nabla\psi$ be its inverse. As we have seen
before, we can write $\Lambda=\exp(-U+V\circ T)$ as$\rho$$\nu$$\Lambda=\det_{2}(I+\nabla_{a}^{2}\phi)\exp\{-\mathcal{L}_{a}\phi-\frac{1}{2}|\nabla\phi|_{H}^{2}\}$

ﬀom this expression and using the fact that
Solvingin the sense of the distributions, we obtain$|\nabla\phi|_{H}^{2}$$\frac{1}{2}|\nabla\phi|_{H}^{2}\leq U-VoT-\mathcal{L}\phi$.
$-\mathcal{L}_{a}\phi\leq-\mathcal{L}\phi$.

Taking the expectation of both sides of this inequality with respect to, we obtain$\rho$(5.8)$\frac{1}{2}d_{H}^{2}(\rho, \nu)\leq E_{\rho}[U]-E_{\nu}[V]+E_{\rho}[(\nabla\phi, \nabla U)]$Interchanging $T$ and
also
(5.9)

$S$

and

$\rho$

and

$\nu$

.

in the inequality (5.8), we obtain$\frac{1}{2}d_{H}^{2}(\rho, \nu)\leq E_{\nu}[V]-E_{\rho}[U]+E_{\nu}[(\nabla\psi, \nabla V)]$.

D. Feyel and A. S.

66$\dot{U}$st\"unel

Adding (5.8) to (5.9) and using the Cauchy-Schwarz inequality completes
the proof of the first part. For the second, using the inequalities (5.8) and(5.9), with the help of the Cauchy-Schwarz inequality and the general
expression of solutions of the second order polynomial equation wethe claim at once.

$get\square $Suppose that $\phi\in D_{2,1}$ is a 1-convex Wiener functional. Let
be, as$(e_{n},n\geq 1)$usual, the sigma algebra generated by, whereis an orthonormal basis of the Cameron-Martin space $H$ . Then
$E[\phi|V_{n}]$ is again 1-convex (cf. [11]), hence
is a measure as it can beconverges toeasily verified. However the sequence
only. Consequently, there is no reason for the limitinto be a measure.
In case this happens, we shall denote the Radon-Nikodym density withrespect to , of the absolutely continuous part of this measure by.
$V_{n}$$\{\delta e_{1}, \ldots, \delta e_{n}\}$$\phi_{n}=$$\mathcal{L}\phi_{n}$$(\mathcal{L}\phi_{n}, n \geq 1)$$D^{/}$$\mathcal{L}\phi$$\mathcal{L}\phi$$\mathcal{L}_{a}\phi$$\mu$Lemma 5.1. Let $\phi\in D_{2,1}$ be 1-convex and let
be defifined asabove and defifine $F_{n}=E[\phi|V_{n}]$ . Then the sequenceissubmartingale, where
denotes the absolutely continuous part of.
the measure$V_{n}$$(\mathcal{L}_{a}F_{n}, n \geq 1)$$a$$\mathcal{L}_{a}F_{n}$$\mu$$\mathcal{L}F_{n}$Proof: Note that, due to the 1-convexity, we have
any $n\in 1N$ . Let
and $f\in D$ be a positive,, we havetest function. Sincefor-measurable$\mathcal{L}_{a}F_{n}\geq \mathcal{L}F_{n}$$X_{n}=\mathcal{L}_{a}F_{n}$$V_{n}$$\mathcal{L}E[\phi|V_{n}]=E[\mathcal{L}\phi|V_{n}]$$E[X_{n+1}f]$$\geq$$\langle \mathcal{L}F_{n+1}, f\rangle$$=$$\langle \mathcal{L}F_{n}, f\rangle$

,

where
denotes the duality bracket for the dual pair $(D^{l}, D)$ . Consequently,$\langle\cdot, \cdot\rangle$$ E[fE[X_{n+1}|V_{n}]]\geq\langle \mathcal{L}F_{n}, f\rangle$

for any positive,-measurable test function , it follows that the absolutely continuous part ofis also dominated by the same conditional
expectation and this proves the submartingale property.
$V_{n}$$f$$\mathcal{L}F_{n}$$\square $Lemma 5.2. Assume that $L\in L\log L$ is a positive random variable
whose expectation is one. Assume further that it is lower bounded byconstant $a>0$ . Let $ T=I_{W}+\nabla\phi$ be the transport map such that$ T\mu=L.\mu$ and let $ T^{-1}=I_{W}+\nabla\psi$ . Then
is a Radon measure on$(W, B(W))$ . If
is upper bounded by $b>0$ , thenis also a Radon
measure on $(W, B(W))$ .
$a$$\mathcal{L}\psi$$L$$\mathcal{L}\phi$

Proof: Let $L_{n}=E[L|V_{n}]$ , then $L_{n}\geq a$ almost surely. Let $T_{n}=$be the transport map which satisfies $ T_{n}\mu=L_{n}.\mu$ and let$I_{W}+\nabla\phi_{n}$67Monge-Amp\‘ere equation and the It\^o calculus$T_{n}^{-1}=I_{W}+\nabla\psi_{n}$be its inverse. We have$L_{n}=\det_{2}(I_{H}+\nabla_{a}^{2}\psi_{n})\exp[-\mathcal{L}_{a}\psi_{n}-\frac{1}{2}|\nabla\psi_{n}|_{H}^{2}]$is 1-convex, it follows
By the hypothesis $-\log L_{n}\leq-\log a$ . Since
ﬀom the finite dimensional results that $\det_{2}(I_{H}+\nabla_{a}^{2}\psi_{n})\in[0,1]$ almost
surely. Therefore we have$\psi_{n}$$\mathcal{L}_{a}\psi_{n}\leq-\log a$besides$\mathcal{L}\psi_{n}\leq \mathcal{L}_{a}\psi_{n}$,as distributions, consequently$\mathcal{L}\psi_{n}\leq-\log a$, we obtaininas distributions, for any $n\geq 1$ . Since, hence $-\log a-\mathcal{L}\psi\geq 0$ as a distribution, henceis a, [10], [28]. This proves the first claim. Note
Radon measure on $W$ ,that whenever is upperbounded, $\Lambda=1/L\circ T$ is lowerbounded, hence
the proof of the second claim is similar to that of the first one.
$\lim_{n}\mathcal{L}\psi_{n}=\mathcal{L}\psi$$D^{J}$$\mathcal{L}\psi\leq-\log a$$\mathcal{L}\psi$$c.f.$$L$$\square $Theorem 5.3. Assume that is a strictly positive bounded randomvariable with $E[L]=1$ . Let $\phi\in D_{2,1}$ be the 1-convex Wiener functionalsuch that$L$$ T=I_{W}+\nabla\phi$and let $ S=I_{W}+\nabla\psi$ beis the transport map realizing the measure
its inverse. Defifine $F_{n}=E[\phi|V_{n}]$ , then the submartingale
be the random variable defifined
converges almost surely to. Let
as$ L.\mu$$(\mathcal{L}_{a}F_{n}, n\geq 1)$$\mathcal{L}_{a}\phi$$\lambda(\phi)$$\lambda(\phi)$$=$$\lim\inf_{n\rightarrow\infty}\Lambda_{n}$$=$$(\lim\inf_{n}\det_{2}(I_{H}+\nabla_{a}^{2}F_{n}))\exp\{-\mathcal{L}_{a}\phi-\frac{1}{2}|\nabla\phi|_{H}^{2}\}$

where$\Lambda_{n}=\det_{2}(I_{H}+\nabla_{a}^{2}F_{n})\exp\{-\mathcal{L}_{a}F_{n}-\frac{1}{2}|\nabla F_{n}|_{H}^{2}\}$

Then it holds true that
(5.10)$E[f\circ T\lambda(\phi)]\leq E[f]$.

D. Feyel and A. S.

68

st\"unelalmost surely. If $E[\lambda(\phi)]=$, in particular, then the inequality in (5.10) becomes an equality and we also havefor any$1$$\dot{U}$$f\in C_{b}^{+}(W)$$\lambda(\phi)\leq\frac{1}{LoT}$$\lambda(\phi)=\frac{1}{L\circ T}$.

Proof: Let us remark that, due to the 1-convexity, $0\leq\det_{2}(I_{H}+$, hence the $\lim\inf$ exists. Now, Lemma 5.2 implies thatis a Radon measure. Let $F_{n}=E[\phi|V_{n}]$ , then we know from Lemma
denote the positive
5.1 thatis a submartingale. Let. Since, we have alsopart of the measure. Hence we find. This implies that
that$\nabla_{a}^{2}F_{n})\leq 1$$\mathcal{L}\phi$$\mathcal{L}^{+}\phi$$(\mathcal{L}_{a}F_{n}, n\geq 1)$$ E[\mathcal{L}^{+}\phi|V_{n}]\geq$$\mathcal{L}^{+}\phi\geq \mathcal{L}\phi$$\mathcal{L}\phi$$E[\mathcal{L}^{+}\phi|V_{n}]\geq \mathcal{L}_{a}^{+}F_{n}$$E[\mathcal{L}\phi|V_{n}]=\mathcal{L}F_{n}$$\sup_{n}E[\mathcal{L}_{a}^{+}F_{n}]<\infty$

conand this condition implies that the submartingale
verges almost surely. We shall now identify the limit of this submartingale. Let
for a Wiener
be the singular part of the measure$G$
is a measure. We havefunction such that$(\mathcal{L}_{a}F_{n}, n\geq 1)$$\mathcal{L}G$$\mathcal{L}_{s}G$$\mathcal{L}G$$E[\mathcal{L}\phi|V_{n}]$$=$$E[\mathcal{L}_{a}\phi|V_{n}]+E[\mathcal{L}_{s}\phi|V_{n}]$$=$$\mathcal{L}_{a}F_{n}+\mathcal{L}_{s}F_{n}$

,

hence
$\mathcal{L}_{a}F_{n}=E[\mathcal{L}_{a}\phi|V_{n}]+E[\mathcal{L}_{s}\phi|V_{n}]_{a}$almost surely, where
denotes the absolutely continuous part
. Note that, ﬀom the Theorem of Jessenof the measureand(cf., for example Theorem 1.2.1 of [29]),
almost surely, hence we have
$E[\mathcal{L}_{s}\phi|V_{n}]_{a}$$E[\mathcal{L}_{s}\phi|V_{n}]$$\lim_{n}E[\mathcal{L}_{s}^{+}\phi|V_{n}]_{a}=0$$\lim_{n}E[\mathcal{L}_{s}^{-}\phi|V_{n}]_{a}=0$$\lim_{n}\mathcal{L}_{a}F_{n}=\mathcal{L}_{a}\phi$,-almost surely. To complete the proof, an application of the Fatou
lemma implies that$\mu$

$E[f\circ T\lambda(\phi)]$$\leq$$E[f]$$=$$E[f\circ T\frac{1}{L\circ T}]$,

for any $f\in C_{b}^{+}(W)$ . Since $T$ is invertible, it follows that$\lambda(\phi)\leq\frac{1}{L\circ T}$

Monge-Amp\‘ere equation and the It\^o calculus

69

almost surely. Therefore, in case $E[\lambda(\phi)]=1$ , we have$\lambda(\phi)=\frac{1}{L\circ T}$,

and this completes the proof.

$\square $Corollary 5.1. Assume that $K$ ,are two positive random vari$[a,b]\subset(0, \infty)$ such that $E[K]=$ables with values in a bounded interval$E[L]=1$ . Let $ T=I_{W}+\nabla\phi$ , $\phi\in D_{2,1}$ , be the transport map pushing$ Kd\mu$ to $ Ld\mu$ ,, $ T(Kd\mu)=Ld\mu$ . We then have
$L$

$i.e$

$L\circ T\lambda(\phi)\leq K$,-almost surely. In particular, if $E[\lambda(\phi)]=1$ , then $T$ is the solution of
the Monge-Amp\‘ere equation.

$\mu$

Proof: Since $a>0$ ,$\frac{dT\mu}{d\mu}=\frac{L}{K\circ T}\leq\frac{b}{a}$.

Hence, Theorem 5.10 implies that$E[f\circ TL\circ T\lambda(\phi)]$$\leq$$E[fL]$$=$$E[f\circ TK]$ ,consequently$L\circ T\lambda(\phi)\leq K$,

the rest of the claim is now obvious.

$\square $For later use we give also the folowing result:
Theorem 5.4. Assume that is a positive random variable of class
Llog $L$ such that $E[L]=1$ . Let $\phi\in D_{2,1}$ be the 1-convex function
corresponding to the transport map $ T=I_{W}+\nabla\phi$ . Defifine $ T_{t}=I_{W}+t\nabla\phi$ ,is absolutely continuous
where $t\in[0,1]$ . Then, for any $t\in[0,1]$ ,with respect to the Wiener measure .
$L$$ T_{t}\mu$$\mu$

be defined as the transport map corresponding to$L_{n}=E[P_{1/n}L_{n}|V_{n}]$ and define. For $t\in[0,1)$ , letas$T_{n,t}=I_{W}+t\nabla\phi_{n}$ . It follows from the finite dimensional results which
is absolutely
are summarized in the beginning of this section, that
Proof: Let$\phi_{n}$$T_{n}$$I_{W}+\nabla\phi_{n}$$ T_{n,t}\mu$70

D. Feyel and A. S.

continuous with respect to . Let
as
Nikodym density and define$\mu$$L_{n,t}$$\dot{U}$st\"unelbe the corresponding Radon-$\Lambda_{n,t}$$\Lambda_{n,t}=\det_{2}(I_{H}+t\nabla_{a}^{2}\phi_{n})\exp\{-t\mathcal{L}_{a}\phi_{n}-\frac{t^{2}}{2}|\nabla\phi_{n}|_{H}^{2}\}$.

Besides, for any $t\in[0,1)$ ,(5.11)$((I_{H}+t\nabla_{a}^{2}\phi_{n})h, h)_{H}>0$,-almost surely for any $0\neq h\in H$ . Since
is of finite rank, (5.11) implies that $\Lambda_{n,t}>0\mu$ -almost surely and we have shown at the beginningof this section$\phi_{n}$$\mu$$\Lambda_{n,t}=\frac{1}{L_{n,t}oT_{n,t}}$-almost surely. An easy calculation shows thatcally concave. Consequently$\mu$$E[L_{t,n}\log L_{t,n}]$$t\rightarrow\Lambda_{n,t}$$=$$E[\log L_{n,t}\circ T_{n,t}]$$=$$-E[\log\Lambda_{t,n}]$$\leq$$E[L_{n}\log L_{n}]$$\leq$$E[L\log L]$ ,is logarithmi-by the Jensen inequality. Therefore$\sup_{n}E[L_{n,t}\log L_{n,t}]<\infty$

and this implies that the sequence $(L_{n,t}, n\geq 1)$ is uniformly integrable
for any $t\in[0,1]$ . Consequently it has a subsequence which converges
weakly into some . Since, ﬀom Theorem 4.2 and from Remark$\lim_{n}\phi_{n}=\phi$, where is the transport map associated to ,
4,
in
$f\in
C_{b}(W)$any, we havefor$L^{1}(\mu)$$L_{t}$$D_{2,1}$$E[f\circ T_{t}]$$L$$\phi$$=$$\lim_{k}E[f\circ T_{n_{k},t}]$$=$$\lim_{k}E[fL_{n_{k},t}]$$=$$E[fL_{t}]$,and this completes the proof.

Let us give an application of this result:$\square $71

Monge-Amp\‘ere equation and the It\^o calculus

Proposition 5.1. Assume that the hypothesis of Theorem 5.4 arevalid. Letwith $\nu_{1}=\nu$ , $t\in[0,1]$ . Then$\nu_{t}=T_{t}\mu$$d_{H}(\nu_{s}, \nu_{t})=|t-s|d_{H}(\mu, \nu)$In particularfunctional$T_{t}\circ S_{s}$for ,is the optimal transport ofis an exact form.

$\in[0,1]$

$t$

$s$

$\nu_{s}$to$\nu_{t}$.
and the Wiener$\nabla\psi_{s}+t\nabla\phi oS_{s}$Proof: It suﬃces to prove the claim for the case $s=1$ . Let, then$\nu_{t}=$$ T_{t}oS\nu$$d_{H}^{2}(\nu_{t}, \nu)$$\leq$$\int|T_{t}\circ S(y)-y|_{H}^{2}d\nu(y)$$=$$\int|T_{t}(x)-T(x)|_{H}^{2}d\mu(x)$$=$$(1-t)^{2}E[|\nabla\phi|_{H}^{2}]$,which means that$d_{H}(\nu_{t}, \nu)\leq(1-t)d_{H}(\mu, \nu)$.

Moreover, from the triangle inequality$d_{H}(\mu, \nu)$$\leq$$d_{H}(\nu, \nu_{t})+d_{H}(\nu_{t}, \mu)$$=$$d_{H}(\nu, \nu_{t})+td_{H}(\mu, \nu)$,

we obtain that $(1 -t)d_{H}(\mu, \nu)\leq d_{H}(l/, \nu_{t})$ . The rest is obvious ﬀom
Theorem 4.2.

$\square $\S 6.

Solution of the Monge-Amp\‘ere equation with the It\^o calculus

To have a better understanding of what will follow, let us give aninterpretation of the Monge-Amp\‘ere equation. Assume that we are given
two probability densities $K$ and and we look for a map $T$ : $W\rightarrow W$
such that$L$$L\circ TJ(T)=K$

almost surely, where $J(T)$ is a kind of Jacobian to be written in terms
whichof $T$ . In Corollary 5.1, we have shown the existence of somegives an inequality instead of the equality. The reason for that originatesfrom the singularity of the second derivative of the potential function .
Although in the finite dimensional case there are some regularity results
about the transport map (cf., [5]), in the infinite dimensional case such$\lambda(\phi)$$\phi$

D. Feyel and A. S.

72$\dot{U}$st\"uneltechniques do not work. All these diﬃculties can be circumvented using
the miraculous renormalization of the It\^o calculus. In fact assume that
$K$ andsatisfy the hypothesis of the corollary. First let us indicatethat we can assume $W=C_{0}([0,1], R)$ (cf., [29], Chapter , to see how
one can pass ﬀom an abstract Wiener space to the standard one) andin this case the Cameron-Martin space $H$ becomes $H^{1}([0,1])$ , whichis the space of absolutely continuous functions on $[0, 1]$ , with a square
integrable Sobolev derivative. Let now
$L$

$II$

$\Lambda=\frac{K}{L\circ T}$,

where $T$ is as constructed above. Then
is a Girsanov measure for themap $T$ . This means that the law of the stochastic process
is equal to the Wiener measure, where $T_{t}(x)$ is defined asunder
the evaluation of the trajectory $T(x)$ at $t\in[0,1]$ . In other words theis a Brownian motion under the probability.
process$T$
be its filtration, the invertibility of implies thatLet$\Lambda.\mu$$(t, x)\rightarrow T_{t}(x)$$\Lambda.\mu$$(t, x)\rightarrow T_{t}(x)$$\Lambda.\mu$$(\mathcal{F}_{t}^{T}, t\in[0,1])$$\vee \mathcal{F}_{t}^{T}=B(W)$.

$t\in[0,1]$

is upper and lower bounded -almost surely, hence also
-almostsurely. The It\^o representation theorem implies that it can be representedas$\Lambda$$\Lambda.\mu$$\mu$$\Lambda=E[\Lambda^{2}]\exp\{-\int_{0}^{1}\dot{\alpha}_{s}dT_{S}-\frac{1}{2}\int_{0}^{1}|\dot{\alpha}_{s}|^{2}ds\}$,

is an $H$ -valued random variable. In fact
where
can
It\^o-Clark
be calculated explicitly using therepresentation theorem (cf.,[28] , and it is given as$\alpha(\cdot)=\int_{0}.\dot{\alpha}_{s}ds$$\alpha$$)$(6.12)$\dot{\alpha}_{t}=\frac{E_{\Lambda}[D_{t}\Lambda|F_{t}^{T}]}{E_{\Lambda}[\Lambda|F_{t}^{T}]}$-almost surely, where
denotes the expectation operator withrespect tois the Lebesgue density of the absolutely conand. From the relation (6.12), it follows thattinuous map$T$is a function of , hence we have obtained the strong solution of theMonge-Amp\‘ere equation. Let us announce all this as$ dt\times\Lambda d\mu$$E_{\Lambda}$$\Lambda.\mu$$ D_{t}\Lambda$$t\rightarrow\nabla\Lambda(t, x)$$\alpha$Theorem 6.1. Assume that $K$ and are upper and lower bounded
densities, let $T$ be the transport map constructed in Theorem 4 1. Then
$T$ is also the strong solutionof the Monge-Amp\‘ere equation in the It\^o$L$$\cdot$73

Monge-Amp\‘ere equation and the It\^o calculus

sense, namely$E[\Lambda^{2}]L\circ T\exp\{-\int_{0}^{1}\dot{\alpha}_{s}dT_{s}-\frac{1}{2}\int_{0}^{1}|\dot{\alpha}_{s}|^{2}ds\}=K$$\mu$-almost surely, where$\alpha$,is given with (6.12).

References[1] P. Appell: “M\’emoire sur d\’eblais et les remblais des syst\‘emes continus ou
discontinus”. M\’emoires pr\’esent\’ees par divers savants \‘a l’Acad\’emie desSciences de l’Institut de Prance. Paris, I. N. 29, 1-208, 1887.
[2] P. Appell: “Le probl\‘eme g\’eom\’etrique des d\’eblais et des remblais”.
M\’emorial des Sciences Math\’ematiques, fasc. XXVII, Paris, 1928.
[3] P. J. Bickel and D. A. Freedman: “Some asymptotic theory for the bootstrap”. The Annals of Statistics, Vol. 9, No. 6, 1196-1217, 1981.
[4] Y. Brenier: “Polar factorization and monotone rearrangement of vector
valued functions”. Comm. pure Appl. Math, 44, 375-417, 1991.
[5] L. A. Caﬀarelli: “The regularity of mappings with a convex potential”.
Jour. Amer. Math. Soc, 5, 99-104, 1992.
[6] C. Dellacherie and P. A. Meyer: Probabilit\’es et Potentiel, Ch. I \‘a IV.
Paris, Hermann, 1975.
[7] N. Dunford and J. T. Schwartz: Linear Operators 2, Interscience 1963.
[8] X. Fernique: “Extension du th\’eor\‘eme de Cameron-Martin aux translations al\’eatoires’’, Comptes Rendus Math\’ematiques, Vol. 335, Issue 1,
65-68, 2002.
[9] X. Fernique: “Comparaison aux mesures gaussiennes, espaces autoreproduisants. Une application des propri\’et\’es isop\’erim\’etriques’’. Preprint.
[10] D. Feyel and A. de La Pradelle: “Capacit\’es gaussiennes”. Annales del’Institut Fourier, t.41, f.1, 49-76, 1991.
[11] D. Feyel and A. S. \"Ust\"unel: “The notion of convexity and concavity on
Wiener space”. Journal of Functional Analysis, 176, 400-428, 2000.
[12] D. Feyel and A. S. \"Ust\"unel: “Transport of measures on Wiener space and
the Girsanov theorem”. Comptes Rendus Math\’ematiques, Vol. 334,
Issue 1, 1025-1028, 2002.
[13] D. Feyel and A. S. \"Ust\"unel: “Monge-Kantorovitch measure transportationand Monge-Amp\‘ere equation on Wiener space”. Preprint, 2003.
[14] W. Gangbo and R. J. McCann: “The geometry of optimal transportation”. Acta Mathematica, 177, 113-161, 1996.
[15] K. It\^o and M. Nisio: “On the convergence of sums of independent Banach
space valued random variables”. Osaka Journ. Math. 5, 35-48, 1968.
[16] L. V. Kantorovitch: “On the transfer of masses”. Dokl. Acad. Nauk. SSSR
37, 227-229, 1942.

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D. Feyel and A. S.

$\dot{U}$st\"unel[17] K. Marton: “Bounding -distance by informational divergence: a method
to prove measure concentration”. Annals of Probability, 24, no.2, 857866, 1996.
[18] R. J. McCann: “Existence and uniqueness of monotone measurepreserving maps”. Duke Math. Jour., 80, 309-323, 1995.
[19] R. J. McCann: “A convexity principle for interacting gases”. Advancesin Mathematics, 128, 153-179, 1997.
[20] G. Monge: “M\’emoire sur la th\’eorie des d\’eblais et des remblais”. Histoirede l’Acad\’emie Royale des Sciences, Paris, 1781.
[21] F. Otto and C. Villani: “Generalization of an inequality by Talagrandand links with logarithmic Sobolev inequality”. Journal of Functional
Analysis, 173, 361-400, 2000.
[22] T. Rockafellar: Convex Analysis. Princeton University Press, Princeton,$\overline{d}$1972.
[23] V. N. Sudakov: “Geometric problems in the theory of infinite dimensional
probability distributions”. Proc. Steklov Inst. Math., 141, 1-178, 1979.
[24] M. Talagrand: “Transportation cost for Gaussian and other product measures”. Geom. Funct. Anal., 6, 587-600, 1996.
[25] H. Tanaka “An inequality for a functional of probability distributions andits applications to Kac’s one-dimensional model of a Maxwellian gas”.
Z. f\"ur Wahrsch. verw. Geb., 27, 47-52, 1973.
[26] H. Tanaka “Certain functional of probability measures on Hilbert spaces”.
Hiroshima Math. Jour., 6, 421-428, 1976.
[27] A. S. \"USt\"une1: “Representation of distributions on Wiener space andStochastic Calculus of Variations”. Journal of Functional Analysis, 70,
126-139, 1987.
[28] A. S. \"Ust\"unel: Introduction to Analysis on Wiener Space. Lecture Notes
in Math. Vol. 1610. Springer, 1995.
[29] A. S. \"Ust\"unel and M. Zakai: Transformation of Measure on Wiener Space.
Springer Monographs in Mathematics. Springer Verlag, 1999.
Denis Feyel
Universit\’e d’Evry-Val-d’Essone,
91025 Evry Cedex,
France.
E-mail: feyel@maths.univ-evry.fr

Ali S\"uleyman \"Ust\"unelENST,
D\’ept. Infres,46, rue Barrault, 75634 Paris Cedex 13,
France.
E-mail: ustunel@enst.fr

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 75-89

Function Spaces and Symmetric Markov Processes
Masatoshi Fukushima
Dedicated to Professor Kiyosi It\^o on the occasion of his 88thbirthday
Abstract.

We exhibit some mutual interactions between potential theory
and the Dirichlet space theory
for concrete function spaces on
associated with symmetric Markov processes. Our first concern isthe role of the Dirichlet form version of the capacitary strong type
inequality in the study of the ultracontractivity of the transition semigroup of time changed symmetric Markov processes. In particular,
we study time changes of symmetric stable processes in relation to-bounds of measures. We next show how the theory on capacity andthe spectral synthesis for the Dirichlet space can be well inherited toa general function space with contractive -norm. A link connecting.
those two topics is a contractive Besov space over a -set of$\mathbb{R}^{n}$

$d$

$p$

$d$

\S 1.

$\mathbb{R}^{n}$Introduction

Since the publication of the seminal work of Beurling and Deny [5],has been
their axiomatic potential theory of the Dirichlet space
unified under one roof with the theory of the symmetric Markov process
M. In particular, any -finite positive measure charging no set ofzero capacity can now be studied in relation to the $trace$ Dirichlet space
on $F$ ofon the support $F$ of and the time changed process$M$ by means of the positive continuous additive functional associatedwith $\mu([13])$ .
In \S 2, we shall see for $\kappa\in(0,1)$ that a simple capacitary isoperimetric inequality$(T, \mathcal{E})$$\sigma$$\mu$$\check{M}$$(f,\check{\mathcal{E}})$

$\mu$

(1)

$\mu(K)^{\kappa}\leq\Theta$

Cap(K),

Received February 28, 2003.

$VX(compact)$ ,76

M. Fhkushima

is equivalent to the ultracontractivity(2)$\check{p}_{t}(x, y)\leq(\frac{H}{t})\frac{1}{1-\kappa}$$t>0$ ,,of the transition function of , with the isoperimetric constant forcontrollingthe measure and the heat constant $H$ for the processeach other. When the (extended) Dirichlet space is the Riesz potential
space
and $M$ is the symmetric -stable process $(0<\alpha<1)$ ,we shall also see in \S 3 that the isoperimetric constant can be replacedby the -bound$\check{M}$$\Theta$$\check{p}_{t}$$\check{M}$$\mu$$ 2\alpha$$\dot{L}^{\alpha,2}(\mathbb{R}^{n})$

$d$

(3)

$v_{d}(\mu)=\sup_{x\in \mathbb{R}^{n},r>0}\frac{\mu(B(x,r))}{r^{d}}$,of the measure . Detailed proof of theorems in \S 2 and \S 3 can be foundin [16].
An important ingredient in proving the above equivalence is thecapacitary strong type inequality
$\mu$

(4)

$\int_{0}^{\infty}$

Cap({x\in X : $|u(x)|\geq t\}$ )$d(t^{2})\leq 4\mathcal{E}(u, u)$$\forall u\in F\cap C_{0}(X)$,which readily ensures the equivalence of (1) to a Sobolev type imbedding of the $trace$ Dirichlet space . We can then invoke the works by$Carlen,Kusuoka$ and Stroock[8] and Bakry, $Coulhon,Ledoux$ and SaloﬀCoste[2] to relate (1) and (2).
In the meantime, potential theory have advanced being modelledon concrete function spaces like Sobolev spaces $W^{r,p}$ , Bessel potential
spaces, Besov spaces
and so on. Imbedding theorems andspectral synthesis have been among important issues in potential theory([1], [4], [23]).
Actually the capacitary strong type inequality was first established, $ 1<p<\infty$ . It was thenby Maz’ $ya[22]$ for the Sobolev space
including the Riesz
extended to a large class of function spaces on
and Bessel potential spaces. It has been also proved in [21], [14] for ageneral function space with contractive -norm $(1 \leq p<\infty)$$\check{F}$$L^{\alpha,p}$$B_{\alpha}^{p,q}$$W^{1,p}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$p$$|||u|||_{p}^{p}$(5)$=\int_{X\times X\backslash d}|u(x)-u(y)|^{p}N(x, dy)m(dx)$$\{$$F_{p}$$=\{u\in L^{p}(X;m):|||u|||_{p}^{p}<\infty\}$,which include as an important example the contractive Besov space, $ 0<\alpha<1,1\leq p<\infty$ , over a -set
defined as$B_{\alpha}^{p,p}(F)$$d$$F\subset \mathbb{R}^{n}$77

Function spaces and Markov processes

6)
$(\{$

$||u;B_{\alpha}^{p,p}(F)||$$B_{\alpha}^{p,p}(F)$$=||u||_{L^{p}(F;\mu)}+(\int\int_{F\times F}\frac{|u(x)-u(y)|^{p}}{|x-y|^{d+\alpha p}}\mu(dx)\mu(dy))^{1/p}$

$=$

{

$u$

is measurable :$||u;B_{\alpha}^{p,p}(F)||<\infty$}.

being taken to be the restriction to $F$ of the -dimensional Hausdorﬀ
measure.
Its Dirichlet space version (4) accompanied by the best constant 4
was proved rather recently by $Vondra\check{c}ek[25]$ . [16] provides an alternative simple proof of (4).
When $p=2$ , the contractive Besov space on a -set is a regular
Dirichlet space on $L^{2}(F;\mu)$ and the properties of the associated jump
type Markov process on $F$ have been studied in [14], [6] and [9]. As weshall see in \S 3, this space is closely related to the Dirichlet space
on $L^{2}(F;\mu)$ of the time changed process of a symmetric stable process onin the sense that the former is continuously imbedded into the latter,although these two spaces are generally diﬀerent because the latter may
involve a killing term in general.
Even when $p\neq 2$ , the function space (5) with contractive -norm
shares with the Dirichlet space a common feature that every normal
contraction operates on it and deserves to be studied on its own light.
We shall see in \S 4 that the well known theory on capacity and spectral
synthesis for the Dirichlet space ([5], [10], [13]) can be well inherited to
the function space (5).
In particular, the spectral synthesis is possible for the contractive
for $ 1<p<\infty$ . As an application, weBesov space on a -setshall get in \S 4 the following criterion for an relatively open set $H\subset F$
has a locally finite positive -dimensional Hausdorﬀ
such that:measure with
$d$

$\mu$

$d$

$(\check{F},\check{\mathcal{E}})$$\mathbb{R}^{n}$

$p$

$d$

$F\subset \mathbb{R}^{n}$$\tilde{d}$$F\backslash H$$\tilde{d}<d$(7)$B_{\alpha,0}^{p,p}(H)=B_{\alpha}^{p,p}(F)\Leftrightarrow\alpha\leq\frac{d-\overline{d}}{p}$,.
being the closure of $B_{\alpha}^{p,p}(F)\cap C_{0}(H)$ in the space
This completes and extends the corresponding results by Caetano, $H=D$ , for[7] and Farkas and Jacob [11]. When $p=2$ , $d=n$ ,, (7) has been shown by $Bogdan,Burdzy$ and Chenan open set[6] giving a complete characterization for almost no sample path of thein finite
censored -stable process on $D$ to approach the boundary
time. Detailed proof of theorems in \S 4 can be found [15].
$B_{\alpha}^{p,p}(F)$$B_{\alpha,0}^{p,p}(H)$$F=\overline{D}$$D\subset \mathbb{R}^{n}$$ 2\alpha$$\partial D$

78

\S 2.

M. Fukushima

Capacitary bounds of measures and time changed processes

Let $(X, m, \mathcal{E}, F)$ be a regular transient Dirichlet space. By this,
we mean that $X$ is a locally compact separable metric space, $m$ is an
is aeverywhere dense positive Radon measure on $X$ , and that
regular transient Dirichlet form on $L^{2}(X;m)$ . The 0-order capacity of acompact set $K\subset X$ is then defined by$(\mathcal{E}, F)$(8)

Cap(K) $=\inf\{\mathcal{E}(u, u) : u\in F\cap C_{0}(X), u(x)\geq 1, x\in K\}$

denotesand extended to any subsets of $X$ as a Choquet capacity.
the extended Dirichlet space. In what follows, any function $u\in F_{e}$ willbe always taken to be quasi-continuous (cf. [13]).
Owing to $Vondra\check{c}ek[25]$ , we then have the capacitary strong type
inequality (4), which in turn implies the following (cf. [1, 7.2]):$F_{e}$

Theorem 1. Let be a Borel measure on $X$ and $\kappa\in(0,1]$ .
(i) If the capacitary isoperimetric inequality (1) holds for some positive
constant , then is a smooth Radon measure and$\mu$$\Theta$

$\mu$

(9)

$||u||_{L^{2/\kappa}}^{2}(x;\mu)\leq S\mathcal{E}(u, u)$,$\forall u\in F_{e}$,.
for some positive constant
$u\in
F\cap
C_{0}(X)$ and(ii) Conversely, if (9) holds for any
for some positive
constant.
(1),
positive
holds
some
constant
thenfor$ S\leq(4/\kappa)^{\kappa}\Theta$$S$$\Theta\leq S$

For a measure on $X$ , we introduce its isoperimetric constant andSobolev constant respectively by
$\mu$

(10)

$\Theta_{\kappa}(\mu)=\sup_{K}\frac{\mu(K)^{\kappa}}{Cap(K)}$$\kappa\in(0,1]$,$\underline{||u||_{L^{\eta}(\mu)}^{2}}$(11)$\eta\in[2, \infty)$$S_{\eta}(\mu)=$.

$\sup_{u\in \mathcal{F}_{\cap}C_{0}(X)}\mathcal{E}(u, u)$

may be called
The supremum in (11) can be taken for all $u\in F_{e}$ .
the Poincar\’e constant of . Theorem 1 can be rephrased as follows:$S_{2}(\mu)$$\mu$

on $X$ and for $\kappa\in(0,1]$ , $0<$
Corollary 1. For a measure
if and only if $ 0<S_{2/\kappa}(\mu)<\infty$ . Moreover,$\mu$$\Theta_{\kappa}(\mu)<\infty$(12)$\Theta_{\kappa}(\mu)\leq S_{2/\kappa}(\mu)\leq(4/\kappa)^{\kappa}\Theta_{\kappa}(\mu)$,$\kappa\in(0,1]$.

Function spaces and Markov processes

79

Let $M=\{X_{t}, P_{x}\}$ be an $m$-symmetric Hunt process on $X$ associatedwith the Dirichlet form and $A=A_{t}$ be a PCAF of $M$ whose Revuz
measure is a given smooth Radon measure . Denote by $F$ and the,andsupport of and $A$ respectively. Then
is a quasi-support of , namely, $if_{-}quasi$ -continuous functions
further, then they coincide $q.e$ . on $F$ . Recall that each element
coincide$u\in F_{e}$ is taken to be quasi-continuous.
We consider the time changed process
defined by$\mathcal{E}$$\tilde{F}$$\mu$$\tilde{F}\subset Fq.e.$$\mu(F\backslash \tilde{F})=0$$\mu$$\overline{F}$$\mu$$\mu- a.e.$$\check{M}=(\check{X}_{t}, P_{x})_{x\in\overline{F}}$$\tau_{t}=\inf\{s>0:A_{5}>t\}$ .

$\check{X}_{t}=X_{\tau_{t}}$is a -symmetric transient right process, whose Dirichlet form
can be described as
on $L^{2}(F;\mu)$ and the extended Dirichlet space
follows (cf. [13, 6.2]) :$\check{M}$$(\check{\mathcal{E}},\check{F})$$\mu$$\check{F}_{e}$(13)$\check{F}_{e}=\{\varphi=u|_{F}\mu-a.e.

(14)

:

u\in F_{e}\}$$\check{F}=\check{F}_{e}\cap L^{2}(F;\mu)$$\varphi=u|_{F}\in\check{F}_{e}$$\check{\mathcal{E}}(\varphi, \varphi)=\mathcal{E}(H_{\overline{F}}u, H_{\overline{F}}u)$

,

where
$x\in X$ ,$H_{\overline{F}}u(x)=E_{x}(u(X_{\sigma_{\overline{F}}}))$

being the hitting
anddenoting the expectation with respect toare regarded
time of the set by the sample path . Two elements of. Sinceis a quasi-support of , theidentical if they coincides
definition (14) of makes sense.
We can restate (14) as follows (the Dirichlet principle):$P_{x}$$E_{x}$$\sigma_{\overline{F}}$$\tilde{F}$$\check{F}_{e}$$X_{t}$$\tilde{F}$$\mu- a.e$$\mu$$\check{\mathcal{E}}$(15)$\check{\mathcal{E}}(\varphi, \varphi)=\inf${$\mathcal{E}(u,$$u)$: $u\in F_{e}$ , $u=\varphi\mu- a.e$ . on $F$ },$\varphi\in\check{F}_{e}$.

The first half of the next theorem is immediate ﬀom (9) and (15).

Theorem 2. Suppose a measuresome $\kappa\in(0,1)$ .
Then the following holds for
(i)

$\mu$

satisfies$\ominus_{\kappa}(\mu)\in(0, \infty)$$S=S_{2/\kappa}(\mu)(\in(\Theta_{\kappa}(\mu), (4/\kappa)^{\kappa}\ominus_{\kappa}(\mu)))$(16)$\forall\varphi\in\check{F}_{e}$$||\varphi||_{L^{2/\kappa}}^{2}(F;\mu)\leq S\check{\mathcal{E}}(\varphi, \varphi)$

for

.

.

on $F$(ii) The transition functionof the time changed processsatisfies the ultracontractivity (2) for $\mu\times\mu-a.e$ . $(x, y)\in F\times F$, where$H$ is some positive constant with$\check{M}$$\check{p}_{t}$(17)$H\leq\frac{1}{1-\kappa}\cdot S$.

80

M. Fhkushima

We know that (1) and (16) are equivalent by Voropoulos [24]. But
we are more concerned with dependence of the isoperimetric constantand the heat constant $H$ .
Simple mutual dependence of $H$ and the constant $N$ appearing in
the Nash type inequality has been well studied in [8]. The Sobolev
inequality (16) can be readily converted by a H\"older inequality into the
Nash type inequality with $N=S$ and we can get the bound (17) easily.
On the other hand, we know that the Sobolev inequality can be derived
by $N$ infrom the Nash type inequality under a certain control ofview of [2, Cor, 4.4, Cor. 7.3], and we can get the following converse to
Theorem 2.
$O-_{\kappa}$$S$

Theorem 3. Suppose that is a smooth Radon measure with $\sup-$port $F$ and that the transition functionof the time changed process$F$with respect to the PCAF with Revuz measure satisfies theonultracontractivity (2) for some $\kappa\in(0,1)$ , $H>0$ . Then(i) The Sobolev inequdity (16) holds for some positive constant with$\mu$$\check{p}_{t}$$\check{M}$

$\mu$

$S$

(18)
(ii)

$S\leq 48e^{2}\frac{1}{\kappa}(\frac{2-\kappa}{1-\kappa})\frac{2-\kappa}{1-\kappa}\cdot H$$\mu$admits an isoperimetric constant(19)$\Theta_{\kappa}(\mu)$.

with a bound$(4/\kappa)^{-\kappa}S\leq\Theta_{\kappa}(\mu)\leq S$

by the constant$S$of (i).

Tierry Coulhon has called author’s attention to the relevance of thecapacitary isoperimetric inequality (1) to the Faber-Krahn inequality.
For an open set $G\subset X$ , we put$F_{G}=${ $u\in F:u=0$$q.e$. on $X\backslash G$

}.

with
Due to the spectral synthesis theory for the Dirichlet space,$L^{2}(G;m)$domaincan be considered as a regular Dirichlet form on
on $G([13, \S 4.4])$ . For a measure on
which is called the part of$X$ , we let$\mathcal{E}$$F_{G}$$(\mathcal{E}, F)$$\mu$$\lambda_{1}(\mu;G)=\inf_{u\in \mathcal{F}_{G}}\frac{\mathcal{E}(u,u)}{||u||_{L^{2}(\mu)}^{2}}(=\inf_{u\in F\cap C_{0}(G)}\frac{\mathcal{E}(u,u)}{||u||_{L^{2}(\mu)}^{2}})$,

which may be regarded, on account of the Dirichlet principle (15), ason thethe first eigenvalue for the part of the $trace$ Dirichlet space
relatively open sunset $F\cap G$ of $F$. Since
is the reciprocal of the$(\check{F},\check{\mathcal{E}})$$\lambda_{1}(\mu;G)$Function spaces and Markov processes

81

Poincar\’e constant $S_{2}(\mu;G)$ defined by (11) for the part form$(\mathcal{E}, F_{G})$,we get from (12)(20)$\frac{1}{\lambda_{1}(\mu,G)}.\leq 4\sup_{K\subset G}\frac{\mu(K)}{Cap(K\cdot G)}$,,

where Cap(K; $G$ ) is defined by (8) with $X$ being replaced by $G$ .
$\in(0,1)$ .
Let us assume thatis finite for some
Cap(X; $G$ ) $\geq Cap(K)$ for $K\subset G$ , we have$\Theta_{\kappa}(\mu)$(21)$\kappa$$\frac{1}{Cap(K\cdot G)},\leq\frac{\Theta_{\kappa}(\mu)}{\mu(K)^{\kappa}}$

Since

.

(20) and (21) lead us to$\frac{1}{\lambda_{1}(\mu\ovalbox{\tt\small REJECT} G)}\leq 4\sup_{K\subset G}\Theta_{\kappa}(\mu)\cdot\mu(K)^{1-\kappa}=4\Theta_{\kappa}(\mu)\cdot\mu(G)^{1-\kappa}$

and
(22)

$\lambda_{1}(\mu;G)\geq\frac{1}{4\Theta_{\kappa}(\mu)}$.

$\frac{1}{\mu(G)^{1-\kappa}}$

for any open set $G\subset X$ of finite -measure.
(22) is called the Faber-Krahn inequality and the above procedureof getting (22) from (1) using the capacitary strong type inequality has
been indicated by Grigor’ $yan[18]$ . Very intimate relationship among
the Faber-Krahn inequality, ultracontractivity (2) and the Nash type
inequality has been studied in [19]. However, in order to recover thecapacitary isoperimetric inequality (1) ﬀom the ultracontractivity (2),one may need to path through Nash type inequality to Sobolev’s one as
being done in this section.
$\mu$

\S 3.

Application to time changes of symmetric stable processes
on -sets$d$

In this section, we consider the symmetric-stable process $M=$
on
for $0<\alpha\leq 1,2\alpha<n$ . The transition function of $M$ isa convolution semigroup $\{l/_{t}, t>0\}$ of symmetric probability measuresonwith$ 2\alpha$$(X_{t}, P_{x})$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\hat{I}J_{t}(x)(=\int_{\mathbb{R}^{n}}e^{i(x,y)}lJ_{t}(dy))=e^{-tc|x|^{2\alpha}}$,being a fixed positive constant. For simplicity, we take $c=1$ . In casethat $\alpha=1$ , $M$ is the -dimensional Brownian motion with variance of
$c$

$n$

M. Fukushima82$\mu_{t}$being equal to 2. $M$ is transient. The Dirichlet form
is given by$(\mathcal{E}, F)$

of $M$ on$L^{2}(\mathbb{R}^{n})$(23)$\mathcal{E}(u, u)$$=$$\int_{\mathbb{R}^{n}}\hat{u}(x)\overline{\hat{v}}(x)|x|^{2\alpha}dx$

$F$

$=$

$\{u\in L^{2}(\mathbb{R}^{n}) :$\{$The extended Dirichlet spacewith the Riesz potential space
the Riesz potential of a measure on\int_{\mathbb{R}^{n}}|\hat{u}(x)|^{2}|x|^{2\alpha}dx<\infty\}$$(F_{e}, \mathcal{E})$of $M$ can then be identified, whereis defined by$\dot{L}^{\alpha,2}(\mathbb{R}^{n})=\{I_{\alpha}*f :$iJ$$\mathbb{R}^{n}$.

$I_{\alpha}*\iota/(x)=\gamma_{\alpha}\int_{\mathbb{R}^{n}}|x-y|^{-(n-\alpha)}\iota/(dy)$,f\in L^{2}(\mathbb{R}^{n})\}$$\gamma_{\alpha}=\frac{\Gamma((n-\alpha)/2))}{\pi^{n/2}2^{\alpha}\Gamma(\alpha/2)}$.

The capacity defined by (8) for the present Dirichlet form coincides withby
the Riesz capacity defined for any compact set$K\subset \mathbb{R}^{n}$(24)$\dot{C}_{\alpha,2}(K)=\inf\{||f||_{L^{2}(\mathbb{R}^{n})}^{2} :We call a closed subset $F$ of
if there exists a positive measureconstants $0<c_{1}\leq c_{2}$ ,f\in L_{+}^{2}(\mathbb{R}^{n}), I_{\alpha}*f(x)\geq 1\forall x\in K\}$a (semi global) -set for $0<d\leq n$
supported by $F$ satisfying, for some$d$$\mathbb{R}^{n}$$\mu$$c_{1}r^{d}\leq\mu(B(x, r))$$\mu(B(x, r))\leq c_{2}r^{d}$.

$\forall x\in F$, $\forall r\in(0,1)$,$\forall x\in F$ $\forall r\in(0, \infty)$,

where $B(x, r)$ denotes the -dimensional ball with center and radius. Such a measure is called a -measure. It is known that the restrictionof the -dimensional Hausdorﬀ measure to a set $F$ is a measure(cf. [20]).
For a measure , we will be concerned with its -bound defined by(3). We consider a measure on a set $F$ with
$n$

$x$

$d$

$r$

$d$

$d$

$d$

$d$

$d$

$\mu$

$d$

$d$

$\mu$

$n$

$-2\alpha<d\leq n$.

Otherwise, $\dot{C}_{\alpha,2}(F)=0$ and can not satisfy the isoperimetric inequality with respect to the present Dirichlet form. Since$\mu$$\dot{C}_{\alpha,2}(B(x, r))=\dot{c}_{\alpha,2}r^{n-2\alpha}$,$\dot{c}_{\alpha,2}=\dot{C}_{\alpha,2}(B(0,1))$,

we can immediately obtain a lower bound of the isoperimetric constantfor by its -bound:
$d$

$\mu$

(25)

$\dot{c}_{\alpha,2}^{-1}v_{d}(\mu)^{\frac{n-2\alpha}{d}}\leq\Theta_{\frac{n-2\alpha}{d}}(\mu)$.

We can also obtain an inequality in the opposite direction:

83

Function spaces and Markov processes

Theorem 4. For any Radon measurewith finite -bound, it holds
$d$

$\mu$

that
(26)

$\Theta_{\frac{n-2\alpha}{d}}(\mu)\leq c(n, \alpha, d)v_{d}(\mu)^{\frac{n-2\alpha}{d}}$

for
(27)

$c(n, \alpha, d)=\frac{4d^{22}\gamma_{\alpha}v_{n}(n-\alpha)^{2}}{(n-2\alpha)^{2}\{d-(n-2\alpha)\}^{2}}$where$v_{n}$

is the volume of the$n$,dimensional unit ball.

in Corollary 1 and using (25) and (26), we get
By setting
for in terms of its
the bound of the Sobolev constant:bound$\kappa=\frac{n-2\alpha}{d}$$S=S_{\frac{2d}{n-2\alpha}}(\mu)$

$d$

$\mu$

$v_{d}(\mu)$(28)$\dot{c}_{\alpha,2}^{-1}v_{d}(\mu)^{\frac{n-2\alpha}{d}}\leq S\leq(4d/(n-2\alpha))^{\frac{n-2\alpha}{d}}c(n, \alpha, d)v_{d}(\mu)^{\frac{n-2\alpha}{d}}$for the constant $c(n, \alpha, d)$ of (27).

By setting$\kappa=\frac{n-2\alpha}{d}$in Theorem 1 and Theorem 2, we havewith $n-2\alpha<d\leq n$ .
Theorem 5. Suppose is a measure on
Then we have the following for satisfying the bounds (28):(i)$d$$\mathbb{R}^{n}$

$\mu$

$S$

(29)

$\forall u\in\dot{L}^{\alpha,2}(\mathbb{R}^{n})$$||u||^{2}L^{\frac{2d}{n-2\alpha}}(\mathbb{R}^{n};\mu)\leq S\mathcal{E}(u, u)$.

be the time changed process on the support $F$ of of $M$ by the
PCAF with Revuz measure . Then its transition function
satisfies(ii) Let$\check{M}$$\mu$$\check{p}_{t}$

$\mu$

(30)

for

$\check{p}_{t}(x, y)\leq(\frac{H}{t})\frac{d}{d-(n-2\alpha)}$$\mu\times\mu- a.e$

(31)

.

$(x, y)\in F\times F$$t>0$ ,,

, where $H$ is some positive constant with

$H\leq\frac{d}{d-(n-2\alpha)}S$.

Actually inequaHty (29) together with the bounds$c_{3}v_{d}(\mu)^{\frac{n-2\alpha}{d}}\leq S\leq c_{4}v_{d}(\mu)^{\frac{n-2\alpha}{d}}$84

M. Fukushima

holding for some positive constants , independent of goes back to
the work of Adams ([23, 1.4.1]). Here we have made these contants
more explicit in (28).
and$c_{3}$$c_{4}$$\mu$$c_{3}$$c_{4}$

We can also derive from Theorem 3 the following converse to Theorem 4.
with
Theorem 6. Suppose that is a smooth Radon measure on$F$support and that the transition function of the time changed process
on $F$ with respect to the PCAF with Revuz measure satisfies thebound (30) for some $d\in(n-2\alpha, n]$ and $H>0$ . Then(i) The inequality (29) holds for some positive constant with$\mathbb{R}^{n}$$\mu$$\tilde{p}_{t}$$\tilde{M}$

$\mu$

$S$

(32)
(ii)

$S\leq\frac{48de^{2}}{n-2\alpha}(\frac{2d-(n-2\alpha)}{d-(n-2\alpha)})\frac{2d-(n-2\alpha)}{d-(n-2\alpha)}\cdot H$

$\mu$

is a

(33)

$d$

measure whosebound$v_{d}(\mu)$satisfies$\frac{n-2\alpha}{4d}(\frac{S}{c(n,\alpha,d)})\frac{d}{n-2\alpha}\leq v_{d}(\mu)\leq(\dot{c}_{\alpha,2}S)^{\frac{d}{n-2\alpha}}$for the constant$S$of (i) and for$c(n, \alpha, d)$of (27).

be as in Theorem 5 andbe the Dirichlet form of
on $L^{2}(F;\mu)$ the $trace$ Dirichlet form of (23) on the -set $F$. Put

Let$\check{M}$

$d$

.

$\mu$

,

$F,\check{M}$$(\check{\mathcal{E}},\check{F})$

$d$

(34)

$\delta=\alpha-\frac{n-d}{2}\in(0,1]$

and consider the Besov space(35)$B_{\delta}^{2,2}(F)$over $F$ defined by$(\varphi, \psi)_{B_{\delta}^{2,2}(F)}$$=$$\int_{F\times F\backslash d}\frac{(\varphi(x)-\varphi(y))(\psi(x)-\psi(y))}{|x-y|^{d+2\delta}}\mu(dx)\mu(dy)$$B_{\delta}^{2,2}(F)$$=$$\{\varphi\in L^{2}(F;\mu) :$\{$$B_{\delta}^{2,2}(F)$(\varphi, \varphi)_{B_{\delta}^{2,2}(F)}<\infty\}$.

is a Dirichlet form on $L^{2}(F;\mu)$ equipped with the norm$||\varphi;B_{\delta}^{2,2}(F)||^{2}=(\varphi, \varphi)_{L^{2}(F;\mu)}+(\varphi, \varphi)_{B_{\delta}^{2,2}(F)}$.

By virtue of a Jonsson-Wallin $trace$ theorem [20, chap. $V$ ], this spaceis related to the Bessel potential spaceas$L_{\alpha,2}(\mathbb{R}^{n})$(36)$B_{\delta}^{2,2}(F)=L_{\alpha,2}(\mathbb{R}^{n})|_{F}$Function spaces and Markov processes

85

both the restriction and extension operators involved being continuous.
Since the present Dirichlet space (23) equipped with -norm is known tobe equivalent to the Bessel potential space, we are led ﬀom the Dirichlet
principle (15) and (36) to the following continuous embedding:$\mathcal{E}_{1}$(37)$B_{\delta}^{2,2}(F)\subset\check{F}_{e}$,$\check{\mathcal{E}}(\varphi, \varphi)\leq C||\varphi;B_{\delta}^{2,2}(F)||^{2}$,$\forall\varphi\in B_{\delta}^{2,2}(F)$,for some positive constant $C$ .
are not necessarily equivNevertheless 0-oder forms and$M$withalent. For instance, let
be the standard Brownian motion on$F$,
be the unit sphere contered at the origin and be the surface measure on . Then we have the following expression of the $trace$
Dirichlet form
for $f\in\check{F}([17])$ :(38)$\check{\mathcal{E}}$$(\cdot, \cdot)_{B_{\delta}^{2,2}(F)}$$\mathbb{R}^{n}$$n$$\Sigma$$\geq 3$$\mu$$\Sigma$$\sigma$$\check{\mathcal{E}}(f, f)$$\check{\mathcal{E}}(f, f)=\frac{1}{\Omega}\int_{\Sigma\times\Sigma\backslash d}(f(\xi)-f(\eta))^{2}\frac{1}{|\xi-\eta|^{n}}\sigma(d\xi)\sigma(d\eta)+v_{0}\int_{\Sigma}f(\xi)^{2}\sigma(d\xi)$,. The first term on the right
where
is the area of
andhand side correponds to the form (35) for $d=n-1$ , $\delta=1/2$ . But theadditional second term appears due to the transience of the Brownian
motion.
$\Omega$$\Sigma$$v_{0}=\frac{n-2}{2}$\S 4. Spectral synthesis for contractive -norms and Besov spaces
Let $X$ be a locally compact separable metric space and $m$ a positive
Radon measure on $X$ with $supp[m]=X$ . Let $N(x, dy)$ be a positive kernel on $(X, B(X))$ such that $N(x, \{x\})=0$ , $x\in X$ , and $N(x, dy)m(dx)$is a symmetric measure over $X\times X-d$ , where $d=\{(x, x) : x\in X\}$ .
and theFor a fixed $ 1\leq p<\infty$ , we introduce the pseudo-normby (5). Denoting the norm of the space $L^{p}(X;m)$ by
function space. , we equipwith the norm$|||\cdot|||_{p}$$F_{p}$$||$$F_{p}$$||_{p}$(39)$|||u|||_{p,1}=|||u|||_{p}+||u||_{p}$$u\in F_{p}\cap L^{p}(X;m)$ .

We assume the regularity of this space in the sense that $F_{p}\cap C_{0}(X)$ isdense inwith norm (39) and in $C_{0}(X)$ with uniform norm.
the family of all open sets in $X$ . We define theDenote bybycapacity of(40),$Cap_{p}(A)=\inf$ {: $u\in F_{p}$ , $u\geq 1$ m-a.e. on $A$ }$F_{p}$$\mathcal{O}$$\Psi$$A\in \mathcal{O}$$A\in \mathcal{O}$$|||u|||_{p}^{p}+||u||_{p}^{p}$and extend it to any set $B\subset X$ by$Cap_{p}(B)=\inf\{Cap_{p}(A) : ^{A}\in \mathcal{O}, B\subset A\}$.

86

M. Fukushima.’ will mean ’except for a set of zero -capacity ’. $Cap_{p}$ quasicontinuousfunction will be called simply quasicontinuous. In what follows, we also
assume that $ 1<p<\infty$ .
Although the space ( , .
is slightly more complicated thanspace, we can well adopt the uniform convexity arguthe ordinary
ment to ensure the unique existence of the equilibrium potential for any
with finite -capacity. Thus $Cap_{p}$ on open sets can be seen to enjoy the continuity along the increasing limit as in [12]. It is also strongly
subadditive as in [21]. Hence $Cap_{p}$ is a Choquet capacity, each element$u\in F_{p}$ has a quasicontinuous version, each set of finite -capacity has
a unique equilibrium potential just as in the case of the Dirichlet space.
We $ako$ have the following nice property:(41)
is quasi-continuous and $u=0$ ra-a.e. on $G(\in \mathcal{O})\Rightarrow u=0q.e$ . on $G$ .
$‘ q.e$$p$$F_{p}$$|||$$|||_{p,1}$$L^{p}$$A\in \mathcal{O}$$p$$\tilde{u}$

$p$

$u$

For

$G\in \mathcal{O}$, we let(42)$F_{p,0}^{G}=\overline{F_{p}\cap C_{0}(G)}^{|||\cdot|||_{p,1}}$,

where $C_{0}(G)$ denotes the family of continuous functions on $X$ whose support is compact and contained in $G$ . We say that the spectral synthesis
if
is possible for$G\in \mathcal{O}$(43)$F_{p,0}^{G}=${$u\in F_{p}$: $\tilde{u}=0q.e$ . on

$X\backslash G$}.

Following the method of [1, 9.2] for the space
and using the contraction property of the space
together with the above
mentioned properties of $Cap_{p}$ , we can prove the next theorem.
$W^{1,p}(\mathbb{R}^{n})$$F_{p}$

Theorem 7. (i) The spectral synthesis is possible foris compact.

$G\in \mathcal{O}$if$X\backslash G$(ii) The spectral synthesis is possible for any
under the next
assumption:(A) There exist non-negative functions $w_{n}\in C_{0}(X)$ increasing to 1
such that$G\in \mathcal{O}$$\sup_{nx}\sup_{\in X}W_{n}(x)<\infty,\lim_{n\rightarrow\infty}\sup_{x\in K}W_{n}(x)=0$

for any compact $K\subset X$ ,

where

(44)

$W_{n}(x)=\int_{X}|w_{n}(x)-w_{n}(y)|^{p}N(x, dy)$$x\in X$ .

87Function spaces and Markov processes

As a consequence of Theorem 7 (i), the next useful identity holds
for any compact set $K\subset X$ :(45)

$Cap_{p}(K)=\inf${$|||u|||_{p}^{p}+||u||_{p}^{p}$: $u\in F_{p}\cap C_{0}(X)$ , $u\geq 1$ on $K$ }.

We now let$0<d\leq n$ ,$<\alpha<1$,$ 1<p<\infty$ ,on a setand consider the contractive Besov space
space
specialexampleofthewith contractive
defined by (6). This is a.
. The associated -capacity of a set $A\subset F$ is denoted-norm
by $Cap_{\alpha,p}(A;F)$ . It can be shown that condition A is satisfied by thisspace. By Theorem 7, the spectral synthesis is therefore possible for any, which immediately
relatively open set $H\subset F$ with respect toimplies the equivalence$d$$B_{\alpha}^{p,p}(F)$$F\subset \mathbb{R}^{n}$$F_{p}$

$|||$

$p$

$|||_{p,1}$$p$$B_{\alpha}^{p,p}(F)$(46)$B_{\alpha,0}^{p,p}(H)=B_{\alpha}^{p,p}(F)\Leftrightarrow Cap_{\alpha,p}(F\backslash H;F)=0$

where$B_{\alpha,0}^{p,p}(H)$$B_{\alpha}^{p,p}(F)$,denotes the closure of $B_{\alpha}^{p,p}(F)\cap C_{0}(H)$ in the space.

On the other hand, the next implications have been proved in [14], a Jonsson-Wallin
by making use of the property (45) of$trace$ theorem ([20]) and the metric properties of the Bessel capacity on$Cap_{\alpha,p}(\cdot;F)$$\mathbb{R}^{n}([1])$

:

(47)

$Cap_{\alpha,p}(\Lambda;F)=0\Rightarrow\prime\mu_{dim}(\Lambda)\leq d-\alpha p$(48)$H_{d-\alpha p}(\Lambda)<\infty\Rightarrow Cap_{\alpha,p}(\Lambda;F)=0$

,

.

Here
and
denote the Hausdorﬀ dimension and -dimensional
Hausdorﬀ measure respectively.
(46),(47) and (48) lead us to the next desired theorem.
$\prime H_{dim}$$H_{\gamma}$$\gamma$

Theorem 8. Assume that $H$ is a relatively open subset of $F$ andthat
has a locally finite positive -dimensional Hausdorﬀ measure$\overline{d}$$F\backslash H$$d-\tilde{d}$with$\tilde{d}<d$. Then $B_{\alpha,0}^{p,p}(H)=B_{\alpha}^{p,p}(F)$ if and only if$\alpha\leq\overline{p}$.

88

M. Fukushima

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[I]

$d$

$d$

$p$-

Function spaces and Markov processes

89, Harwood[20] A. Jonsson and H. Wallin, Function spaces on subsets of
Academic Pub., 1984[21] V. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetricinequalities for Markov operators, Potential Anal. 1(1992), 61-82[22] V.G. Maz’ya, On certain integral inequalities for functions of many variables, J. Soviet Math. 1(1973) 205-237[23] V.G. Maz’ya, Sobolev spaces, Springer-Verlag, 1985[24] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Func.
Anal 63(1985), 240-260[25] Z. Vondracek, An estimate for the -norm of a quasi continuous function
with respect to a smooth measure, Arch. Math. 67(1996), 408-414$\mathbb{R}^{n}$$L^{2}$

Faculty of Engeneering
Kansai University
Suita, Osaka 564-8680
JapanAdvanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 91-109

Gauge Theorems for Stieltjes Exponentials
Ronald Getoor
Abstract.
A gauge theorem for the Stieltjes exponential of a right continuous additive functional satisfying a general Kato type condition isestablished. Results for the ordinary exponential are then obtained
as corollaries.

\S 1.

Introduction

In a series of recent papers Chen and Song [CS02], [CS03] andChen [C02] have made remarkable progress in establishing the gauge
and conditional gauge theorem under quite general hypotheses. The
gauge theorem has the following structure: Given a multiplicative function $M$ $=(M_{t})$ and a terminal time of a strong Markov process $X$
the gauge function $g(x):=E^{x}(M_{\tau})$ is bounded on $\{g<\infty\}$ under suitable hypotheses on $X$ and $M$ . Usually $X$ is assumed to satisfy some
irreducibility hypothesis which then implies that is either bounded or
identically infinite. Also $M$ usually is of the form $M_{t}=\exp(A_{t})$ where $A$
is an additive functional. See the above cited papers and also the bookof Chung and Zhao [CZ95] for some history of the subject. Before theabove cited papers $A$ was usually assumed to be continuous and often
of the form $A_{t}=\int_{0}^{t}q(X_{s})ds$ where is a function on the state space of$X$ . See however [CR88], [So93] and [St91] for notable exceptions.
In their papers Chen and Song and Chen consider both continuousand a class of discontinuous additive functionals. The arguments in thetwo cases are similar in structure but somewhat diﬀerent in detail. It
turns out that by modiping slightly their approach one can prove agauge theorem for arbitrary right continuous additive functionals in a
unified way, assuming only that the underlying process $X$ is a Borel right$\tau$$g$$q$

Received January 27, 2003.
Keywords and Phrases: gauge theorem, additive functional, Stieltjes$\exp$ onent $ial$ .
Mathematics Subject Classification (2000): Primary $60J40$ , Secondary
60555, 60557

92

R. Getoor

process. The present paper is devoted to spelling this out in some detail.
We obtain a slightly sharper result even in the case considered by Chen
and Song, but our emphasis is the generality and the simplicity of theresult obtained. We work directly with Stieltjes exponentials of additive
functionals obtaining a gauge theorem for such exponentials. Resultsfor ordinary exponentials then appear as simple corollaries. Our results
are general enough to apply to infinite dimensional processes since we
do not assume any absolute continuity condition.
One of the main purposes of the Chen, Song papers was to prove agauge theorem in enough generality that it could be applied to prove a
conditional gauge theorem. A critical hypothesis for their result for a
conditional gauge theorem is that $X$ be in strong duality with anotherBorel right process $X$ . Under this duality hypothesis, the hypotheses andargument in section 3 of [CSOO] can be adapted to prove a conditional
gauge result for Stieltjes exponentials along the lines of this paper. We
leave the precise formulation to the interested reader.
We close this introduction with some words on notation. If $(F, F, \mu)$is a measure space, then we use $F$ also to denote the class of all(resp.
, then)valued measurable functions. If A{. For
valued) functions indenotes the class of bounded (resp. [0,$f\in F$ , $\mu(f)$ denotes the integral. If
is a second measurable
space and $K=K(x, dy)$ is a kernel ﬀom $(F, F)$ to$K(x, A)$ is inforfor each
and $K(x, \cdot)$ is a measure oneach $x\in F$ ), then we write
for the measure $A\rightarrow\int\mu(dx)K(x, A)$
and $Kf$ for the function $x\rightarrow\int K(x, dy)f(y)$ . The symbol “ $:=$ ” stands
for “is defined to be”.
$\overline{\mathbb{R}}=$$[-\infty, \infty]$$\subset F$$\mathcal{F}$$p\mathcal{M}$$\mathcal{M}$$\infty]$$\int fd\mu$$b\mathcal{M}$$(E, \mathcal{E})$$(E, \mathcal{E})(i.e.$$A\in \mathcal{E}$$\mathcal{F}$$ x\rightarrow$$(E, \mathcal{E})$$\mu K$\S 2.

Preliminaries

Throughout this paper $(P_{t}, t \geq 0)$ will denote a Borel right semi, and $X=(X_{t}, P^{x})$ will denote thegroup on a Lusin state spacecanonical realization of
as a right continuous strong Markov prois excessive provided
cess. A (positive) -finite measure $m$ on$mP_{t}\leq m$ for all. Sinceis a right semigroup, it follows that$mP_{t}\uparrow m$ setwise as. See [DM87; XII, 36-37]. We fix an excessive measure $m$ to serve as a background measure. In general we shalluse the standard notation for Markov processes without special mention. See, for example, [BG68], [DM87], [S88] and [G90]. In particular,, $q\geq 0$ , denotes the resolvent and $U:=U^{0}$ the potential kernel ofor $X$ . We assume only thatis sub-Markovian andso a point is adjoined to $E$ as an isolated point to serve as a cemetary
and $\zeta:=\inf\{t:X_{t}=\triangle\}$ is the lifetime of $X$ and $P^{x}(\zeta>t)=P_{t}1(x)$ .
$(E, \mathcal{E})$$(P_{t})$$(E, \mathcal{E})$$\sigma$$t$$\geq 0$$t$$(P_{t})$$\downarrow 0$$U^{q}:=\int_{0}^{\infty}e^{-qt}P_{t}dt$$(P_{t})$$\triangle$$(P_{t})$Gauge theorems for Stieltjes exponentials

93

As usual a function on $E$ is extended to by $f(\triangle)=0$ unless explicitly stated otherwise. Thus for example, if $f\geq 0$$\triangle$$f$$U^{q}f(x)=E^{x}\int_{0}^{\zeta}e^{-qt}f(X_{t})dt=E^{x}\int_{0}^{\infty}e^{-qt}f(X_{t})dt$.

We shall assume throughout that $X$ is transient or equivalently that $U$
is proper. More precisely we shall assume:, $0<$(2.1) Transience Assumption. There exists a function$b\leq 1$ with $Ub\leq 1$ ; reducing
if necessary we may also suppose that$ m(b)<\infty$ .
$b\in \mathcal{E}$$b$Recall that a set
is $m$ -polar (resp. $m$ -semipolar) provided$\{t : X_{t}\in B\}$ is empty (resp. at most countable) $P^{m}a.s$ .
Heredenotes the -algebra of nearly Borel sets. A nearly Borel set $N$ is mis absorbing for $X$ . By $[GS$inessential provided it is $m$ -polar and$84;(6.12)]$ any $m$ -polar set is contained in a Borel $m$ inessential set. A
property or statement $P(x)$ is said to hold quasi-everywhere $(q.e.)$ or
for quasi-every provided it holds for all outside some $m$ -polar set.
The exceptional set may then be supposed to be $m$ -inessential. We also
write $a.e$ . for ra-a. .
$B\in \mathcal{E}^{7l}$$\mathcal{E}^{n}$$\sigma$$E\backslash N$

$x$

$x$

$e$

A positive additive functional (PAF) is an(2.2) Definition.
, foradapted increasing process $A=(A_{t}; t\geq 0)$ with values inand a Borel $m$ inessential set
which there exist a defifining set$A$(called an exceptional set for ) such that;(i) $P^{x}(\Omega_{A})=1$ for $dl$
(ii)
all
for
is right continuous on
the mapping(iii) For all$A_{0}(\omega)=0_{i}$[ with[0,
fifinite valued on [0,$A_{s+t}(\omega)=A_{t}(\omega)+A_{s}(\theta_{t}\omega)$$t\geq0$;,(iv) For all
and ,$t\geq0$$A_{t}([\triangle])=0$,
where
is the dead path identically(v) For all
equal to .
.
(vi) For,for$(\mathcal{F}_{t})-$$[0, \infty]$$\Omega_{A}\in F$$N_{A}$$x\not\in N_{A}$$t\geq 0_{i}$$\theta_{t}\Omega_{A}\subset\Omega_{A}$$\omega\in\Omega_{A}$$t\rightarrow A_{t}(\omega)$$\infty[,$$\zeta(\omega)$$\omega\in\Omega_{A}$$s$$[\triangle]$$\triangle$$\omega\in\Omega_{A}$$A_{t}(\omega)=A_{\zeta(\omega)-}(\omega)$$t\geq\zeta(\omega)$

We letdenote the class of all PAFs. If in (2.2-iii) right continuous is replaced by continuous, then $A$ is a positive continuous additive
for the class of such functionals.
functional (PCAF) and we write$A$
Two PAFs
and $B$ are $m$ -equivalent provided $P^{m}(A_{t}\neq B_{t})=0$ for
all $t\geq 0$ . One can check that $A$ and $B$ are $m$ -equivalent if and only
if they have a common defining set
and a common exceptional set$N$ such that $A_{t}(\omega)=B_{t}(\omega)$ for all $t\geq 0$ and. See the argument just below Definition 3.1 in $[FG96]$ . Equality between elements$\overline{A}^{+}$$\overline{A}_{c}^{+}$$\Lambda$$\omega\in\Lambda$

R. Getoor

94

of
An
and

$\overline{A}^{+}$

will mean $m$ -equivalence unless explicitly mentioned otherwise.
is a PCAF
may be decomposed as $A=A^{c}+A^{d}$ where
$A$
–As-.
is the sum of the jumps of ,$A\in\overline{A}^{+}$$A^{c}$$\triangle A_{s}=A_{s}$$A_{t}^{d}=\sum_{0<s\leq t}\triangle A_{s}$on. Of course
and $ N_{A}=\phi$ are
Because of (vi),allowed in Definition 2.2. By restricting $X$ to the Borel absorbing set
is empty. In
one may usually reduce to the case in which
order to keep this exposition as simple possible I will restrict attention
to this case. To be precise we define$\triangle A_{\zeta}=0$$\Omega_{A}=\Omega$$\Omega_{A}$$N_{A}$$E\backslash N_{A}$(2.3)$A^{+}=\{A\in\overline{A}^{+} :N_{A}=\phi\}$.

Our results will be stated for $A\in A^{+}$ , but the interested reader should. If $A\in A^{+}$ , thenhave no diﬃculty in formulating them for$P^{x}(\Omega_{A})=1$ for all. Hence $A$ is almost perfect as defined in [S88,p. 173].
If $A\in A^{+}$ , then its characteristic (Revuz) measure
is defined by$A\in\overline{A}^{+}$$x$$\mu_{A}$

(2.3)

for

$\mu_{A}(f):=\uparrow\lim_{t\rightarrow 0}\frac{1}{t}E^{m}\int_{0}^{t}f(X_{s})dA_{s}$$f\in p\mathcal{E}^{n}$. Moreover(2.6)$\mu_{A}(f)=\uparrow\lim_{q\rightarrow\infty}q\cdot mU_{A}^{q}f$

where$U_{A}^{q}f(x):=E^{x}\int_{0}^{\zeta}e^{-qt}f(X_{t})dA_{t}$doesis the -potential operator of $A$ . As usual $U_{A}:=U_{A}^{0}$ . Clearly$m$
not charge -polars.
We shall also need the Stieltjes exponential $Exp(A)$ of $A\in A^{+}:$$q$$\mu_{A}$(2.6)$Exp(A_{t}):=e^{A_{t}^{c}}\prod_{0<s\leq t}(1+\triangle A_{s})$.

:Clearly $Exp(A)$ is increasing, right continuous and finite on {the compensated powers,
and $Exp(A_{0})=1$ . On
$A_{t}^{(0)}=1$
of $A$ are defined by
and for $ t<\zeta$$(t, \omega)$$t<\zeta(\omega)$$\omega\in\Omega_{A}\}$$ 0\leq$$\Omega_{A}$$A^{(n)}$$A_{t}^{(n)}=n\int_{]0,t]}A_{s-}^{(n-1)}dA_{S}$It is well-known that;

$Exp(A_{t})=\sum_{n\geq 0}\frac{1}{n!}A_{t}^{(n)}$$n$$\geq 1$.

, see [DD70, p189]. Recall

that a terminal time for $X$ is a stopping time which satisfies
on $\{t<\tau\}$ . A straightforward induction argument yields the following
result. See [SSOO] for much more general results.
$\tau$$ t+\tau\circ\theta_{t}=\tau$Gauge theorems for Stieltjes exponentials

95

be either a terminal time or(2.7) Lemma. Let $A\in A^{+}$ and letconstant time. Let. If $E^{x}(A_{\tau-})\leq C<1$ for all , then$E^{x}[Exp(A_{\tau-})]\leq(1-C)^{-1}$for all .

$a$$\sigma$$\tau=\sigma\wedge\zeta$

$x$

$x$

, $\tau(B):=\inf\{t>0$ :If$B$ . Evidently $\tau(B)\leq\zeta$ .
$B\in \mathcal{E}^{n}$$X_{t}\not\in B$] denotes the exit time ﬀom(2.8) Proposition. Let $A\in A^{+}$ and suppose that $A$ has boundedjumps;
that is, there exists $ 0\leq c<\infty$ such that $\sup_{t}\triangle A_{t}\leq ca.s$ . Then there
exists an increasing sequence $(G_{n})$ of fifinely open nearly Borel sets with
$E=\cup G_{n}$ ,. and for each , $\mu_{A}(G_{n})<\infty$ and$\tau(G_{n})\uparrow\zeta a.s$

$ U_{A}1_{G_{n}}\leq$$n$$(2c+1)n$ .

Proof. Suppose first that $c<1$ . Define$M_{t}:=Exp(-A_{t})=e^{-A_{t}^{c}}\prod_{0<s\leq t}(1-\triangle A_{s})$.

is right continuous and decreasing, $M_{t}>0$ if $ t<\zeta$
Then $a.s.$ ,and $M_{0}=M_{0+}=1$ . It is well-known and easily verified that $M_{t+s}=$
and $d(M_{t}^{-1})=M_{t}^{-1}dA_{t}$ . Define$t\rightarrow M_{t}$$M_{t}M_{s}\circ\theta_{t}$(2.8)$g(x):=E^{x}\int_{0}^{\zeta}M_{t}b(X_{t})dt$

where

$b$

is the function in (2.1). Clearly $g>0$ , and$U_{A}g$. $\int_{0}^{\zeta}M_{t}^{-1}\int_{t}^{\zeta}M_{s}b(X_{s})dsdA_{t}$$=$E. $\int_{0}^{\zeta}g(X_{t})dA_{t}=E$$=$

E. $\int_{0}^{\zeta}M_{s}b(X_{s})\int_{0}^{s}M_{t}^{-1}dA_{t}ds=Ub-g$ .

Hence $0<g\leq Ub$ and $U_{A}g\leq Ub$ . It is easily checked that is excessive
relative to $(X, M)$
the $M$ subprocess of $X-$ and consequently isnearly Borel and finely continuous. Thus the sets $G_{n}:=\{g>\frac{1}{n}\}$ form
an increasing sequence of finely open nearly Borel subsets of $E$ with$\cup G_{n}=E$ . Let. Sincebe the exit time ﬀom
is finely closed,. on. Hence
,
$g$

–

$g$

$G_{n}$$\tau_{n}$$g(X_{\tau_{n}})\leq\frac{1}{n}$$a.s$$G_{n}^{c}$$\{\tau_{n}<\zeta\}$$\frac{1}{n}\geq E.[g(X_{\tau_{n}})]=E.[M_{\tau_{n}}^{-1}\int_{\tau_{n}}^{\zeta}M_{t}b(X_{t})dt]\geq E$.
$[\int_{\tau_{n}}^{\zeta}M_{t}b(X_{t})dt]$.

, so we must have $\lim_{n}\tau_{n}=\zeta a.s$ .
But $b>0$ and $M_{t}>0$ on [0,Since $U_{A}g\leq Ub$ one has $U_{A}1_{G_{n}}\leq nUb\leq n$ . Therefore $\mu_{A}(G_{n})\leq$$ n\mu_{A}Ub\leq n\cdot m(b)<\infty$ where the second inequality comes ﬀom (the$\zeta[a.s.$R. Getoor

96

proof of) the lemma at the bottom of page 508 in $[Re70]$ . This estabhaslishes (2.8) when $c<1$ . If $c\geq 1$ , let $A_{t}^{*}=(2c)^{-1}A_{t}$ so that$A$
jumps bounded by . Let be defined as in (2.9) but with replaced. Since $U_{A}=2cU_{A^{*}}$ and $\mu_{A}=2c\mu_{A^{*}}$ , $G_{n}:=\{g>\frac{1}{n}\}$ has thebydesired properties.
$A^{*}$$\frac{1}{2}$$g$$A^{*}$$\blacksquare$

The proof of (2.8) is easily modified to prove the following:(2.10) Proposition. Let
$a.s$

.

$P^{x}$

for$x\in E\backslash N_{A}$and suppose that

$A\in\overline{A}^{+}$$\sup_{t}\triangle A_{t}\leq c<\infty$. Then there exists an increasing sequence $(G_{n})$ of.
is $m$ -polar,
fifinely open nearly Borel sets such that, and for each , $\mu_{A}(G_{n})<\infty$ and $U_{A}1_{G_{n}}\leq(2c+1)n$for.
on$\tau(G_{n})\uparrow\zeta a.s$

$E\backslash \cup G_{n}$$P^{x}$$x\in E\backslash N_{A}$$n$$E\backslash N_{A}$replaces
From time to time we will spell out the situation when
. Usually it is just a matter of keeping track of the exceptional set
as illustrated in (2.10).
$\overline{A}^{+}$$A^{+}$$N_{A}$\S 3.

Kato Classes

In this section we introduce some Kato classes of additive functionals. The definitions are modifications of those in [C02]. Let
denote$||f||_{\infty}=mess\sup_{x}|f(x)|$; that is for,. We
the norm in; that is $||f||_{qe}=\inf\{\beta$ :shall also use the $q.e$ . supremum norm for$|f|\leq\beta q.e.\}$ . Clearly. Recall thatis quasi-finely
continuous (qfc) provided it is finely continuous oﬀ an $m$-polar set which
may be assumed to be $m$ -inessential. Since an $m$ -null finely open set is$m$ -polar, it follows that if
is $qfc$ , then $||f||_{\infty}=||f||_{qe}$ .
$||\cdot||_{\infty}$$L^{\infty}(m)$$f\in \mathcal{E}^{n}$$f\in \mathcal{E}^{n}$$||f||_{qe}\geq||f||_{\infty}$$f\in \mathcal{E}^{n}$$f$(resp.
(3.1) Definition. Let $ 0<\beta<\infty$ . Then
) consists of
thosethat have bounded jumps as defifined in (2.10) and suchthat there exist a positive measure on $E$ , a set
with $\nu(K)<\infty$
and $\delta=\delta(\nu, K)>0$ with the following property:$\overline{K}_{\beta}$$\overline{K}_{\beta}^{*}$$A\in\overline{A}^{+}$$K\in \mathcal{E}$

$iJ$

(3.2)

If

$B\subset \mathcal{E}^{n}$with$B\subset K$$||U_{A}1_{B\cup K^{c}}||_{\infty}\leq\beta$and$\nu(B)<\delta$, then(resp. $||E.(A_{\tau(B\cup K^{c})-}||_{qe}\leq\beta)$ .

,Remarks. For andis excessive for $X$ restricted to.
and hence $qfc$ . But $E.(A_{\tau(D)-})\leq E$ $\int 1_{D}(X_{t})dA_{t}=U_{A}1_{D}$ and. Replacing by
therefore
one may suppose that is finite
when convenient.
Once again to keep the exposition simple we are going to eliminate
the exceptional sets.
$D\in \mathcal{E}^{n}$$U_{A}1_{D}$$E\backslash N_{A}$$\overline{K}_{\beta}\subset\overline{K}_{\beta}^{*}$$iJ$$\iota/|_{K}$$\iota/$97

Gauge theorems for Stieltjes exponentials) provided(3.3) Definition. If $ 0<\beta<\infty$ , then $A\in K_{\beta}$ (resp.
$A\in A^{+}$ and has bounded jumps as in (2.8) and there exist, $K$ andas in (3.1) such that $B\subset K$ , $\nu(B)<\delta$ imply$K_{\beta}^{*}$$\delta$$\nu$(resp.

$\sup_{x\in E}U_{A}1_{B\cup K^{c}}(x)\leq\beta$$\sup_{x\in E}E^{x}(A_{\tau(B\cup K^{c})-})\leq\beta$).

We are going to work under (3.3) and leave the straightforward extension to the more general situation to the interested reader. Obviously. It is convenient to define
if $ 0<\beta<\gamma$ , thenand$K_{\beta}\subset K_{\gamma}$(3.4)$K_{\beta}^{*}\subset K_{\gamma}^{*}$$K_{0}:=\bigcap_{\beta>0}K_{\beta}$

and$K_{0}^{*}:=\bigcap_{\beta>0}K_{\beta}^{*}$.

It will turn out that theare the appropriate classes for the gauge
theorem. Moreover in an important special case a suﬃcient conditionis the dual predictable prois that
that
where
jection of $A$ . We now describe this result. Let $A\in A^{+}$ have boundedjumps. Then there exists a unique predictable element $A^{p}\in A^{+}-$ thedual predictable projection of $A-$ such that for any positive predictableprocess$K_{\beta}^{*}$$A^{p}$$A^{p}\in K_{\beta}^{*}$$A\in K_{\beta}^{*}$$(Z_{t})$(3.5)$E^{x}\int_{0}^{\infty}Z_{t}dA_{t}=E^{x}\int_{0}^{\infty}Z_{t}dA_{t}^{p}$.

See [S88, 31].
(3.6) Proposition. Let $A\in A^{+}$ have bounded jumps with bound(2.8). If the dual predictable projectionof $A$ is continuous,.
if and only if$A^{p}$

$c$

as in

$A^{p}\in K_{\beta}^{*}$$A\in K_{\beta}^{*}$

Proof. If $T$ is a stopping time, $1_{]0,T]}(t)$ is predictable. Therefore since
is continuous$A^{p}$$E^{x}(A_{T})-c\leq E^{x}(A_{T-})\leq E^{x}(A_{T})=E^{x}(A_{T}^{p})=E^{x}(A_{T-}^{p})$,and this establishes (3.6).

$\blacksquare$

The next two propositions are taken from [C02]. We give proofs forthe convenience of the reader. For $A\in A^{+}$ , $u_{A}:=U_{A}1=E.(A_{\zeta-})$ is
the potential (function) of $A$ .
(3.7) Proposition. Suppose$A\in K_{\beta}$, $\beta>0$ . Then $\sup_{x\in E}u_{A}(x)<\infty$ .

and be as in Definition 3.3. Then $K$ contains
Proof. Let ,at most a finite number of points
with $\nu(\{x_{j}\})>\delta$ . It
follows ﬀom a result of Saks (see [DS58, p308]) thatcanbe written as the disjoint union of a finite number,,of sets$\nu$$K\in \mathcal{E}$$\delta$$\{x_{1}, \ldots, x_{n}\}$$K\backslash \{x_{1}, \ldots x_{n}\}$$B_{1}$$\ldots$$B_{k}$

98

R. Getoor. Rom (2.8) there exists
withand$U_{A}1_{G_{j}}\leq(2j+1)c$ for each
. Let $F=\{x_{1}, \ldots, x_{n}\}$ . Since $E=\cup G_{j}$ ,there exists an with
. Henceinwith$\mathcal{E}$$\nu(B_{j})\leq\delta$$(G_{j})$$G_{j}\uparrow E$

$j$

$p$

$1_{F}\leq 1_{G_{\ell}}$$u_{A}=U_{A}1_{K^{c}}+U_{A}1_{F}+\sum_{j=1}^{k}U_{A}1_{B_{j}}\leq(k+1)\beta+(2\ell+1)c$.

$\blacksquare$

depends on what appears to be an arbitrary
The definition ofofmeasure.
The
choice thenext proposition gives an intrinsic criterion, at least when $m$ is a reference measure. If :,for $A$ to be in
let $||f||=\sup_{x\in E}|f(x)|$ .
$K_{\beta}$$\nu$$E\rightarrow\overline{\mathbb{R}}$$f$$K_{\beta}$(3.8) Proposition. Le $A\in A^{+}$ and $\beta>0$ . (i) If $A\in K_{\beta}$ , then for.
every decreasing sequence $(Dn)\subset \mathcal{E}^{n}with\cap D_{n}=\phi$ ,$\lim_{n}||U_{A}1_{D_{n}}||\leq\beta$(ii) If $m$ is a reference measure and if for every decreasing sequence$(Dn)\subset \mathcal{E}^{n}with\cap D_{n}=\phi$ ,and $A$ has bounded jumps,.
then$\lim_{n}||U_{A}1_{D_{n}}||<\frac{\beta}{2}$$A\in K_{\beta}$(i) Suppose $\beta>0$ and. Let. Then there exists
ansuch that $\nu(K\cap D_{n})\leq\delta$ for $n\geq N$ . Thus for $\geq N$ ,and sofor $n\geq N$ . (ii) Let $A\in A^{+}$ havebounded jumps. By (2.8) there exists an increasing sequence $(G_{n})$ offinely open sets inwith $\mu_{A}(G_{n})<\infty and\cup G_{n}=E$ . Let.
$Then\cap D_{n}=\phi$ and so. Fix anwithand put $K=G_{n}$ . We claim that there exists a $\delta>0$ such that if$B\subset K$ and $\mu_{A}(B)<\delta$ , then
. Suppose no such $\delta>0$
exists. Then for each
there exists $B_{n}\subset K$ with $\mu_{A}(B_{n})\leq 2^{-n-1}$. Let. Thenandis a decreasing$\mu_{A}(F_{n})\leq
2^{-n}$$F:=\cap
F_{n}$$\mu_{A}(F)=0$. If. Hencesequence with, then$0=\mu_{A}(F)=\uparrow\lim qmU_{A}^{q}1_{F}$ and so $mU_{A}^{q}1_{F}=0$ for $q>0$ . Letting
Proof,$A\in K_{\beta}$$ D_{n}\downarrow\phi$$N$$ U_{A}1_{D_{n}}\leq$$n$$||U_{A}1_{D_{n}}||\leq\beta$$U_{A}1_{(D_{n}\cap K)\cup K^{c}}$$\mathcal{E}^{n}$$D_{n}=E\backslash G_{n}$$\lim_{n}||U_{A}1_{D_{n}}||<\frac{\beta}{2}$$||U_{A}1_{D_{n}}||<\frac{\beta}{2}$$n$$||U_{A}1_{B\cup K^{c}}||\leq\beta$$n$$||U_{A}1_{B_{n}\cup K^{c}}||>\beta$$(F_{n})$$F_{n}=\bigcup_{k\geq n}B_{k}$$ q\rightarrow\infty$

we see that $U_{A}1_{F}=0a.e$ . $m$ and thus everywhere sinceisexcessive and $m$ is a reference measure. Consequently, $\lim_{n}||U_{A}1_{F_{n}}||<Q2$ . Choose with $||U_{A}1_{F_{n}}||<\rho 2$ .
and sinceNowso$U_{A}1_{F}$$q\downarrow 0$$U_{A}1_{F_{n}}=U_{A}1_{F_{n}\backslash F}$$ F_{n}\backslash F\downarrow\phi$$n$$B_{n}\subset F_{n}$$\beta<||U_{A}1_{F_{n}\cup K^{c}}||\leq||U_{A}1_{F_{n}}||+||U_{A}1_{K^{c}}||<\beta$

and this contradiction establishes (3.8).

$\blacksquare$

Remarks. We emphasize that the measure constructed in the proof of(3.8) is $\nu=\mu_{A}$ . The only place in the proof that the fact that $m$ is a

99

Gauge theorems for Stieltjes exponentials

reference measure is used is concluding that $U_{A}1_{F}=0$ ﬀom $U_{A}1_{F}=0$$a.e$ . and of course then $q.e$ .
Consequently the proof is easily adaptedusing (2.10) in place of (2.8) to show:(3.9) Proposition. Let
have bounded jumps as in (2.10). Iffor
every decreasing sequence $(D_{n})\subset \mathcal{E}^{n}with\cap D_{n}=\phi$ ,$A\in\overline{A}^{+}$$\lim_{n}||U_{A}1_{D_{n}}||_{\infty}<$$Q2$, then$A\in\overline{K}_{\beta}$.

(3.10) Proposition. Suppose(resp.

$A^{1}+A^{2}\in K_{\beta_{1}+\beta_{2}}$$A^{j}\in K_{\beta_{j}}$$K_{\beta_{1}+\beta_{2}}^{*}$(resp.

$K_{\beta_{j}}^{*}$) for $j=1,2$ . Then).

andserve for $A^{j}\in K_{\beta_{j}},j=1,2$ . We may suppose
Proof. Let ,thatis carried by $K_{j},j=1,2$ . Define $\nu=\nu_{1}+\nu_{2}$ , $K=K_{1}\cup K_{2}$ and. Then$\nu_{j}$$K_{j}$$\delta_{j}$$\nu_{j}$$\delta=\delta_{1}\wedge\delta_{2}$$\nu(K)$$=$$\nu(K_{1}\cap K_{2}^{c})+\nu(K_{1}^{c}\cap K_{2})+\nu(K_{1}\cap K_{2})$$\leq 2[\nu_{1}(K_{1})+\nu_{2}(K_{2})]<\infty$.

Suppose $B\subset K$ with $\nu(B)<\delta$ . Thenthat $B\cup K^{c}\subset(B\cap K_{j})\cup K_{j}^{c}$ for $j=1,2$ and so$\nu_{j}(B)\leq\delta_{j}$for $j=1,2$ . Note

$U_{A_{1}+A_{2}}1_{B\cup K^{c}}\leq\sum_{j=1}^{2}U_{A_{j}}1_{(B\cap K_{j})\cup K_{j}^{c}}\leq\beta_{1}+\beta_{2}$.

Of course $A_{1}+A_{2}$ has bounded jumps. The same argument works when.
theare replaced by$\blacksquare$$K_{\beta}$\S 4.

$K_{\beta}^{*}$

Gauge Theorems

Gauge theorems are usually stated for fluctuating additive functionals. Formally let $A:=A^{+}-A^{-}$ and introduce the obvious notion of
for $j=1,2$ , then $A_{1}=A_{2}$ proequality: If $A_{j}=A_{j}^{+}-A_{j}^{-}$ ,. Then it is known that $A\in A$ canvided $A_{1}^{+}+A_{2}^{-}=A_{1}^{-}+A_{2}^{+}$ in, $A^{-}\in A^{+}$ having a combe written uniquely as $A=A^{+}-A^{-}$ withandmon defining set
and such that the measures. Actually the only thing that we[ are orthogonal foron [0,
have no common discontinuities. Of
shall use is that $a.s.$ ,andcourse $A\in A$ can be decomposed as $A=A^{c}+A^{d}$ where $A^{c}\in A$ iscontinuous and $A^{d}\in A$ is purely discontinuous,$A_{j}^{\pm}\in A^{+}$$A^{+}$$A^{+}$$dA_{t}^{+}(\omega)$$\Omega_{A}$$\omega\in\Omega_{A}$$\zeta(\omega)$$A^{+}$$A^{-}$$A_{t}^{d}=\sum_{0<s\leq t}\triangle A_{s}=\sum_{0<s\leq t}\triangle A_{s}^{+}-\sum_{0<s\leq t}\triangle A_{s}^{-}$$dA_{t}^{-}(\omega)$

R. Getoor

100

with

$\sum_{0<s\leq t}|\triangle A_{s}|<\infty$

if $ t<\zeta$ . In particular$t\rightarrow A_{t}$is of boundedvariation on compact intervals in [0, $\zeta[a.s$ . Fix $A=A^{+}-A^{-}\in A$ .
Define for $ t<\zeta$ ,$L_{t}^{+}:=Exp(A_{t}^{+})$, $L_{t}^{-}:=Exp(A_{t}^{-})$ .

It turns out that the appropriate multiplicative functional to consider is(4.1)$L_{t}:=L_{t}^{+}/L_{t}^{-}=e^{A_{t}^{c}}\prod_{0<s\leq t}\frac{(1+\triangle A_{s}^{+})}{(1+\triangle A_{s}^{-})}$.

.
Clearly $a.s.$ ,is increasing and finite on [0, [ andHenceand sowhere $\infty/\infty=0$ by convention.
is right continuous on [0, $\zeta[and$ is of bounded variation
Moreover$a.s$ .
on compact subintervals of [0,Henceforth we shall omit the”qualifierin places where it is obviously required such as in thepreceding two sentences. Note that $L_{0}=L_{0+}=1$ . The function$t\rightarrow L_{t}^{\pm}$$\zeta$$L_{\zeta-}^{\pm}=L_{\zeta}^{\pm}$$\triangle A_{\zeta}^{\pm}=0$$L_{\zeta-}=L_{\zeta}$$t\rightarrow L_{t}$$\zeta[,$$\zeta‘ a.s$

(4.2)

$g(x):=E^{x}[L_{\zeta}]=E^{x}[L_{\zeta-}]$is called the gauge of $A$ (or

$L$

).

(4.3) Proposition. The gauge is nearly Borel measurable and fifinely
continuous. If $F:=\{g<\infty\}$ , then $F$ is absorbing.
$g$Remark. Since
map ﬀom $E$ to$g$may take the value$[0, \infty]$$+\infty$,$g$

is finely continuous as a.

Proof. Since, is universally measurable and because
is amultiplicative functional,
if $ t<\zeta$ . Let$M_{t}=(L_{t}^{-})^{-1}$ . Then $M$ is a decreasing, right continuous multiplicative
functional that is strictly positive for $ t<\zeta$ . Now$L_{\zeta}\in \mathcal{F}$

$L$

$g$

$L_{\zeta}\circ\theta_{t}=L_{\zeta-}\circ\theta_{t}=L_{\zeta-}/L_{t}$(4.4)$E^{x}[g(X_{t})M_{t}]$$=$$E^{x}(M_{t}L_{\zeta-}/L_{t} ; t<\zeta)$$=$$E^{x}[L_{\zeta-}/L_{t}^{+}; t<\zeta]\uparrow g(x)$. Therefore is excessive relative to $(X, M)-$ the $M$ -subprocessasof $X-$ and hence is nearly Borel and finely continuous. In particular$F=\{g<\infty\}$ is finely open and nearly Borel. The computation in (4.4),, holds with replaced by a stopping
aside ﬀom taking the limit astime $T$ . Hence $Ex(MTg(XT))\leq g\{x$ ) for any stopping time $T$ . Let$D=\{g=\infty\}=E\backslash F$ . Then $D$ is finely closed and so $ g\circ X_{T(D)}=\infty$. on $\{T(D)<\zeta\}$ where $T(D):=\inf\{t>0 : X_{t}\in D\}$ is the hitting
time of $D$ . Hence if $x\in F$$t\downarrow 0$

$g$

$g$

$t\downarrow 0$

$t$

$a.s$

$\infty>g(x)\geq E^{x}(M_{T(D)}goX_{T(D);}T(D)<\zeta)$

Gauge theorems for Stieltjes exponentials

101and since $M_{T(D)}>0$ on $\{T(D)<\zeta\}$ , this forces $P^{x}(T(D)<\zeta)=0$ .
Hence $F$ is absorbing.
$\blacksquare$

Of course $g(x)\leq E^{x}(L_{\zeta}^{+})$ and so we may obtain bounds on by. Thus in what follows we are going to assume that
estimating$A\in A^{+}$ . For $A\in A^{+}$ let $c(A)$ denote the infimum of thesuch that$A\inA^{+}$. Thenhas bounded jumps provided $c(A)<$. We come now to the main result of this section. The proof isborrowed ﬀom Chen and Song [CS02].
$g$$E^{x}(L_{\zeta}^{+})$$c$$\sup_{t<\zeta}\triangle A_{t}\leq ca.s$$\infty$(4.5) Theorem. Suppose
is bounded on $\{g<\infty\}$ .

$A\in K_{\beta}^{*}$

with $\beta<1$ . Then the gauge $g(x)=$$E^{x}(L_{\zeta})$. Since $\nu(K)<\infty$
Proof. Let , $K$ and be as in (3.3) for
we may choose $M$ large enough that $\nu(K\cap\{M<g<\infty\})<\delta$ . Let$B:=K^{c}\cup\{M<g<\infty\}=K^{c}\cup(K\cap\{M<g<\infty\})$ . Then
E. $(A_{\tau(B)-})\leq\beta$ . Consequently by (2.7)$\delta$$\nu$$A\in K_{\beta}^{*}$(4.6)

E. $(L_{\tau(B)-})\leq\gamma:=(1-\beta)^{-1}<\infty$ .

Fix an . Then$x$$g(x)$$=$$E^{x}[L_{\tau(B)-;}\tau(B)=\zeta]+E^{x}[L_{\zeta-;}\tau(B)<\zeta]$$\leq$$\gamma+E^{x}[L_{\tau(B)}g(X_{\tau(B)});\tau(B)<\zeta]$.

Let $F=\{g<\infty\}$ and suppose that $x\in F$ . But $F$ is absorbing and so$g(X_{t})<\infty a.s$ .
on [0, [. Hence $g\circ X_{\tau(B)}\leq M$ on $\{\tau(B)<\zeta\}a.s$ .
since is finely continuous. Therefore $g(x)\leq\gamma+ME^{x}[L_{\tau(B)} ; \tau(B)<\zeta]$
on $F$ . Since $a.s.$ $L_{\tau(B)}\leq(1+c)L_{\tau(B)-}$ where $c=c(A)$ , we see that$g\leq\gamma+M\gamma(1+c)$ on $F=\{g<\infty\}$ .
$P^{x}$$P^{x}$$\zeta$$g$)$\blacksquare$with $\beta<1$ , thenRemark. If one only assumes that
. It follows that is finely continuous on
is only defined on
and is bounded on $\{g<\infty\}\cap(E\backslash N_{A})$ . Under the hypotheses of (4.5),$\{g=\infty\}=\{g>M\}$ where $M$ $=\sup\{g(x) : g(x)<\infty\}$ . Thus both$\{g<\infty\}$ and $\{g=\infty\}$ are finely open. Therefore if $E$ cannot be written
as the disjoint union of two finely open nearly Borel sets one of which isabsorbing, in particular if $E$ is finely connected, then is either bounded
or identically infinite. This is the classical form of a gauge theorem.
$(4.7)$$A\in\overline{K}_{\beta}^{*}$$E\backslash N_{A}$$g$$E\backslash N_{A}$

$g$

$g$

$g$

(4.8) Corollary. Let $A\in A^{+}$ and defifine$B_{t}:=A_{t}^{c}+\sum_{0<s\leq t}(e^{\triangle A_{s}}-1)$.

Then $B\in A^{+}$ . If
for some $\beta<1$ , then $g_{A}(x):=E^{x}(e^{A_{\zeta}})$ isfifinely continuous and bounded on $\{g_{A}<\infty\}$ .
$B\in K_{\beta}^{*}$

102

R. Getoor

Proof. Sinceon [0, [ onit is clear that B$\in A^{+}$and has the same defining set as $A$ . Now (4.8) is evident because$e^{A_{t}}=$$Exp(B_{t})$$\zeta$$\sum_{0<s\leq t}\triangle A_{s}<\infty$$\Omega_{A}$.

$\blacksquare$

If $ A=A^{+}-A^{-}\in$ $A$ and one defines $B_{t}^{\pm}:=A_{t}^{\pm c}+$, then $B_{t}:=B_{t}^{+}-B_{t}^{-}\in A$ and $e^{A_{t}}=Exp(B_{t}^{+})/Exp(B_{t}^{-})$ .

Remark.

$\sum_{0<s\leq t}(e^{\triangle A_{s}^{\pm}}-1)$

Therefore $g_{A}(x):=E^{x}(e^{A_{\zeta}})$ is nearly Borel, finely continuous and $\{g_{A}<$is absorbing. Herewith $\infty\cdot 0=0$ as customary.
Suppose $A\in A^{+}$ is purely discontinuous $(A=A^{d})$ and all of its
jumps are totally inaccessible. In this $cxe$ the dual predictable projecof $A$ has an especially nice form which we now describe. Let
tion$e^{A_{\zeta}}=e^{A_{\zeta}^{+}}e^{-A_{\zeta}^{-}}$$\infty\}$$A^{p}$(4.9)$J:=\{(t, \omega) :X_{t-}(\omega)\neq X_{t}(\omega), X_{t-}(\omega)\in E\}$be the set of totally inaccessible discontinuities of $X$ . Heredenotesthe left limit in the Ray topology. A L\’evy system $(N, H)$ for $X$ consistsof a kernel $N=N(x, dy)$ on $E$ with $N(x, \{x\})=0$ and a PCAF, $H$ ,with empty exceptional set and bounded one potential such that if $ F\in$with $F\geq 0$ and $Z=(Z_{t})$ with $Z_{t}\geq 0$ is predictable, then$X_{t-}$$(\mathcal{E}^{n}\otimes \mathcal{E}^{n})$(4.10)$E^{x}\sum_{s\in J}Z_{s}F(X_{s-}, X_{s})=E^{x}\int_{0}^{\infty}Z_{t}NF(X_{t})dH_{t}$where $NF(x)=\int F(x, y)N(x, dy)$ . If $A\in A^{+}$ is purely discontinuouswith totally inaccessible jumps, then there exists such an $F$ vanishingon the diagonal such that
(4.11)$A_{t}=A_{t}^{F}=\sum_{s\in J,s\leq t}F(X_{s-}, X_{s})$.

See \S 73 of [S88]. If $X$ is a special standard process andis the left
$E$
limit in the original topology of , thenandare indistinguishmay be replaced byable on [0, [ and soin (4.11) and is
automatically in when $X_{s-}(=X_{s-}^{0})\neq X_{s}$ , $ s<\zeta$ . See [S88, (47.10)].
Moreover if $A=A^{F}$ then$X_{t-}^{0}$$X_{t-}$$X_{s-}$$\zeta$$X_{t-}^{0}$$X_{s-}^{0}$

$s$

$J$

$(NF*H)_{t}:=\int_{0}^{t}NF(X_{s})dH_{s}$is the dual optional projection of. Suppose $F$ is bounded. Since$NF*H$ is continuous, (3.6) implies thatwhenever $ NF*H\in$. In particular if $NF*H\in K_{\beta}$ . The next proposition treats animportant special case. It is the case considered in [C02]. It is animmediate consequence of (3.6), (3.10), (4.5) and (4.8).
$A^{F}$$A^{F}\in K_{\beta}^{*}$$K_{\beta}^{*}$

103

Gauge theorems for Stieltjes exponentials(4.12) Proposition. Let $A\in A^{+}$ have the form $A=A^{c}+A^{F}$ wherewith $F$ bounded. If
is as inis continuous andand $NF*H\in K_{\gamma}^{*}$ with $\beta+\gamma<1$ , then is bounded on $\{g<\infty\}$ where. If $G=e^{F}-1$ and $NG*H\in K_{\gamma}^{*}$ with $\beta+\gamma<1$ ,is defifined inis defifined (4-8).
is bounded on $\{g_{A}<\infty\}$ where
then$A^{F}$$A^{c}$$(4\cdot 11)$$A^{c}\in K_{\beta}^{*}$$g$$(4\cdot 2)$$g$$g_{A}$$g_{A}$\S 5.

Additional Conditions for the Gauge to be Bounded

In this section we develop some conditions that are equivalent to theboundedness of the gauge function. We follow a well-trodden path thatwas originally broken by Chung and Rao [CR88] and explored further byChen and Song and Chen in their papers. The direct results using theStieltjes exponential appear to be new. In what follows $ A=A^{+}-A^{-}\in$$A$ as in the beginning of section 4 andis defined in (4.1). The gauge
is defined in (4.2). We begin with the following proposition which isthe part of Lemma 9 in [CR88] and Lemma 7 in [C02] that carries overto the present situation.
$L_{t}$$g$(5.1) Proposition. Suppose that$a.s$ . and thatis bounded. Let

for $\beta<1$ with. Defifine$A^{+}\in K_{\beta}^{*}$$E.(A_{\zeta}^{-})$$\epsilon>0$$\tau_{n}:=\inf\{t:A_{t}^{+}>n\epsilon\}$$ A_{\zeta}^{+}<\infty$, $n\geq 1$

where as usual the infifimum of the empty set isbounded, then$+\infty$.

If the gauge

$g$

is

$\lim_{n}\sup_{x}E^{x}[L_{\tau_{n} ^{;} }\tau_{n}<\zeta]=0$, then E.
Remark. If
soa.s. in this case.
$A^{+}\in K_{\beta}$$(A_{\zeta}^{+})$

is bounded according to (3.7) and$ A_{\zeta}^{+}<\infty$,, we shall give it for
Proof. Since the proof is the same for all$K$which is the only case used later. Let , , be as in the requirement. Thus E.
when $B\subset K$ with $\nu(B)<\delta$ .
that. Here and in the remainder
Since
it follows that”
where it is obviously required.
of this section we omit the qualifier
Therefore by Egorov’s theorem, since $ E^{x}(L_{\zeta})<\infty$ ,almost uniformly on $K$ with respect to . Recall $\nu(K)<\infty$ . Henceand angiven
there exists a closed set $D\subset K$ with
$N$ such that if $n\geq N$ , then$\epsilon=1$$\epsilon>0$$\delta$$\nu$$A^{+}\in K_{\beta}^{*}$$ A_{\zeta}^{+}<\infty$$[A_{\tau(B\cup K^{c})-}^{+}]\leq\beta$$\{\tau_{n}<\zeta\}\downarrow\phi$$‘‘ a.s.$$E^{x}[L_{\zeta}; \tau_{n}<\zeta]\downarrow 0$$\nu$$\epsilon>0$(5.2)$\nu(K\backslash D)<\delta$$\sup_{x\in D}E^{x}[L_{\zeta;}\tau_{n}<\zeta]<\epsilon$

.

104

R. Getoor

Now $D^{c}=K^{c}\cup K\backslash D$ and $\tau(D^{c})=T(D)\wedge\zeta$ where $T(D)$ is the hitting
iftime of $D$ . But
and so $ E.(A_{T(D)-}^{+})=E.(A_{\tau(D^{c})-}^{+})\leq$$\beta<1$ . Thus by (2.7)$A_{t}^{+}=A_{\zeta-}^{+}$$ t\geq\zeta$

E. $[L_{T(D)-}^{+}]\leq(1-\beta)^{-1}<\infty$ .

(5.3)

Define $B_{t}=A_{t}^{+c}+\sum_{0<s\leq t}\log(1+\triangle A_{s}^{+})$ . Note that$e^{B_{t}}=L_{t}^{+}$and soSince, and hence, $B$ has boundedjumps, $\sup_{x}E^{x}[\exp(B_{T(D)})]<\infty$ . Consequently it follows ﬀom Corollary 4.2 in [SSOO] that there exists $p>1$ with $\sup_{x}E^{x}[\exp(pB_{T(D)})]<$. Finally $e^{pB_{t}}=[Exp(A_{t}^{+})]^{p}=(L_{t}^{+})^{p}$ and so$E^{x}[\exp(B_{T(D)-})]\leq(1-\beta)^{-1}$ .

$A^{+}$$\infty$(5.4)$\sup_{x}E^{x}[(L_{T(D)}^{+})^{p}]=M^{J}<\infty$.

$>\max(N, c(A^{+}))$ where $c(A^{+})$ is defined in the paragraphLet
above (4.5). Then$n$

$E^{x}[L_{\zeta} ; \tau_{3n}<\zeta]=E^{x}[L_{\zeta} ; \tau_{n}\leq T(D), \tau_{3n}<\zeta]$$+E^{x}[L_{\zeta}; T(D)<\tau_{n}, \tau_{3n}<\zeta]=I+II$.

If $T(D)<\tau_{n}$ , then $T(D)+\tau_{n^{0}}\theta_{T(D)}\leq\tau_{3n}$ sinceandon. Therefore writing $T_{D}=T(D)$ when convenient,$\sup_{s}\triangle A_{s}^{+}<n$$\tau_{n}<\tau_{3n}$$\{\tau_{3n}<\zeta\}$

(5.5)

$=$

$E^{x}[L_{\zeta}; T(D)<\tau_{n}<\tau_{3n}<\zeta]$$\leq$$E^{x}[L_{T(D)}L_{\zeta^{0}}\theta_{T(D);}T(D)+\tau_{n^{0}}\theta_{T(D)}<\zeta_{)}T(D)<\zeta]$$=$$E^{x}[L_{T(D)}E^{X(T_{D})}[L_{\zeta}; \tau_{n}<\zeta];T(D)<\zeta]$

$II$

.

Noting thateven if $ T(D)\geq\zeta$ because
in that case, and that $L_{T(D)}\leq L_{T(D)}^{+}$ one obtains ﬀom (5.4),$L_{\zeta}=L_{T(D)}L_{\zeta^{0}}\theta_{T(D)}$$I$$\zeta\circ\theta_{T(D)}=0$$\leq||g||E^{x}[L_{T(D)}^{+ }\leq T(D), ]\leq||g||E^{x}[(L_{T(D)}^{+})^{p}]^{1/p}P^{x}[\tau_{n}\leq T(D)]^{1/q}$

where $1/p+1/q=1$ . Moreover $\{\tau_{n}\leq T(D)\}\subset\{A_{T(D)}^{+}\geq n\}$ and so$P^{x}[\tau_{n}\leq T(D)]\leq\frac{1}{n}[E^{x}[A_{T(D)-}^{+}]+c(A^{+})]\leq\frac{1}{n}[\beta+c(A^{+})]$.

Consequently I approaches zero uniformly in. On theas$D$otherhand $X(T_{D})\in D$ on $\{T(D)<\zeta\}$ since
is closed. Thus from
(5.2), (5.3) and (5.5)
$x$

$n$

$II$ $\leq\epsilon E^{x}[L_{T(D)}^{+}]\leq\epsilon(1-\beta)^{-1}[1+c(A^{+})]$$\rightarrow\infty$.

Gauge theorems for Stieltjes exponentials

105

Combining these estimates we find that
(5.6)$\lim_{n}\sup_{x}E^{x}[L_{\zeta;}\tau_{n}<\zeta]=0$

Next observe that
Jensen’s inequality implies that.

. Therefore if $c=\sup_{x}E^{x}(A_{\zeta}^{-})$ ,$E^{x}(L_{\zeta})\geq e^{-c}$ . Hence$L_{\zeta}\geq(L_{\zeta}^{-})^{-1}\geq e^{-A_{\zeta}^{-}}$$E^{x}[L_{\tau_{n}} ; \tau_{n}<(]\leq e^{c}E^{x}[L_{\tau_{n}}E^{X(\tau_{n})}(L_{\zeta});\tau_{n}<\zeta]=e^{c}E^{x}[L_{\zeta}; \tau_{n}<\zeta]$

and combining this with (5.6) completes the proof of Proposition 5.1$\blacksquare$

We come now to the main result of this section. This should becompared with Corollary 2.16 in [C02].

satisfy the hypotheses of (5.1) and(5.7) Theorem. Let
andsuppose in addition thathas bounded jumps. Then the following are
equivalent:(i) $\sup_{x}E^{x}[L_{\zeta}]<\infty$ ;
;
(ii)
;
(ii)
(iv) $\sup_{x}E^{x}[\sup_{t<\zeta}L_{t}]<\infty j$$A^{+}$$A^{-}$$A^{-}$$\sup_{x}E^{x}\int_{0}^{\zeta}L_{t-}dA_{t}^{+}<\infty$$\sup_{x}E^{x}\int_{0}^{\zeta}L_{t}dA_{t}^{+}<\infty$

Proof. Since $L_{t}=L_{t-}(1+\triangle A_{t}^{+})(1+\triangle A_{t}^{-})^{-1}\leq[1+c(A^{+})]L_{t-}$ and$L_{t-}\leq[1+c(A^{-})]L_{t}$ , the equivalence of (ii) and (iii) is clear.
Also
and so$dL_{t}^{+}=L_{t-}^{+}dA_{t}^{+}$$\int_{[0,t[}L_{s-}dA_{s}^{+}=\int_{[0,t[}(L_{s-}^{-})^{-1}dL_{s}^{+}\geq(L_{t-}^{-})^{-1}[L_{t-}^{+}-1]\geq[L_{t-}-1]$

for$ t\leq\zeta$$t\in[0,$$\zeta[$. Taking $ t=\zeta$ , (ii) implies (i). Also taking the supremum over

,

(5.8)

$\sup_{t<\zeta}L_{t}=\sup_{t<\zeta}L_{t-}\leq 1+\int_{[0,\zeta[}L_{t-}dA_{t}^{+}$,and so (ii) implies (iv). Clearly (iv) implies (i). Thus it suﬃces to showthat (i) implies (ii) to complete the proof of (5.7). Therefore suppose
choose $N>c(A^{+})$ such thatthat (i) holds. Using (5.1) with$\epsilon=1$$\lambda$$:=\sup_{x}E^{x}[L_{\tau_{N} ^{;} }\tau_{N}<\zeta]<1$

Defineandstrong Markov property,
and hence$\tau_{N}^{0}=0$$\tau_{kN}\leq\tau_{N}^{k}$.

for $k\geq 0$ . Then using the. We claim that
. This is obvious when $k=1$ . Assume$\tau_{N}^{k+1}=\tau_{N}^{k}+\tau_{N_{\mathcal{T}_{N}}^{0\theta k}}$$\sup_{x}E^{x}[L_{\tau_{N}^{k}} ; \tau_{N}^{k}<\zeta]\leq\lambda^{k}$$\lim_{k}\tau_{N}^{k}\geq\zeta$

R. Getoor

106

that it holds for a fixed $k\geq 0$ . Writing $A^{+}(t)=A_{t}^{+}$ for typographicalsimplicity we have$\tau_{N}^{k+1}=\tau_{N}^{k}+\sup\{t:A^{+}(\tau_{N}^{k}+t)-A^{+}(\tau_{N}^{k})>N\}$, $A^{+}(\tau_{N}^{k})\geq kN$ and soIfclaim. Now$\tau_{N}^{k+1}\geq\tau_{(k+1)N}$$\tau_{kN}\leq\tau_{N}^{k}$.

. This establishes the$E^{x}\int_{0}^{\zeta}L_{t-}dA_{t}^{+}=\sum_{k\geq 0}E^{x}\int_{[\tau_{N}^{k},\tau_{N}^{k+1}[}L_{t-}dA_{t}^{+};$$\tau_{N}^{k}<\zeta]$$=\sum_{k\geq 0}E^{x}[L_{\tau_{N}^{k}}E^{X(\tau_{N}^{k})}\int_{[0,\tau_{N[}}L_{t-}dA_{t}^{+}; \tau_{N}^{k}<\zeta]$.

But$E^{x}\int_{[0,\tau_{N[}}L_{t-}dA_{t}^{+}\leq E^{x}\int_{[0,\tau_{N[}}L_{t-}^{+}dA_{t}^{+}$$=E^{x}[L_{\tau_{N-}}^{+}-1]\leq E^{x}e^{A^{+}(\tau_{N}-)}\leq e^{N}$,and combining these estimates we obtain$E^{x}\int_{0}^{\zeta}L_{t-}dA_{t}^{+}\leq e^{N}\sum_{k\geq 0}E^{x}[L_{\tau_{N}^{k }}\tau_{N}^{k}<\zeta]\leq e^{N}(1-\lambda)^{-1}$.

Hence (i) implies (ii) establishing (5.7).

$\blacksquare$

Remark. Suppose in addition to the hypotheses in (5.7), that $E$
can not be written as the disjoint union of two finely open sets one ofwhich is absorbing as in (4.7). Then the condition (5.7-i) is equivalent tofor at least one $x\in E$ . But in view of (5.8), the remaining
conditions in (5.7) are equivalent to the corresponding condition withreplaced by for at least one $x\in E$ .
$(5.9)$$ E^{x}(L_{\zeta})<\infty$$\sup_{x}$We next give a suﬃcient condition that the gauge is bounded.
The integral condition in the following result should be compared to theconditions in Theorem 5.7.
$g$

(5.10) Theorem. Let$a.s$with $\beta<1$ and suppose that, then is bounded.

$A^{+}\in K_{\beta}$. If $\sup_{x}E^{x}\int_{0}^{\zeta}L_{t}dt<\infty$$\zeta<\infty$$g$

In the course of the proof we shall need the following lemma.

(5.11) Lemma. Let$A\in K_{\beta}$

with$\beta<\infty$

If $\beta<1$ , then there exist $ C<\infty$ and$Ce^{t\lambda}$.

. Then$\lambda>0$$\lim_{t\downarrow 0}\sup_{x}E^{x}(A_{t})\leq\beta$.

such that $\sup_{x}E^{x}[L_{t}]\leq$107

Gauge theorems for Stieltjes exponentials

Proof. Proposition 2.3 in [C02] asserts that$\lim\sup_{x}E^{x}\int_{0}^{\infty}e^{-\alpha t}dA_{t}\leq$$\alpha\rightarrow\infty$. The proof in [C02] works perfectly well for right continuous $A$ . If$\eta>0$ ,$\beta$$E^{x}\int_{0}^{\infty}e^{-\alpha t}dA_{t}=\alpha E^{x}\int_{0}^{\infty}e^{-\alpha t}A_{t}dt$$=\int_{0}^{\infty}e^{-t}E^{x}(A_{t/\alpha})dt\geq E^{x}[A_{\eta/\alpha}]e^{-\eta}$

This implies that$\lim_{t\rightarrow 0}\sup_{x}E^{x}(A_{t})\leq\beta e^{\eta}$

and letting.

fall to zero yields$\eta$

the first assertion in (5.11). If $\beta<1$ , then it follows from (2.7) that there
exists $t>0$ such that $\sup_{x}E^{x}(L_{t})<\infty$ . Since $Q_{t}f(x):=E^{x}[f(X_{t})L_{t}]$defines a semigroup, it is well-known and easily checked that this implies
the final assertion in (5.11).
We now turn to the proof of (5.10). By (5.11), there exist $ C<\infty$. Since,andsuch that$\blacksquare$$\lambda>0$$g(x)$$E^{x}[L_{t}^{+}]\leq Ce^{\lambda t}$$\zeta<\infty$$=$$\sum_{n\geq 0}E^{x}[L_{\zeta}; n <\zeta\leq n+1]$$=$$\sum_{n\geq 0}E^{x}[L_{n}E^{X(n)}[L_{\zeta}; \zeta\leq 1]; ; n<\zeta]$$\leq$$\sum_{n\geq 0}E^{x}[L_{n}E^{X(n)}[L_{1}^{+}]; ; n <\zeta]\leq Ce^{\lambda}\sum_{n\geq 0}E^{x}[L_{n}; n<\zeta]$

If $n\leq t<n+1$ , then writing.

$c=Ce^{\lambda}$$E^{x}[L_{n+1} ; n+1<\zeta]\leq E^{x}[L_{n+1} ; t<\zeta]$$=E^{x}[L_{t}E^{X(t)}[L_{n+1-t}];t<\zeta]\leq cE^{x}[L_{t}; t<\zeta]$.

Consequently$\sum_{n\geq 0}E^{x}[L_{n}; n<\zeta]\leq 1+cE^{x}\int_{0}^{\zeta}L_{t}dt$

and hence$g$is bounded.

,$\blacksquare$

Remarks. The proof of (5.10) is just the argument on page 831 of[CR88]. Under the hypotheses in the first sentence of (5.10), the proof
implies that $ g(x)<\infty$ . Noteshows that for fixed,that in (5.10) it is not assumed thatis bounded. If one assumes
in addition thatand E.
are bounded, then the proof of The$x$$ E^{x}\int_{0}^{\zeta}L_{t}dt<\infty$$E.(A_{\zeta}^{-})$$E.(\zeta)$$(A_{\zeta}^{-})$

orem 6 in [CR88] may be modified to show that$\sup_{x}E^{x}\int_{0}^{\zeta}L_{t-}dt<\infty$

108

is necessary for

R. Getoor

to be bounded. This requires showing first that$\lim E^{x}[L_{n};n<\zeta]=0$ uniformly in, which may be proved by an
$g$

$x$

$ n\rightarrow\infty$

argument that is similar to, but simpler than, the proof of (5.1). We
leave the details to the interested reader.

References[BG68] R.M. Blumenthal and R. K. Getoor, Markov Processes and Potential
Theory, Academic Press, New York, 1968.
Z-Q. Chen, Gaugeability and conditional gaugeability, Trans. Amer.
[C02]
Math. Soc, 354(2002), 4639-4679.
[CS02] Z-Q. Chen and R. Song, General gauge and conditional gauge theorems, Ann. Probab., 30(2002), 1313-1339.
[CS03] Z-Q. Chen and R. Song, Conditional gauge theorem for nonlocal Feynman-Kac transforms, Probab. Theory Related Fields,
125 (2003), 45-72.
[CR88] K. L. Chung and K. M. Rao, General gauge theorem for multiplicative
functionals, Trans. Amer. Math. Soc, 306(1988), 819-836.
[CZ95] K. L. Chung and Z. Zhao, From Brownian Motion to Schr\"odinger’s
equation, Springer, Berlin, 1995.
[DD70] C. Dol\’eans-Dade, Quelques applications de la formule de changement
de variables pour les semimartingales, Z. Wahrsch, 16(1970), 181194.
[DM87] C. Dellacherie and P. -A. Meyer, Probabilit\’es et Potentiel, Chapitres
XII- XVI, Hermann, Paris, 1987.
[DS58] N. Dunford and J. T. Schwartz, Linear Operators: Part I, Interscience, New York, 1958.
[FG96] P.J. Fitzsimmons and R.K. Getoor, Smooth measures and continuous
additive functionals of right Markov processes. In It\^o’s Stochastic
Calculus and probability Theory, Springer, Tokyo, 1996, pp.31-49.
[G90]
R. K. Getoor, Excessive Measures, Birkh\"auser, Boston, 1990.
[GS84] R. K. Getoor and M. J. Sharpe, Naturality, standardness and weakduality, Z. Wahrsch., 67(1984), 1-62.
[Re70] D. Revuz, Measures aassoci\’ees aux fonctionnelles additives de Markov
I, Trans. Amer. Math. Soc, 148(1970), 501-531.
M. Sharpe, General Theory of Markov Processes, Academic Press,[S88]
Boston, 1988.
[So93] R. Song, Probabilistic approach to the Dirichlet probem of perturbed
stable processes, Prob. Th. Rel. Fields 95(1993), 371-389.
[SS00] W. Stummer and K. -Th Sturm, On exponentials of additive functionals, Stochastic Process. Appl., 85 (2000), 45-60.
[St91] K.-Th. Sturm, Gauge theorems for resolvents with application toMarkov processes, Probab. Theory Related Fields, 89(1991), 387406.

Gauge theorems for Stieltjes exponentials
Department of Mathematics
University of California
La Jolla, CA 92093-0112
U. S. A.
email: rgetoor@ucsd.edu

109Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 111-119

A Frontier of White Noise Analysis,in line with It\^o Calculus
Takeyuki Hida

Dedicated to Professor K. It\^o
Abstract.
This note discusses two topics; one is the notion of the multiple
Wiener integral and the other is the L\’evy-It\^o decomposition of a L\’evy
process.
Both have been taken up by Professor K. It\^o showing the significance in stochastic analysis. The extensive development of thestochastic analysis at present largely depends on these discoveries byhim.
It seems to be a good time to remind his profound ideas and to
discuss some of future directions in probability theory.

AMS Subject Classification: $60H40$\S 1.

Introduction

The White Noise Analysis has extensively developed in the last quarter of a century and now it is the time to be in search of further directionsof research which will be still in line with the It\^o’s original contribution
toward stochastic analysis.
Two directions are proposed in this line.
(1) Analysis of white noise functionals parameterized by a contour
or a surface, namely some kinds of random fields.
ofThose fields are assumed to live in the Hilbert space
white noise functionals (or of Brownian functionals). In orderto analyze those functionals we shall appeal to the variational
calculus for the random fields, and there is requested to introduce a suitable class of generalized white noise functionals. A
good tool from the stochastic analysis for this purpose is thein terms of
direct sum decomposition of the Hilbert space$(L^{2})$$(L^{2})$

Received May 2, 2003.

T. Hida112

the subspaces of multiple Wiener integrals (see [2]) or of those
involving homogeneous chaos.
We then come to introduce classes of generalized white noisefunctionals, where we should note that our generalization canbe done for each subspace given by the decomposition mentioned above. Representations of those functionals can be givenby the so-called -transform which looks like an infinite dimensional analogue of the Laplace transform.
To discuss a variational calculus for random fields in questionsome more probabilistic interpretation is necessary; however
the decomposition of the basic Hilbert space is the milestoneof the further study of random fields formed by white noisefunctionals.
(2) Reduction of random complex systems.
Given a random, evolutional complex system, we first try toobtain the innovation of the system. Under some reasonableconditions, we may assume that the innovation comes ﬀom aL\’evy process. Some concrete results on this topic can be seenin [5].
Then, we are naturally led to the decomposition of the L\’evy
process established by It\^o (see [1]). The decomposition itself
was obtained earlier, by taking a relaxed view of rigor. Whennonlinear functionals of innovation are considered, finer andrigorous results on the decomposition are necessary, so thatthe results in [1] are quite helpful. This fact can be seen assoon as we come to the analysis of nonlinear functionals ofPoisson process (which is an elemental process) or of compound
Poisson noise. It is noted that interesting results are obtainedby viewing dissimilarity to the Gaussian case.
$S$

\S 2.

Topic (1)

Decomposition of the white noise spaceof generLet be the standard white noise measure on a space$L^{2}(E^{*},\mu)=(L^{2})$bethe space of white
alized functions on , and let
is thenoise functionals. One of the basic tools for the analysis on
direct sum decomposition due to It\^o and Wiener:
2.1.

$E^{*}$$\mu$$\mathbb{R}$$(L^{2})$$(L^{2})=\oplus_{r\iota\geq 0}H_{n}$,is the space of multiple Wiener integrals or homogeneous chaos
whereof degree .
$H_{n}$$n$A frontier of white noise analysis, in line with It\^o calculus

113

Another tool is the so-called -transform (see e.g. I. Kubo and S.
Takenaka [8] or [4, Chapt. 2] defined by
$S$

$)$

$(S\varphi)(\xi)=\exp[-\frac{1}{2}||\xi||^{2}]\int_{E^{*}}\exp[\langle x, \xi\rangle]\varphi(x)d\mu(x)$,where is an-functional. The -transform gives a nice representation of white noise functionals. Indeed, the transform plays fundamentalroles in two ways; one is that a visualized representation of is given,and the other is that it helps to introduce generalized white noise functionals which are very important in advanced white noise analysis.
$S$$(L^{2})$$\varphi$$\varphi$

2.2.

The isomorphism$H_{n}\cong L^{2}(\mathbb{R}^{n})^{\wedge}$), where(up todenotes the subspace of
only symmetric functions. This isomorphism extends to$\sqrt{n^{I}}.$$L^{2}(\mathbb{R}^{n})$$L^{2}(\mathbb{R}^{n})^{\wedge}$$H_{n}^{-}\cong H^{-\frac{n+1}{2}}(\mathbb{R}^{n})^{\wedge}$

containing

,

where
is the subspace of the Sobolev space of order $m$ involvof generalized functionals
ing symmetric functions. Define a spaceby,$H^{m}(\mathbb{R}^{n})^{\wedge}$$(L^{2})^{-}$$(L^{2})^{-}=\oplus c_{n}H_{n}^{-}$.
is a certain sequence of positive numbers such thatwhere
For details, we refer to the literature $[4, Chapt.3.A]$ . It is noted that
depends on the singularity of functionals to be
the choice of the
discussed.
$c_{n}\rightarrow 0$$c_{n}$$\{c_{n}\}$Another space
triple$(S)^{*}$of generalized functional is defined by a Gel’fand$(S)\subset(L^{2})\subset(S)^{*}$,

which is an infinite dimensional analogue of the triple to define the
Schwartz distributions. The Rigorous definition and the properties ofare found in [4, Chapt.4]. Also see [9, Chapt.4].
$(S)^{*}$The spaces
andof generalized white noise functionals
are basic concept of the white noise analysis; indeed, the choice of thesespaces is one of the advantages of our analysis. By using these spaces, we can carry on the analysis of white noise functionals
andin suﬃciently large class and find significant applications in other fields
like quantum dynamics. Actually, to be surprising enough, we can findan interesting application to the non-commutative geometry (e.g. [10]).
$(L^{2})^{-}$$(L^{2})^{-}$$(S)^{*}$$(S)^{*}$

114

\S 3.

T. HidaTopic (2)3.1. Elemental processes involved in innovation
We shall discuss the analysis of stochastic processes and random
fields. The approach is done in line with
$Reduction\rightarrow Synthesis\rightarrow Analysis$.

The step of the reduction is realized by taking the innovation for the
random complex phenomena in question. Then, we are given a (Gaussian) white noise $and/or$ a Poisson noise, both of which are elemental
generalized stochastic processes with independent values at every time
point, as is illustrated just below. The two cases have much similarity-spaces; however dissimilarity is also interesting.
in the analysis on
Hence, they are discussed separately, except those properties in common.
-parameter in genIn the multi-dimensional parameter case, sayeral, innovations should still be generalized stochastic processes having.
independent values at every point in$L^{2}$$\mathbb{R}^{d}$$\mathbb{R}^{d}$

To fix the idea, we first take the one-dimensional parameter case.
Then, under mild, and in fact reasonable assumptions, we may assume
that innovation is the time derivative of some L\’evy process with stationary independent increments. Let it be denoted by $L(t)$ . We may assume
it has no non-random part. Then, the L\’evy-It\^o decomposition gives us
an expression of the form
$L(t)=cB(t)+X(t)$ ,where is a constant, $B(t)$ is a Brownian motion, and $X(t)$ is a compound Poisson process which consists of independent Poisson processes
with diﬀerent heights of jumps (see [1]). Thus, we can conclude that a
Brownian motion and each Poisson process being a component of the$X(t)$ are all elemental L\’evy processes.
$c$After the reduction follows the step of synthesis which means theconstruction of the functionals of (Gaussian) white noise and Poissonnoises of diﬀerent heights of jumps. It is, formally speaking, easy toform a general space of functionals of the innovation, however we need
profound background. Actually rigorous interpretation can be seen e.g.
in [6].
The collection of -transforms offunctionals
producing Kernel Hilbert space which is isomorphic to$S$$(L^{2})$$F$$\{\varphi\}$forms a Re. In short,$(L^{2})$

A frontier of white noise analysis, in line with It\^o calculus

115

has a representation in terms of a functional (indeed, non-randomfunctional) of a smooth function . We can therefore appeal to the classical theory of functional analysis in order to carry on the calculus ona$\varphi$$\xi$$(L^{2})$.

3.2. Random fields
Coming to random fields, a certain class of them is introduced. Let$C$ be a class of manifolds $C$ in:$\mathbb{R}^{d}$$C=$ { $C$ ; convex,$C^{\infty}$-manifold,$\approx S^{d-1}$},

a diﬀeomorphism.
The topology is introduced to $C$ by using the Euclidean distance.

$where\approx means$

Given a random field $X(C)=X(C, x)$ , $x\in E^{*}(\mu)$ , indexed by $ C\in$C. The -transform is now of the form
$S$$(SX(C))(\xi)=\exp[-\frac{1}{2}||\xi||^{2}]\int_{E^{*}}\exp[\langle x, \xi\rangle]X(C, x)d\mu(x)$,and it defines a $U$ functional$(SX(C))(\xi)=U(C, \xi)$ ,$\xi\in E$.

Suppose that $X(C)$ is a linear functional of a multiple Wiener integral of degree . Then, the associated $U$ -functional is expressed in the
form
$n$

$U(C, \xi)=\int F(C, u)\xi(u)^{n\otimes}du^{n}$.

Thus, we are ready to apply the classical theory of calculus of variations to the functional $U(C, \xi)$ of variable $C$ (see [5, Chapt.6]).

We then come to the case of a Poisson noise.
The case of functionals of a single Poisson process $P(t)$ , we introduce
the $U$ -transform following the paper by K. Sait\^o and A. Tsoi [14]. It is
given by the formula$(U\varphi)(\xi)=C_{P}(\xi)^{-1}\int\exp[i<x, \xi>]\varphi(x)d\mu_{P}(x)$whereis the probability distribution of Poisson noiseis the characteristic functional of . Namely,$\mu_{P}$$\dot{P}$$C_{P}(\xi)$$C_{P}(\xi)=\exp[\lambda\int(e^{i\xi(u)}-1)du]$.

$\dot{P}$,, and where

T. Hida

116

For simplicity, the intensity$\varphi$$\lambda$

is taken to be 1 in this subsection.

Fact. Under the $U$ -transform, a discrete chaos (in Wiener’s sense)of degree has a representation of the form$n$$\int\cdots\int F(u)\Pi_{1}^{n}(e^{i\xi(u_{j})}-1)du^{n}$,where $u=(u_{1}, \cdots, u_{n})$ . (For proof see [14].)

Generalized Poisson noise functionals can be introduced by using
this representation. We can further play a similar game for the analysisof them to the Gaussian case.
Generalization to the multi-dimensional parameter case is juststraightforward. Also, a random field $X(C)$ of functional of Poissonnoise is defined, and its variation can be discussed in a similar mannerto the Gaussian case.

3.3.

Stochastic variational equation

First note that, as a generalization of the infinitesimal equation,
there is given a stochastic variational equation for a random field $X(C)$ :$\delta X(C)=\Phi(X(C’), C^{/}<C, Y(s), s\in C, C, \delta C)$ ,where $\{Y(s), s\in C\}$ is the innovation, and where $C’<C$ denotes thatlies inside of $C$ .
As was explained before, we apply -transform in the Gaussian case,and we are given a variational equation for $U$ -functional. When we come, a diﬃculty arises. This can beback to a random function by using
illustrated as follows.
$C^{/}$$S$$S^{-1}$

To fix the idea, we consider a Gaussian random field $X(C)$ which has
a causal representation in terms of white noise. Namely, it is a linear
functional of expressed in the form:$x$$X(C)=\int_{(C)}F(C, u)x(u)du^{d}$ ,where $F(C, u)$ is smooth in $(C, u)$ , and where (C) denotes the domain
enclosed by the ovaloid $C$ . Then, $U$ -functional can be expressed as$U(C, \xi)=\int_{(C)}F(C, u)\xi(u)du^{d}$ .

117

A frontier of white noise analysis, in line with It\^o calculus

The variation of $U$ is easily computed to have$\delta U(C, \xi)=\int_{C}F(C, s)\xi(s)\delta n(s)ds+\int_{(C)}\delta F(C, u)\xi(u)du^{d}$,where theis the variation of $F$ in the variable $C$ which is defined in
the classical sense, and where $ds$ is the surface element over $C$ .
$\delta F$

In what follows, the representation of $X(C)$ in the above form isassumed to be canonical. Namely, the sigma-field $B_{C}(X)$ generated by$X(C’)$ with $C’$ inside of $C$ is equal to the sigma-field $B_{C}(x)$ generated
by $x(u)$ , being inside of $C$ for every $C$ .
$u$

Theorem. The variation $\delta X(C)$ is given by$\delta X(C)=\int_{C}F(C, s)x(s)\delta n(s)ds+\int_{(C)}\delta F(C, u)x(u)du^{d}$ .

The innovation is obtained from the first term of the right side.

Proof. This result may be thought of as an easy consequence of thedefinition of $\delta X(C, x)$ , but not quite. A rigorous proof needs, in addition
to the functional analysis, the following considerations on the restrictionof the parameter and the choice of $C$ . Hence, what follow in Subsection
3.4 are to be included in the proof. For details, see [5].

3.4.
Other variations
Restriction of the parameter.
i) Gaussian case.
. For
the
Given anparameter white noise$x\inE^{*}$
asubject,,
isdefinedwhichistostochastic bilinear form
Gaussian distribution $N(0, ||f||^{2})$ . Take to be an indicator function
such that$\mathbb{R}^{d}$$(E^{*}, \mu)$$f\in L^{2}(\mathbb{R}^{d})$$\langle x, f\rangle$$f$$f_{t}(u)=\chi_{I(t)}(u)$, $t=(t_{1}, t_{2}, \cdots, t_{d})$ , $I(t)=\Pi_{j}[0, t_{j}]$ .

Then, $ X(I(t))=\langle x, f_{t}\rangle$ is a Brownian sheet, that is,$E[X(I(t))]=0$ , $E[X(I(t))\cdot X(I(s))]=\Pi_{j}(t_{j}\wedge s_{j})$ .

-dimensional parameter
Set $t_{d}=1$ . Then, we are given an-dimensional paramBrownian sheet. It is now ready to have aneter white noise by applying partial derivatives. Similarly, much lower
dimensional parameter Brownian sheet and white noise can be derived.
The idea is the same for the restriction of the parameter to a hypersurface $C$ which is a smooth ovaloid.
$\mathbb{R}^{d-1}$$\mathbb{R}^{d-1}$

T. Hida118

Proposition. The restriction of the parameter of white noise canbe done with the help of Brownian sheet.

Poisson case
For the Poisson noise the same trick can be applied as is easily
shown. Just remind that the characteristic functionalof a Poissonparameter is
noise with$ii)$

$C_{P}(\xi)$$\mathbb{R}^{d}$$C_{P}(\xi)=\exp[\lambda\int(e^{i\xi(t)}-1)dt^{d}]$,where $\lambda>0$ is the intensity. The associated measure onby .

$E^{*}$

is denoted$\mu_{p}$Take to be $I(t)$ as above, and form a stochastic bilinear form
, $x\in E^{*}(\mu_{p})$ , which is to be a Poisson sheet. A restriction of
the parameter to a hyperplane defines a lower dimensional parameter
Poisson noise. Also a restriction to a hypersurface is given.
$\xi$$\langle x, f_{t}\rangle$

As in the Gaussian case we can state a proposition. Since it issimilar, so that we omit.
It is noted that observation of on a Poisson sheet, we can see invariance of Poisson noise under some transformations of the parameter
space. This will be reported later.
Choice of the $C$ .

One may ask why the parameter $C$ for a random field should betaken to be a smooth ovaloid (convex, closed, without boundary). For
one thing, the deformation of $C$ will be done by members of a known
transformation group acting on. For another reason, a white noise$C$parameterized by a point in should be defined. Indeed, as soon as thevariational calculus is applied we are naturally led to define an innovationor white noise with parameter set $C$ , as we have seen in Subsection 3.3.
For this purpose a Gel’fand triple of functions spaces on $C$ should be
defined. Our assumptions for $C$ guarantees the existence of a Gel’fand
triple to be requested.
$\mathbb{R}^{d}$With the considerations mentioned above, we can deal with variational equations, not only for a Gaussian type field reviewed in thissection, but also more general non-Gaussian fields and even in the casewhere $X(C)$ is a nonlinear functional of Poisson noise.
The above considerations are more significant if functionals of Poisson noise are discussed. For example we take a causal and linear functional of a homogeneous chaos, sample function of which is denoted byA frontier of white noise analysis, in line with It\^o calculus
$x$

119

. It is expressed in the form
$X(C)=\int_{(C)}F(C, u)x(u)du^{d}$ .

Note that, the canonical property (which is an easy generalization of
the notion on canonical property discussed in $[3, II [8]])$ of the above representation follows easily under the assumption that the kernel $F(C, u)$never vanishes. Hence, the innovation can immediately be obtained.

References[1] K. It\^o, On stochastic processes (I), Japanese J. of Math. 18 (1942), 261301.
[2] K. It\^o, Multiple Wiener integrals, J. Math. Soc. Japan, 3 (1951), 157-169.
[3] L. Accardi, H.-H. Kuo, N. Obata, K. Sait\^o, Si Si and L. Streit eds, Selected
papers of Takeyuki Hida. World Sci. Pub. Co. Pte. Ltd, 2001.
[4] T. Hida, H.-H. Kuo, J. Potthoﬀ and L. Streit, White noise. An infinite
dimensional calculus. Kluwer Acad. Pub. 1993.
[5] T. Hida and Si Si, An innovation approach to random fields. Applicationof white noise theory. World Scientific Pub. Co. Pte. Ltd. 2003.
[6] T. Hida and Si Si, L\’evy processes, revisited. Levico Conf. 2003.
[7] John R. Klauder and E. C. G. Sudarshan, Fundamentals of quantum optics,
Benjamin Inc, 1968.
[8] I. Kubo and S. Takenaka, Calculus on Gaussian white noise. I,II,III, IV.
Proc. Japan Academy. 56 (1980), 376-380, 411-416; 57 (1981) 433-437;
58 (1982), 186-189.
[9] H.-H. Kuo, White noise distribution theory. CRC Press, 1996.
[10] R. L\’eandre and H. Ouerdiane, Connes-Hida calculus and BismutQuillen superconnections. Les pr\’epublications de l’Institut \’Elie Cartan.
2003/no. 17.
[11] P. L\’evy, Th\’eorie de l’addition des variables al\’eatoires. Gauthier-Villars,
1937.
[12] P. L\’evy, Processus stochastiques et mouvement brownien. GauthierVillars, 1948. 2\‘eme \’ed. 1965.
[13] P. L\’evy, Probl\‘emes concrets d’analyse fonctionelle. Gauthier-Villars,
1951.
[14] K. Sait\^o and A. Tsoi, The L\’evy Laplacian acting on Poisson noise functionals. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2 (1999) 503-510.
Meijo University
Nagoya 468-8502, Japan

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 121-140

Integral Representation of Linear Functionals on
Vector Lattices and its Application to $BV$ Functions
on Wiener Space
Masanori Hino
Abstract.
We consider vector lattices $D$ generalizing quasi-regular Dirichlet
spaces and give a characterization for bounded linear functionals on$D$to have a representation by an integral with respect to smoothmeasures. Applications to $BV$ functions on Wiener space are also
discussed.

\S 1.

Introduction

Let $X$ be a compact Hausdorﬀ space and $C(X)$ the Banach spaceof all continuous functions on $X$ with supremum norm. The Riesz representation theorem says that every bounded linear operator on $C(X)$
is realized by an integral with respect to a certain finite signed measure
on $X$ . As a variant of this fact, Fukushima [7] proved that, for any
and for $\in F$ ,is represented
quasi-regular Dirichlet form
as an integral by a smooth signed measure if and only if$(\mathcal{E}, F)$$|\mathcal{E}(u, v)|\leq C_{k}||v||_{L^{\infty}}$$u$$\mathcal{E}(u, \cdot)$

for all $v\in F_{b,F_{k}}$ ,$k\in \mathbb{N}$,. As its apholds for some nest
and some constants
plications, he gave a characterization for additive functionals of function
to be semimartingales, and also proved the smoothness
type forof the measures associated with $BV$ functions ([7, 8, 9]).
In this paper, we show a corresponding result in the ﬀamework ofvector lattices generalizing quasi-regular Dirichlet spaces. The proof issimilar to that in [7] but based on a purely analytical argument, unlike[7] where probabilistic methods are used together. Typical examples$\{F_{k}\}_{k\in N}$$C_{k}$$k\in \mathbb{N}$$(\mathcal{E}, F)$

Received February 27, 2003.
Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists,
15740089, 2003.

122

M. Hinowhich lie in this ﬀamework are first order Sobolev spaces derived ﬀom agradient operator and ﬀactional order Sobolev spaces by a real interpolation method with diﬀerentiability index between 0 and 1. Using such
results, we can improve the smoothness of the measure associated with
$BV$ functions on Wiener space, discussed in $[8, 9]$ .
We can find several studies closely related to this article regarding
the Riesz representation theorem, e.g., in $[22, 14]$ . Their ﬀameworks are
based on Markovian semigroups and the function spaces are derived ﬀom
their generators, which seems to be suitable for complex interpolation
spaces. Ours is based on the lattice property instead and fits for realinterpolation spaces.
The organization of this paper is as follows. In Section 2, we give ageneral ﬀamework and preparatory lemmas, which are slight modifications of what have been developed already in the case of Dirichlet spaces
or in the framework of the nonlinear potential theory. We also give someexamples there. In Section 3, the representation theorems are proved.
In Section 4, we discuss some applications to $BV$ functions on Wiener
space.

\S 2.

Framework and main results

Let $X$ be a topological space and a Borel measure on $X$ . Let $L^{0}(X)$be the space of all -equivalence classes of real-valued Borel measurable
functions on $X$ . We will adopt a standard notation to describe function) and .
.
spaces and their norms, such as $L^{p}(X)$ (or simply$D$
We suppose that a subspace of $L^{0}(X)$ equipped with norm .
satisfies the following.
$\lambda$$\lambda$$L^{p}$$||$$||_{L^{p}}$$||$$||_{D}$(A1)
is a separable and uniformly convex Banach space.
(A2) (Consistency condition) If a sequence in $D$ converges to 0 in $D$ ,then its certain subsequence converges to 0.
$(D, ||\cdot ||_{D})$$\lambda- a.e$

Since $D$ is assumed to be uniformly convex, it is reflexive and the BanachSaks property holds: every bounded sequence in $D$ has a subsequence
whose arithmetic means converge strongly in $D$ (see [16, 19, 13] for the
proof). The following lemma is proved by a standard argument.
Lemma 2.1. Let
be a sequence bounded in D. Then, there
exists a subsequencesuch thatconverges to some weakly
in $D$ and the arithmetic means
converge to strongly inconverges
D. Moreover,. If furthermore, then it holds that $g\in D$ andto someweakly converges toin D.
$\{f_{n}\}_{n\in N}$$\{f_{n_{k}}\}_{k\in N}$$f$$f_{n_{k}}$$(1/k)\sum_{j=1}^{k}f_{n_{j}}$$f$$||f||_{D}\leq\lim\inf_{k\rightarrow\infty}||f_{n_{k}}||_{D}$$g\lambda- a.e.$$f_{n}$$f_{n}$$g$

Integral representation

123

By virtue of the Banach-Alaoglu theorem and the reflexivityof ,is weakly relatively compact in D. Using the Banach-Saks
property together, we can prove the first claim. The second one follows
ﬀom the Hahn-Banach theorem. The last one is a consequence of the
Q.E.D.
consistency condition (A2).

Proof.

$D$$\{f_{n}\}_{n\in N}$

We further assume the following.
(A3) (Normal contraction property) For every $f\in D,\check{f}:=0\vee f\wedge 1$.
belongs to $D$ and$D_{b}:=D\cap
L^{\infty}$(A4) For every,
belongs to D. Moreover,in$\sup\{||f^{2}||_{D}|||f||_{D}+||f||_{L^{\infty}}\leq 1\}$ is finite.
The condition (A4) is equivalent to the following:(A4)’ for every and in , $fg$ belongs to D. Moreover, $\sup\{||fg||_{D}|$$||g||_{D}+||g||_{L^{\infty}}\leq 1\}$ is finite.
$||f||_{D}+||f||_{L^{\infty}}\leq 1$ ,Indeed, it is clear that (A4)’ implies (A4). To show the converse implication, use the identity $fg=((f+g)/2)^{2}-((f-g)/2)^{2}$ and thesubadditivity of the norm . .
We introduce a suﬃcient condition for (A3) and (A4).
$||\check{f}||_{D}\leq||f||_{D}$$f^{2}$$f$$f$$D_{b}$$g$$||$$||_{D}$

Lemma 2.2. Under (A1) and (A2), the following condition (C)implies (A3) and (A4):(C) when is a bounded and infifinitely diﬀerentiate function onwith $\chi(0)=0$ andfor some $c>0$ , then $\chi\circ v\in D$
and $||\chi ov||_{D}\leq c||v||_{D}$ for every $v\in D$ .

$\mathbb{R}$$\chi$$||\chi^{/}||_{\infty}\leq c$

To show (A3), apply Lemma 2.1 with a sequenceandconverges
so thatfor every , and$\chi(x)=0\vee
x\wedge1$. (A4) is similarly proved.
pointwise to
Q.E.D.

Proof.

$\{\chi_{n}ov\}_{n\in N}$$||\chi_{n}||_{\infty}\leq 1$$||\chi_{n}^{J}||_{\infty}\leq 1$$n$$\chi_{n}$For $f\in L^{0}(X)$ , we set $f_{+}(z)=f(z)\vee 0$ and $f_{-}(z)=-(f(z)\wedge 0)$ .
Lemma 2.3. Let $f\in D$ . Then $f_{+}\in D$ and$||f_{+}||_{D}\leq||f||_{D}$Proof. Define $f_{n}:=0\vee f\wedge n=n(0\vee(f/n)\wedge 1)$ ,$||f_{n}||_{D}\leq n||f/n||D=||f||_{D}$by (A3) and$f_{n}\rightarrow f_{+}$

finishes the proof.

.

$n\in \mathbb{N}$. Thenpointwise. Lemma 2.1
Q.E.D.

Lemma 2.4. For every $f\in D$ , $(-a)\vee f\wedge a\rightarrow 0$ weakly in $D$ as
and $(-a)\vee f\wedge a\rightarrow f$ weakly in $D$ as.

$a\downarrow 0$$ a\rightarrow\infty$

Proof.

It is enough to notice that$||(-a)\vee f\wedge a||_{D}=||0\vee f\wedge a-0\vee(-f)\wedge a||_{D}\leq 2||f||_{D}$$(-a)\vee f\wedge a\rightarrow 0$

pointwise as$a\downarrow 0$$(-a)\vee f\wedge a\rightarrow f$

pointwise as$ a\rightarrow\infty$

,

,

124

M. HinoQ.E.D.

and to use Lemma 2.1.

For a measurable set $A$ , we let $D_{A}:=$ { $f\in D|f=0\lambda- a.e$ . on}of increasing sets in $X$ is
and $D_{b,A}:=D_{A}\cap L^{\infty}$ . A sequence
is dense in D. A nest
called a nest if eachis closed and$(\lambda-)regular$
if, for all , any open set $O$ withis called. A subset $N$ of $X$ is called exceptional if there is asatisfies. When $A$ is a subset of $X$ , wenestsuch thatsay that a statement depending on $z\in A$ holds quasi everywhere ( $q.e$ . infor a certain exceptional set
abbreviation) if it does for every$N$ . For a nest
$C(\{F_{k}\})$
the set of all functions, we denote by$X$on such that is continuous on each . A function on $X$ is said tosuch that $f\in C(\{F_{k}\})$ .
be quasi-continuous if there is a nest
We say that a Borel measure on $X$ is smooth if it does not charge
any exceptional Borel sets and there exists a nestsuch that$X$$\mu(F_{k})<\infty$ for all. A set function onwhich is given bywith finite total mass is called afor some smooth measuresandsignedsmooth measure with attachedsignedsmoothmeasure.
A
fifinitenest
is a map ﬀom{ $A\subset X|$ $A$ is a Borel set of some }, fortosuch that is represented as $\iota/(A)=\iota/_{1}(A)-l/_{2}(A)$ ,satisfying
andfor eachsome smooth Borel measure. When we want to emphasize the dependency of $D$ ,$i=1,2$ andwe write $D$-nest, $D$-smooth, and so on.
We further assume the following quasi-regularity conditions.
(QR1) There exists a nest consisting of compact sets.
(QR2) There exists a dense subset of $D$ whose elements have quasicontinuous -modifications.
in $D$ and an ex(QR3) There exists a countable subset
has a quasi-continuous
ceptional set $N$ such that each.
separates the points ofmodification
andEvery quasi-regular symmetric Dirichlet form
satisfies all conditions $(A1)-(A4)$ and $(QR1)-(QR3)$ (and (C)) when letting $D=F$ and$||f||_{D}=(\mathcal{E}(f, f)+||f||_{L^{2}}^{2})^{1/2}$ . We give other examples in the last part ofthis section.
$X\backslash A$$\{F_{k}\}_{k\in N}$$F_{k}$$\bigcup_{k=1}^{\infty}D_{F_{k}}$$k$$\{F_{k}\}_{k\in N}$$0$$\lambda(O\cap F_{k})=$$O\subset X\backslash F_{k}$$N\subset\bigcap_{k=1}^{\infty}(X\backslash F_{k})$$\{F_{k}\}_{k\in N}$$z\in A\backslash N$$f$$\{F_{k}\}_{k\in N}$$F_{k}$$f$$f$$\{F_{k}\}_{k\in N}$$\mu$$\{F_{k}\}_{k\in N}$

$k$

$iJ$

$iJ$

$\nu_{1}$$=\nu_{1}-\iota/_{2}$$\nu_{2}$$\iota/$$\mathcal{R}:=$$\{F_{k}\}_{k\in N}$$F_{k}$$A\in \mathcal{R}$$\mathbb{R}$$\iota/$$\iota/_{1}$$\iota/_{i}(F_{k})<\infty$$\nu_{2}$$k\in \mathbb{N}$$\lambda$$\{\varphi_{n}\}_{n\in N}$$\lambda-$$\varphi_{n}$$\tilde{\varphi}_{n}$$X\backslash N$$\{\tilde{\varphi}_{n}\}_{n\in N}$$(\mathcal{E}, F)$

$C$

$C$

Lemma 2.5. There exist some $\rho\in D$ and some countable subset$=\{h_{n}\}_{n\in N}$of such that $0\leq\rho\leq 1\lambda- a.e.$ , $h_{n}\geq 0\lambda- a.e$ . for all ,$-C:=\{h-\hat{h}|h,\hat{h}\in C\}$ is dense in $D$ , and,for each$c_{n}\in(0,\infty)$.
. for some$D_{b}$$n$$n\in \mathbb{N}$$\rho\geq c_{n}h_{n}$$\lambda- a.e$Letbe a countable dense subset of D. Denote by
the set of all arithmetic means of finite number of functions
in $\{(f_{m})_{+}\wedge M, (f_{m})_{-}\wedge M\}_{m\in N}$ , $M\in N$ . Then $-C$ is dense in $D$ by

Proof

$C$

$\{f_{m}\}_{m\in N}$$=\{h_{r\iota}\}_{n\in N}$

$C$

125

Integral representation

Lemma 2.4 and the Banach-Saks property. Define$\rho=\sum_{n=1}^{\infty}\frac{h_{n}}{2^{n}(||h_{n}||_{D}+||h_{n}||_{L^{\infty}}+1)}$Then$\rho$.

and Csatisfy the conditions in the claim.

Q.E.D.

We will fix and Csatisfying the statement in the lemma above. Notethat we can always takeif $1\in D$ .
Take a strictly increasing and right-continuous function :with $\xi(0)=0$ . For an open set $O\subset X$ , we define$\rho$$\rho\equiv 1$$\xi$$[0, \infty)\rightarrow$$[0, \infty)$(1){$cap_{\xi}(O)=\inf$ $\xi(||f||_{D})|f\in D$and$f\geq\rho\lambda- a.e$

. on $O$ }.

For any subset $A$ of $X$ , we define the capacity of $A$ by{$cap_{\xi}(A)=\inf$ $cap_{\xi}(O)|O\supset A$, $O$ : open}.

It should be noted that $cap_{\xi}(A)\leq\xi(||\rho||_{D})<\infty$ for every $A\subset X$ .
The following lemma is proved in the same way as in [10].

$e_{O}$

Lemma 2.6. For every open set $O$ , there exists a unique function
in $D$ attaining the infifimum in (1). Moreover, $0\leq e_{O}\leq 1\lambda- a.e$ .

The uniqueness follows from the uniform convexity of D.
The existence is deduced by the Banach-Saks property and (A2). The
Q.E.D.
last claim is a consequence of (A3).

Proof.

We will discuss some basic properties of the capacity.
be a sequence of open sets such that
Lemma 2.7. Let. Then, there exists a sequencesuch that$\{O_{n}\}_{n\in N}$$\{n_{k}\}\uparrow\infty$$cap_{\xi}(O_{n})\rightarrow 0$$e_{O_{n_{k}}}\rightarrow 0\lambda- a.e$

Proof.

.

Since$||e_{O_{n}}||_{D}\rightarrow 0$, the claim is clear from (A2).

Q.E.D.

Lemma 2.8. If $cap_{\xi}(A)=0$ , then $A\cap\{\rho>0\}\dot{u}$ a -null set.
$\lambda$

Take a decreasing open setssuch thatand, we have $\rho=0\lambda- a.e$ . on
. Since. on
Q.E.D.
by virtue of Lemma 2.7. This implies the assertion.

Proof.

$cap_{\xi}(O_{n})\rightarrow 0$$\bigcap_{k=1}^{\infty}O_{k}$$\{O_{n}\}_{n\in N}$$e_{O_{n}}\geq\rho\lambda- a.e$$A\subset O_{n}$$\bigcap_{k=1}^{\infty}O_{k}$be a sequence of increasing closed sets.
Lemma 2.9. Let
Then the following are equivalent.
(i)
is a nest.
.
(ii)$\{A_{k}\}_{k\in N}$$\{A_{k}\}_{k\in N}$$\lim_{k\rightarrow\infty}cap_{\xi}(X\backslash A_{k})=0$

126

M. Hino

Proof. Suppose (i) holds. Take a sequence

in $D$ such that
, we have$\{f_{k}\}_{k\in N}$

in D. Since $\rho-f_{k}=\rho\lambda- a.e$ . on.
as
Next, suppose (ii) holds. It suﬃces to prove that each $h\in C$ can. Since $\rho\geq ch\lambda- a.e$ . forbe approximated in $D$ by functions in. onfor each . Let
some $c>0$ , it holds that$f_{k}=(h-c^{-1}e_{X\backslash A_{k}})_{+}\in D_{A_{k}}$ .
By Lemma 2.7, there exists a sequence. Therefore,
diverging to infinity such that,. as
. On the other hand,which is bounded in . From Lemma 2.1, we can take arithmetic means, so that they
, which belong toof some subsequence of
Q.E.D.
converge to in D.
and$f_{k}\in D_{A_{k}}$$X\backslash A_{k}$$ f_{k}\rightarrow\rho$$ k\rightarrow\infty$$cap_{\xi}(X\backslash A_{k})\leq\xi(||\rho-f_{k}||_{D})\rightarrow 0$$\bigcup_{k}D_{A_{k}}$$\{k^{J}\}$$k$$X\backslash A_{k}$$e_{X\backslash A_{k}}\geq ch\lambda- a.e$$f_{k’}\rightarrow h$$e_{X\backslash A_{k’}}\rightarrow 0\lambda- a.e$$ k^{/}\rightarrow\infty$$\lambda- a.e$$||f_{k}||_{D}\leq||h||_{D}+c^{-1}||e_{X\backslash A_{k}}||_{D}$$k$$\{f_{k’}\}$$\bigcup_{k}D_{A_{k}}$$h$As is seen ﬀom this lemma, any choices of , and are consistent with
the notion of nest. From now on, we treat only the case $\xi(t)=t$ and.
write cap in place of
$C$

$\xi$

$\rho$$cap_{\xi}$Lemma 2.10. For any sequence of subsets.
that cap$\{A_{k}\}_{k\in N}$

in $X$ , it follows$(\bigcup_{k=1}^{\infty}A_{k})\leq\sum_{k=1}^{\infty}cap(A_{k})$Proof.

When$O_{1},$$\ldots$,$O_{k}$

are open sets, it is easy to see the inequal. Indeed, since.

ity cap $(\bigcup_{j=1}^{k}O_{j})\leq\sum_{j=1}^{k}cap(O_{j})$on$\bigcup_{j=1}^{k}O_{j}$cap$\sum_{j=1}^{k}e_{O_{j}}\geq\rho\lambda- a.e$

, we have$(\bigcup_{j=1}^{k}O_{j})\leq||\sum_{j=1}^{k}e_{O_{j}}||_{D}\leq\sum_{j=1}^{k}||e_{O_{j}}||_{D}=\sum_{j=1}^{k}cap(O_{j})$.

for eachsuch that
Now, let $\in>0$ . Take an open set. Sinceand $cap(O_{k})<cap(A_{k})+\in 2^{-k}$ . Let$||\rho||_{D}<\infty$ , Lemma 2.1 assures the existence of a subsequence
converges to weakly in $D$ and theofand $e\in D$ such thatconverge to in D. Since $e\geq\rho\lambda- a.e$ . onarithmetic means ofby using (A2), we have$k\in \mathbb{N}$$O_{k}$$U_{k}=\bigcup_{j=1}^{k}O_{j}$$O_{k}\supset A_{k}$$||e_{U_{k}}||_{D}\leq$$\{e_{U_{k’}}\}$$\{e_{U_{k}}\}$$e$$e_{U_{k’}}$$\{e_{U_{k’}}\}$$e$$\bigcup_{k=1}^{\infty}O_{k}$cap$(\bigcup_{k=1}^{\infty}A_{k})$

Since

$\in$

$\leq$

cap$=$$\lim_{k\rightarrow\infty}cap(U_{k})\leq\lim_{k\rightarrow\infty}\sum_{j=1}^{k}cap(O_{j})\leq\in+\sum_{k=1}^{\infty}cap(A_{k})$$(\bigcup_{k=1}^{\infty}O_{k})\leq||e||_{D}\leq\lim_{k\rightarrow},\inf_{\infty}||e_{U_{k’}}||_{D}$is arbitrary, we obtain the claim.

.

Q.E.D.

The following series of lemmas are now proved in a standard way as in
the case of quasi-regular Dirichlet spaces; see e.g. $[11, 15]$ for the proof.

127

Integral representation

Lemma 2.11. Suppose that $f\in D$ has a quasi-continuous
cation . Then, we havefor each$cap(\{\tilde{f}>\lambda\})\leq\lambda^{-1}||f||_{D}$$\overline{f}$$\lambda- modififi-$$\lambda>0$.

is a sequence of quasi-contiLemma 2.12.
(i) Whensuch that
nuous functions, there exists a nest
$C(\{F_{k}\})$
for every .
(ii) If $f_{n}\in D$ has a quasi-continuous $\lambda- modifification$and con$D$
verges to, thenhas a quasi-continuousinas$\lambda- modifification$andand there exists a sequencebelongs to $C(\{F_{k}\})$ andnestsuch that every. In particular,conconverges to uniformly on each.
verges to(iii) Every $f\in D$ has a quasi-continuous $\lambda- modifification$ .
$\{f_{n}\}_{n\in N}$$\{F_{k}\}_{k\in N}$$ f_{n}\in$$n$$\tilde{f}_{n}$

$f$

$f$

$ n\rightarrow\infty$$\tilde{f}$$\{n\iota\}\uparrow\infty$$\tilde{f}_{n}$$\{F_{k}\}_{k\in N}$$\overline{f}$$a$$\tilde{f}_{n_{l}}$$\tilde{f}_{n_{l}}$$F_{k}$$\tilde{f}q.e$$\tilde{f}$such that
Lemma 2.13. There exists a regular nest
is a separable and metrizable compact space with respect to the relative
topology for any .
$K_{k}$$\{K_{k}\}_{k\in N}$$k$is a regular nest and $ f\in$Lemma 2.14. Suppose that$O$
$C(\{F_{k}\})$ .
.
If $f\geq 0\lambda- a.e$ . on an open set , then $f\geq 0$ on$\{F_{k}\}_{k\in N}$$O\cap\bigcup_{k}F_{k}$Lemma 2.15. If and, then $u_{1}=u_{2}q.e$ .
$u_{1}$$u_{2}$

are quasi-continuous functions and $u_{1}=$$u_{2}\lambda- a.e.$

In what follows, always means a quasi-continuous -modificationof a function , a particular version of which is sometimes chosen to suitthe context.
We can also prove the next two propositions as in [10] (see also [21,Section 2]) by using Lemma 3.1 below together, though they are notused later in this article.
$\tilde{f}$$\lambda$$f$Proposition 2.16. For any subset $A$ of $X$ , there exists a unique. on $A$ } minimizing the norm
element
in the set {. Moreover, $0\leq e_{A}\leq 1\lambda- a.e$ . and cap({/ $=||e_{A}||_{D}$ .
$e_{A}$$f\in D|\tilde{f}\geq\tilde{\rho}q.e$$||f||_{D}$Proposition 2.17. cap is a Choquet capacity.
We remark that the assumption (A4) is not necessary so far. The
following are our main theorems, which are stated in [7] in the case ofquasi-regular Dirichlet spaces.
Theorem 2.18. Under $(A1)-(A4)$ and $(QR1)-(QR3)$ , for a bounded linear functional $T$ on $D$ , the next two conditions are equivalent.
and positive constants(i) There exist a nest,such that for each$\{C_{k}\}_{k\in N}$$\{F_{k}\}_{k\in N}$$k\in \mathbb{N}$$|T(v)|\leq C_{k}||v||_{L^{\infty}(X)}$for all$v\in D_{b,F_{k}}$

.

128

M. HinoThere exists a signed smooth measurenestsuch that

(ii)

$\nu$

with some attached$\{F_{k}^{/}\}_{k\in N}$

for$T(v)=\int_{X}\tilde{v}(z)\nu(dz)$Moreover, the measure$\nu$$dl$ $v\in k=1\cup\infty D_{b,F_{k}’}$.

is uniquely determined.

Theorem 2.19. Under $(A1)-(A4)$ and $(QR1)-(QR3)$ , for a bounded linear functional $T$ on $D$ and a positive constant $C$ , the next two
conditions are equivalent.
(i) $|T(v)|\leq C||v||_{L^{\infty}(X)}$ for all $v\in D_{b}$ .
(ii) There exists fifinite signed smooth measure on $X$ such thatthe total variation of is dominated by $C$ and
$a$

$\nu$

$\nu$

for all$T(v)=\int_{X}\tilde{v}(z)\nu(dz)$$v\in D_{b}$.

is uniquely determined. Moreover, if (C) in Lemma 2. 2
In addition,holds, we may replacein (i) by that satisfifies the following:$D$
is a -dense subspace ofsuch that, for $each\in>0$ , there$|\chi|\leq
1+\in$ , $0\leq\chi^{/}\leq 1$ , $\chi(x)=x$
is afunction on with.
on [-1, 1], andfor every$\nu$$\mathcal{L}$$D_{b}$$D_{b}$$\mathcal{L}$$(\mathcal{L})$$C^{\infty}$$\mathbb{R}$$\chi$$v\in \mathcal{L}$$\chi\circ v\in \mathcal{L}$

Before ending this section, we give a few examples of $D$ other thanquasi-regular Dirichlet spaces. Suppose that $X$ is a separable Banach
space and $H$ a separable Hilbert space which is continuously and densely
imbedded to $X$ . The inner product and the norm of $H$ will be denoted, respectively. The topological dual
of $X$ isby
andidentified with a subspace of $H$ . Let be a finite Borel measure on $X$ .
When $K$ is a separable Hilbert space, we denote by $L^{p}(X\rightarrow K)$ the.
space consisting of $K$ -valued functions on the measure space
$X$
by
on
Define function spaces
and$X^{*}$$||\cdot||_{H}$$\langle\cdot, \cdot\rangle_{H}$$\lambda$$L^{p}$$(X, \lambda)$$FC_{b}^{1}$$FC_{b}^{1}$$(FC_{b}^{1})x*$$=$$\{u:X\rightarrow \mathbb{R}|$$=$$\{G:X\rightarrow X^{*}|$$(FC_{b}^{1})_{X^{*}}$$\ell_{1}u(,z)\ldots=forsom$$\ell_{m}\in X^{*}f(\ell_{1}(z)em\in \mathbb{N}’,\cdots f’\in C_{b}^{1}(\mathbb{R}^{m}) \ell_{m}(z)),\}$

’

,

,

$m\in \mathbb{N}\}$$\ell_{1}g_{1}G(,’ z.\cdot).\cdot.\cdot=,’\sum_{\ell_{m}\in g_{m}\in}j=1g_{j}(mz)\ell_{j}X^{*}fors’ oFC_{b}^{1}$,

meis the set of all bounded functions onthat havewherebounded and continuous first-order derivatives. Let $u\in FC_{b}^{1}$ and$X^{*}\subset H\subset X$ . We define
by $\partial_{\ell}u(z)=\lim_{\epsilon\rightarrow 0}(u(z+\in\ell)-u(z))/\in$ .
$f$$C_{b}^{1}(\mathbb{R}^{m})$$\mathbb{R}^{m}$$\ell\in$$\partial_{\ell}u$129Integral representation

The $H$ -derivative
relation$\nabla u$

is a unique map from $X$ to $H$ that satisfies the$\langle\nabla u(z), \ell\rangle_{H}=\partial_{\ell}u(z)$,$\ell\in X^{*}\subset H$.

We assume that, if $u\in FC_{b}^{1}$ and $v\in FC_{b}^{1}$ coincide on a measurable
set $A$ , then $\nabla u=\nabla v\lambda- a.e$ . on $A$ . Let $p\geq 1$ . We also assume that
to $L^{p}(X\rightarrow H)$ . We denote byis closable as a map from$W^{1,p}$and extend the domain of
the domain of the closure of$W^{1,p}$$W^{1,p}$naturally. The space
is a separable Banach space with
to$||f||_{W^{1,p}}=||f||_{L^{p}}+||\nabla
f||_{L^{p}(X\rightarrow
H)}$
.
norm
$L^{p}$$(\nabla, FC_{b}^{1})$$(\nabla, FC_{b}^{1})$$\nabla$Proposition 2.20. Suppose also $p>1$ . Then, when we regard $W^{1,p}$
as $D$ , the conditions $(A1)-(A4)$ , $(QR1)-(QR3)$ , and (C) are satisfified.
Prom the results of [21], (A1) and (QR1) hold. Since $X$
is separable,
is also separable with respect to the weak* topology(see Corollary after Proposition 8 in Chapter $FV$ of [5] for the proof).
and
,
When
is a countable dense set ofassure the validity of (QR3). The remaining conditions are easily
Q.E.D.
checked.

Proof.

$X^{*}$$X^{*}$$\{\ell_{n}\}_{n\in N}$$\varphi_{n}(\cdot)=\arctan\ell_{n}(\cdot)$$ N=\emptyset$

In order to give another example, we introduce real interpolation
spaces. Letandbe separable Banach spaces. We assume thatis continuously imbedded tofor simplicity. For parameters$\theta\in(0,1)$, we define the spaceby all elements
and$f\in B_{1}$ such that there exist some-valued measurable functions$(j=0,1)$ satisfyingon$B_{0}$$B_{1}$$B_{1}$$B_{0}$$ q\in$$(1, \infty)$$(B_{0}, B_{1})_{\theta,q}$$u_{j}(t)$$B_{j}$$[0, \infty)$(2)$u_{0}(t)+u_{1}(t)=fa.e$ .

We set the norm of$t$, $\int_{0}^{\infty}(t^{j-\theta}||u_{j}(t)||_{B_{j}})^{q}\frac{dt}{t}<\infty(j=0,1)$ .

$f\in(B_{0}, B_{1})_{\theta,q}$by$||f||_{(B_{0},B_{1})_{\theta,q}}=\inf_{u_{0},u_{1}}[\max_{j=0,1}(\int_{0}^{\infty}(t^{j-\theta}||u_{j}(t)||_{B_{j}})^{q}\frac{dt}{t})^{1/q}]$,satisfying (2). Rom
andwhere the infimum is taken over all pairs
is a Banach space, we
the general theory of real interpolation,
is dense, andhave continuous imbeddings. Keeping the notation in the previous example, we haveinthe following proposition.
$u_{0}$$u_{1}$$(B_{0}, B_{1})_{\theta,q}$$B_{0}L\rightarrow(B_{0}, B_{1})_{\theta,q}\simeq\rightarrow B_{1}$$B_{0}$$(B_{0}, B_{1})_{\theta,q}$

Proposition 2.21. Let $p\in(1, \infty)$ , $q\in(1, \infty)$ , and $\theta\in(0,1)$ . Then
$D:=(W^{1,p}, L^{p})_{\theta,q}$
satisfifies $(A1)-(A4)$ , $(QR1)-(QR3)$ and (C).

130

M. Hino

Proof. In general, we can prove thatis uniformly convex
is, in the same way as Proposition $V.I$ of [3]. Therefore, $D$ isorif
uniformly convex. The separability, (QR1) and (QR2) come from thoseof $W^{1,p}$ . (QR3) is proved in the same way as the case of $W^{1,p}$ . (A2)is clearly true. We will prove (C). Let
be as in (C) in Lemma 2.2.
Let $f\in D$ and takeandsatisfying (2). Set $v_{0}(t)=\chi\circ u_{0}(t)$ and$v_{1}(t)=\chi\circ f-\chi\circ u_{0}(t)$ . Then $v_{0}(t)+v_{1}(t)=\chi\circ f$ and it is easy to. Thissee that $||v_{0}(t)||_{W^{1,p}}\leq c||u_{0}(t)||_{W^{1,p}}$ and$\chi\circf\inD$
implies that
Q.E.D.
and.
$B_{0}$$(B_{0}, B_{1})_{\theta,q}$$B_{1}$$\chi$$u_{1}$$u_{0}$$||v_{1}(t)||_{L^{p}}\leq c||u_{1}(t)||_{L^{p}}$$||\chi\circ f||_{D}\leq c||f||_{D}$Proof of Theorems 2.18 and 2.19{?}3.

First, we will prove that (ii) implies (i) in Theorem 2.18. We takeand $ C_{k}=|\nu|(F_{k})<\infty$ . Let $v\in D_{b,F_{k}}$ and $M$ $=||v||_{L^{\infty}(X)}$ .
everyWe can take a quasi-continuous -modification so that
where. Then $|T(v)|\leq M|\nu|(F_{k})=C_{k}M$ . Therefore, (i) holds.
Next, we will prove that (i) implies (ii) in Theorem 2.18. Take aso that. Define.
nest$F_{k}=F_{k}^{/}$$\lambda$

Lemma 3.1.

$\{E_{k}^{(2)}\}_{k\in N}$

Clearly,

Proof.

DefineTake a sequencehave

is a sequence of increasing closed sets.
. Thenweakly in $D$ by Lemma 2.4.
so thatconverges to,$k\in \mathbb{N}$$\{k_{j}\}\uparrow\infty$$ m\rightarrow\infty$

is a nest.

$\{E_{k}^{(2)}\}_{k\in N}$$\rho_{k}=\rho\wedge(1/k)$

0 in $D$ as$E_{k}^{(2)}=E_{k}^{(1)}\cap\{\tilde{\rho}\geq 1/k\}$$\tilde{\rho}\in C(\{E_{k}^{(1)}\})$$\{E_{k}^{(1)}\}_{k\in N}$$\rho_{k}\rightarrow 0$$\hat{\rho}_{m}:=(1/m)\sum_{j=1}^{m}\rho_{k_{j}}$. Since$\hat{\rho}_{m}+e_{X\backslash E_{k_{m}}^{(1)}}\geq\rho\lambda- a.e$$cap(X\backslash E_{k_{m}}^{(2)})\leq||\hat{\rho}_{m}||_{D}+||e_{X\backslash E_{k_{m}}^{(1)}}||_{D}\rightarrow 0$$\{E_{k}^{(2)}\}_{k\in N}$$|\tilde{v}|\leq M$$\tilde{v}$

as

. on

$X\backslash E_{k_{m}}^{(2)}$$ m\rightarrow\infty$, we. Therefore,

is a nest.

Q.E.D.

Define $E_{k}^{(3)}=F_{k}\cap K_{k}\cap E_{k}^{(2)}$ ,, whereis what appeared in
Lemma 2.13. Then
is a regular nest consisting of separable, let
and metrizable compact sets. Givenbe a countable. The totality of every union of finite elements inopen basis of. Take a countable family
ofwill be denoted by$X$$\in>0$such that for every
open sets in,and, somesatisfies thatand cap(O) $<cap(V_{k,n})+\in$ .
$=\{(-M)\vee\tilde{\varphi}_{n}\wedge M|n\in$
Recall the condition (QR3) and set,. The functions of separate the points of. Fix a$V$
$D$
countable subset
of such that $\{f\in D|||f||_{L\infty(X)}\leq M\}$ is a dense. Take a nest
set of $\{f\in D|||f||_{L\infty(X)}\leq M\}$ for eachsothatfor each ,and the quasi-continuous$k\in \mathbb{N}$$K_{k}$$\{E_{k}^{(3)}\}_{k\in N}$$k\in \mathbb{N}$$\{U_{k,n}\}_{n\in N}$$E_{k}^{(3)}$$\{U_{k,n}\}_{n\in N}$$\{V_{k,n}\}_{n\in N}$$k\in \mathbb{N}$$\mathcal{O}$$O\in \mathcal{O}$$n\in \mathbb{N}$$O\supset V_{k,n}$$S$$\mathbb{N}$$S$$M\in \mathbb{N}\}$$X\backslash N$$M\in \mathbb{N}$$\hat{F}_{k}\subset E_{k}^{(3)}$$k$$N\subset\bigcap_{k\in N}(X\backslash \hat{F}_{k})$$\{\hat{F}_{k}\}_{k\in N}$

131

Integral representation-modifications of all elements in $S\cup D\cup\{e_{O}|O\in \mathcal{O}\}$ belong to. Denote by $A$ the algebra generated by.
. From the StoneNote that all functions in $A$ and belong to
Weierstrass theorem, $\{f|_{\hat{F}_{k}}|f\in A\}$ is dense inwith uniform
topology for any .
$\lambda$$C(\{\hat{F}_{k}\})$$S\cup\{1\wedge M\tilde{\rho}|M \in \mathbb{N}\}$$C(\{\hat{F}_{k}\})$$\tilde{\rho}$$C(\hat{F}_{k})$$k$Lemma 3.2. There exist a nest
satisfying the following:$\{F_{k}^{/}\}_{k\in N}$

and functions$\{\psi_{n}\}_{n\in N}$

in$\bigcup_{k}D_{\hat{F}_{k}}$

(i)
(ii)

for all;belongs to
the quasi-continuous $\lambda- modifification$
all ;$0\leq\psi_{n}\leq 1\lambda- a.e$ . on $X$ andon
for all$F_{k}’\subset\hat{F}_{k}$$k$$\tilde{\psi}_{n}$$C(\{F_{k}^{/}\})$

for

$n$

(iii)

$\tilde{\psi}_{n}=1$$F_{n}^{/}$$n$.

such that $||\eta_{n}-\rho||_{D}<$Take a sequence$1/(n2^{n+1})$ ,. By Lemma 2.11, there exists an open setso$cap(G_{n})<2^{-n}$
.
for
each
Take
that
and
. Thenanda nestsuch that. Define
on
on
since
,
on
and $\psi_{n}=0\vee 2n\eta_{n}\wedge 1$ . Then
sets,
2.10,
is a sequence of increasing closed
and by Lemma
we have$cap(X\backslash F_{k}’)\leq cap(X\backslash E_{k})+\sum_{n=k}^{\infty}cap(G_{n})\rightarrow 0$ as
Q.E.D.
.

Proof.

$\{\eta_{n}\}_{n\in N}\subset\bigcup_{k}D_{\hat{F}_{k}}$$n\in \mathbb{N}$$G_{n}$$G_{n}\supset\{|\tilde{\eta}_{n}-\tilde{\rho}|>1/(2n)\}$$\{E_{k}\}_{k\in N}$$\overline{\eta}_{n}\geq 1/(2n)$$E_{n}\backslash G_{n}$$n$$E_{k}\subset\hat{F}_{k}$$\{\overline{\eta}_{n}\}_{n\in N}\subset C(\{E_{k}\})$$\tilde{\rho}\geq 1/n$$E_{n}$$F_{k}^{J}=E_{k}\backslash \bigcup_{n=k}^{\infty}G_{n}$$\psi_{n}\in\bigcup_{k}D_{\hat{F}_{k}},\overline{\psi}_{n}=1$$F_{n}’$$\{F_{k}^{J}\}_{k\in N}$$ k\rightarrow\infty$:
Now, fix. Defineso thatand take, the statement (i)by $T_{n}(f)=T(\psi_{n}f)$ . Sinceof Theorem 2.18 implies $|T_{n}(f)|\leq C_{m}||\psi_{n}f||_{L^{\infty}(X)}\leq C_{m}||f||_{L^{\infty}(X)}$ .
such that
For an arbitrary $f\in C(\hat{F}_{m})$ , we can take
. Then,$n\in \mathbb{N}$$m\in \mathbb{N}$$T_{n}$$\psi_{n}\in D_{\hat{F}_{m}}$$D_{b}\rightarrow$$\mathbb{R}$$\psi_{n}f\in D_{b,\hat{F}_{m}}\subset D_{b,F_{m}}$$\{f_{j}\}_{j\in N}\subset A$$\lim_{j\rightarrow\infty}||f_{j}-f||_{C(\hat{F}_{m})}=0$$|T_{n}(f_{i})-T_{n}(f_{j})|\leq C_{m}||\psi_{n}(f_{i}-f_{j})||_{L^{\infty}(X)}\leq C_{m}||f_{i}-f_{j}||_{L^{\infty}(\hat{F}_{m})}\rightarrow 0$

as. The limit of$|\hat{T}_{n}(f)|\leq C_{m}||f||_{C(\hat{F}_{m})}$ . Therefore,$ i\geq j\rightarrow\infty$$\{T_{n}(f_{j})\}_{j\in N}$, denoted by$\hat{T}_{n}(f)$, satisfies

is a bounded linear functional on. On account of the Riesz representation theorem, there exists anonsuch thatassociated finite signed measure
to a measure on $X$ by letting
for every $f\in C(\hat{F}_{m})$ . We extend$\nu_{n}(A):=\nu_{n}(A\cap\hat{F}_{m})$.
$\hat{T}_{n}$$C(\hat{F}_{m})$$\hat{F}_{m}$$\nu_{n}$$\hat{T}_{n}(f)=\int_{\hat{F}_{m}}fd\nu_{n}$$\nu_{n}$Lemma 3.3. The measure$\nu_{n}$charges no exceptional sets.

restricted on
is regular, it isSince the measure
enough to prove that $|\nu_{n}|(K)=0$ for any compact setof nullfromcapacity. Take such $K$ . Then we can take a sequence

Proof.

$|\nu_{n}|$$\hat{F}_{m}$$K\subset\hat{F}_{m}$$\{O_{j}\}_{j\in N}$$\mathcal{O}$M. Hino132

so thatfor all and $\lim_{j\rightarrow\infty}cap(O_{j})=0$ . Indeed, for each ,take an open set $O$ such that $K\subset O$ and cap(O) $<1/j$ . Since $K$ iscompact, there is a set $V$ insuch that $K\subset V\subset O$ . Chooseso thatand cap(Oj)\leq cap(V)+l/j.
Let $f_{j}:=e_{O_{j}}$ . Since
is a regular nest, Lemma 2.14 implies. We may also assume thatonthat
everywhere. Since. as, we can suppose
by taking a subsequence if necessary. Since
is bounded, the arithmetic means
inof a further subsequence of
converge strongly in. Take a sequencesuch
converges
. as
.
that
. Define$X$$K$
Then $0\leq f\leq 1$ on , $f=1$ on , and $f=0\lambda- a.e$ . by the way ofconstruction.
Given $h\in A$ , we have
$K\subset O_{j}$

$j$

$j$

$\{V_{k,m}\}_{k\in N}$$V\subset O_{j}$$O_{j}\in \mathcal{O}$$\{\hat{F}_{k}\}$$K\subset O_{j}\cap\hat{F}_{m}$$\tilde{f}_{j}=1$$0\leq\overline{f}_{j}\leq 1$$\lim_{j\rightarrow\infty}||f_{j}||_{D}=0$$f_{j}\rightarrow 0\lambda- a.e$$\{\tilde{f_{j}}\}_{j\in N}$$ j\rightarrow\infty$$\{\hat{f}_{j}\}_{j\in N}$$L^{2}(|\nu_{n}|)$$\{\tilde{f_{j}}\}_{j\in N}$$L^{2}(|\nu_{n}|)$$\hat{f}_{j_{1}}$$l$$|\nu_{n}|- a.e$(3)$\{j_{l}\}\uparrow\infty$$f(z)=\lim\inf_{l\rightarrow\infty}\hat{f}_{j_{l}}(z)$$\rightarrow\infty$$\int_{\hat{F}_{m}}\hat{f}_{j_{I}}hd\nu_{n}=\hat{T}_{n}(\hat{f}_{j_{l}}h|_{\hat{F}_{m}})=T_{n}(\hat{f}_{j_{\iota}}h)=T(\psi_{n}\hat{f}_{j_{l}}h)$.

byWhen tends to , the left-hand side of (3) converges tothe dominated convergence theorem. On the other hand,is bounded in $D$ by (A4)’. Since they converge to 0
, they also$D$
by Lemma 2.1. Therefore, the right-handconverge weakly to 0 inside of (3) converges to 0 as
. Namely,. Since$\{h|_{\hat{F}_{m}}|h\in A\}$ is dense in, we conclude that $fd\nu_{n}=0$ , therefore,$|\nu_{n}|(K)=0$ .
Q.E.D.
$l$$\infty$$\int_{\hat{F}_{m}}fhd\nu_{n}$$\{\psi_{n}\hat{f}_{j_{l}}h\}_{l\in N}$$\lambda- a.e.$$l$$\rightarrow\infty$$\int_{\hat{F}_{m}}fhd\nu_{n}=0$$C(\hat{F}_{m})$Lemma 3.4. For all $f\in D_{b}$ ,$T_{n}(f)=\int_{X}\tilde{f}d\nu_{n}$Proof. We can take a sequence.

from $D$ so thatconverges to in $D$ and,is bounded inconverges to. Note that
. Then,outside some Borel exceptional set, whileas$L^{\infty}(X)$$\{f_{j}\}_{j\in N}$$T_{n}(f_{j})$$\tilde{f}$$\tilde{f}_{j}$$f$$f_{j}$$N_{0}$$T_{n}(f_{j})\rightarrow T_{n}(f)$$\{f_{j}\}_{j\in N}$$\overline{f}_{j}|_{\hat{F}_{m}}\in C(\hat{F}_{m})$$ j\rightarrow\infty$$=$$\hat{T}_{n}(\tilde{f}_{j}|_{\hat{F}_{m}})=\int_{X}\tilde{f}_{j}d\nu_{n}=\int_{X\backslash N_{0}}\tilde{f}_{j}d\nu_{n}$$ j\rightarrow\infty\rightarrow$$\int_{X\backslash N_{0}}\tilde{f}d\nu_{n}=\int_{X}\tilde{f}d\nu_{n}$by means of the dominated convergence theorem.

For any ,$k$$l$$\in \mathbb{N}$, we have$\tilde{\psi}_{k}d\nu_{l}=\tilde{\psi}_{l}d\nu_{k}$Q.E.D.

. Indeed, For $f\in A$ ,$\int_{X}f\tilde{\psi}_{k}d\nu_{l}=T_{l}(f\psi_{k})=T(\psi_{l}f\psi_{k})=T_{k}(f\psi_{l})=\int_{X}f\tilde{\psi}_{l}d\nu_{k}$

.

133

Integral representation

Therefore, we can define a signed smooth measure$(n=1,2, \ldots)$ , which is well-defined by the fact that
, we have $\psi_{k}f=f$ andfor any$\nu$

by $\nu=\nu_{n}$ on. Thenon$F_{n}^{/}$$\overline{\psi}_{n}=1$$F_{n}’$$f\in D_{b,F_{k}’}$$ T(f)=T(\psi_{k}f)=T_{k}(f)=\int_{X}\tilde{f}d\nu_{k}=\int_{X}\overline{f}d\nu$.

Thus, (ii) holds.
In order to prove the uniqueness of , it is enough to show that
, whereif
for all
is a nest attached, takewith . Following the same procedure as in the proof of, the function space $A$ , and the sequence
the nests
and. For any
and $f\in A$ , we haveof functions$\nu\equiv 0$$\nu$$v\in\bigcup_{k}D_{b,F_{k}}$$\int_{X}\tilde{v}d\nu=0$$\{F_{k}\}_{k\in N}$$(i)\Rightarrow(ii)$$\nu$$\{\hat{F}_{k}\}_{k\in N}$$\{F_{k}^{/}\}_{k\in N}$$n\in \mathbb{N}$$\{\psi_{n}\}_{n\in N}$$ f\psi_{n}\in D_{b,\hat{F}_{n}}\subset$,, therefore. Since $\{f|_{\hat{F}_{n}}|f\in A\}$ is dense in.
. In particular, $\nu=0$ on
becauseon
we have.
This implies that
The implicationof Theorem 2.19 is proved in the same way
as in Theorem 2.18. Because of the result and the proof ofof Theorem 2.18, Theorem 2.19 (i) implies that there exists a finite
signed smooth measure with some attached nestsuch that$C$$T(v)=\int_{X}\tilde{v}(z)\nu(dz)$for allandthe total variation is dominated by. It is easy to show that this identity holds for all $v\in D_{b}$by an approximation argument.
The uniqueness of is clear from the corresponding result of Theorem 2.18. The final claim is also deduced by an approximation argument
and the use of Lemma 2.1.
This completes the proof of Theorems 2.18 and 2.19.
$C(\hat{F}_{n})$$\int_{X}f\tilde{\psi}_{n}d\nu=0$$D_{b,F_{n}}$$\tilde{\psi}_{n}d\nu=0$$\tilde{\psi}_{n}=1$$F_{n}^{/}$$F_{n}^{/}$$\nu\equiv 0$$(ii)\Rightarrow(i)$$(i)\Rightarrow(ii)$$\{F_{k}^{J}\}_{k\in N}$$\nu$$v\in\bigcup_{k}D_{b,F_{k}’}$$\nu$

Application to $BV$ functions on Wiener space\S 4.

First, we will review some results of [9]. Let $E$ be a separable Banach
space and $H$ a separable Hilbert space which is continuously and densely
imbedded to $E$ . We use the notations in the end of Section 2 with letting$X=E$ . Define a Gaussian measure on $E$ by the following identity:$\mu$$\int_{E}\exp(\sqrt{-1}\ell(z))\mu(dz)=\exp(-||\ell||_{H}^{2}/2)$,$\ell\in E^{*}\subset H$.

When $Y$ is a separable Hilbert space and is a nonnegative measurable function on $E$ , we denote by $L^{p}(E\rightarrow Y;\rho)$ in this section thespace consisting of $Y$ -valued functions on $E$ with underlying mea. We omit
andsure
and from the notation when, respectively, and write simply. We also set
for$\rho$$L^{p}$$\rho d\mu$$\rho\equiv 1$$E\rightarrow Y$$Y=\mathbb{R}$$\rho$$L^{p}$$L^{p}(E\rightarrow \mathbb{R};1)$

134

M. Hino$L^{\infty-}=\bigcap_{p>1}L^{p}$and denote by$L_{+}^{p}$

the set of all nonnegative functions

in $IP$ .
If $u\in FC_{b}^{1}$ and $v\in FC_{b}^{1}$ coincide on a measurable set $A$ , then$\nabla u=\nabla v\mu- a.e$ . on $A$ . See Proposition 1.7.1.4 of [4] for the proof.
For $p\geq 1$ , $Cl_{p}(E)$ denotes the set of all functions insuch$L^{p}(E\rightarrow
H;\rho)$thatis closable as a map ﬀomto. A
simple example for such is a function which is uniformly away ﬀom 0.
Suppose $\rho\in Cl_{p}(E)$ . We write $W^{1,p}(\rho)$ instead of $W^{1,p}$ when regarding$(E, \rho d\mu)$ as
in Section 2. When $p>1$ , $W^{1,p}(\rho)$ satisfies all theconditions $(A1)-(A4)$ , $(QR1)-(QR3)$ and (C).
Let
be the topological support of the measure. Since$Y;\rho)$ is identified with, we abuse the notation and$W^{1,p}(\rho)$ is also regarded as a function space on. When $\rho\in Cl_{2}(E)$ ,an associated Dirichlet formon
is defined by$\rho$$(\nabla, FC_{b}^{1})$$L_{+}^{1}$$L^{p}(\rho)$$\rho$$(X, \lambda)$$F^{\rho}$$\rho d\mu$$ L^{0}(E\rightarrow$$L^{0}(F^{\rho}\rightarrow Y;\rho)$$F^{\rho}$$(\mathcal{E}^{\rho}, W^{1,2}(\rho))$$\mathcal{E}^{\rho}(f, g)=\int_{F^{\rho}}\langle\nabla f, \nabla g\rangle_{H}\rho d\mu$,$L^{2}(F^{\rho};\rho)$$f$, $g\in W^{1,2}(\rho)$ .

This is a quasi-regular Dirichlet form and a finite signed measure on
is smooth with respect to
if and only if is $W^{1,2}(\rho)$ smooth
For each $G\in(FC_{b}^{1})_{E^{*}}$ , the (formal) adjoint
is defined by thefollowing identity:$\nu$$F^{\rho}$$\mathcal{E}^{\rho}$$\nu$$\nabla^{*}G$$\int_{E}(\nabla^{*}G)ud\mu=\int_{E}\langle G, \nabla u\rangle_{H}d\mu$

for all $u\in FC_{b}^{1}$ .

Denote by $L(\log L)^{1/2}$ the space of all functions on $E$ such that$\Phi\circ|f|\in L^{1}$ , where $\Phi(x)=x((\log x)\vee 0)^{1/2}$ .
We say that a real$E$measurable function on
is of bounded variation $(\rho\in BV(E))$ if$\rho\in L(\log L)^{1/2}$ and$f$$\rho$$ V(\rho):=\sup_{G}\int_{E}(\nabla^{*}G)\rho d\mu<\infty$,

where $G$ is taken over all functions insuch that $||G(z)||_{H}\leq 1$
$z\in
E$
.
for everybe the Ornstein-Uhlenbeck semigroup, which is associLet
ated with . It is strongly continuous, analytic and contractive on
for any $p\in(1, \infty)$ .
We recall some results discussed in [9].
$(FC_{b}^{1})_{E^{*}}$$\{T_{t}\}_{t>0}$$\mathcal{E}^{1}$$L^{p}$(i) For $\rho\in BV(E)$ ,Theorem 4.1.
forevery $t>0$ .
(ii) $BV(E)$ is a vector lattice. Namely, it is a vector space, andalso belongs to $BV(E)$ .
for each $\rho\in BV(E)$ ,$||\nabla T_{t}\rho||_{L^{1}}\leq V(\rho)$$\rho_{+}$

135

Integral representation(iii)

belongs to $BV(E)$ if and only if $\rho\in L^{1}$ andA function
in $W^{1,1}(:=W^{1,1}(1))$ such that
there exists a sequence.
is bounded in andinas
Each $\rho\in BV(E)$ has a unique fifinite Borel measure andon $E$ such that $||\sigma||_{H}=1$
unique $H$ -valued Borel function. and for every $G\in(FC_{b}^{1})_{E^{*}}$ ,$\rho$$\{\rho_{n}\}_{n\in N}$$||\rho_{n}||_{W^{1,1}}$(iv)$L^{1}$$n$$ n\rightarrow\infty$$\rho_{n}\rightarrow\rho$

$a$

$\nu$

$\sigma$$\nu- a.e$$\int_{E}(\nabla^{*}G)\rho d\mu=\int_{E}\langle G, \sigma\rangle_{H}d\nu$

The measure
then

$\nu$

.

is $W^{1,2}(|\rho|+1)$ -smooth. If moreover $\nu\in Cl_{2}(E)$ ,and is $W^{1,2}(\rho)$ -smooth.

$\nu|_{E\backslash F^{\rho}}=0$$\nu$

for in the theorem above. In thisIn what follows, we will write
section, we improve the result for the smoothness of . In view of theproof of Theorem 4.1 (iv) (Theorem 3.9 of [9]), the smoothness ofis, whereis a unique
derived from the smoothness of for eachfinite signed measure on $E$ satisfying$\nu$$\nu_{\rho}$$\nu_{\rho}$$\nu_{\rho}$$\ell\in E^{*}$$\nu_{\ell}$$\int_{E}\partial_{\ell}u(z)\rho(z)\mu(dz)=-2\int_{E}u(z)\nu_{\ell}(dz)$Therefore, applying Theorem 2.19 withfunctional(4)$I_{\ell}$:$\nu_{\ell}$,$\mathcal{L}=FC_{b}^{1}$$u\in FC_{b}^{1}$.

, if we show that the$FC_{b}^{1}\ni u\mapsto\int_{E}\partial_{\ell}u(z)\rho(z)\mu(dz)\in \mathbb{R}$extends continuously on $D$ , where $D$ satisfies $(A1)-(A4)$ , $(QR1)-(QR3)$ ,and (C), and has
as a dense set, then we can say that , hence ,is $D$ -smooth. It is obvious that
extends to a continuous functional on$W^{1,p}(|\rho|+1)$ (and $W^{1,p}(\rho)$ if furthermore) for every $p\geq 1$ . Also,iffor some $q\in(1, \infty)$ , then extends to a continuous functional$W^{1,q/(q-1)}(:=W^{1,q/(q-1)}(1))$
onby H\"older’s inequality. Therefore, wehave the following results.
$FC_{b}^{1}$$\nu_{\ell}$$\nu_{\rho}$$I_{\ell}$$\rho\in Cl_{p}$$\rho\in L^{q}$$I_{\ell}$

Proposition 4.2. Let $\rho\in BV(E)$ . Then,
is $W^{1,p}(|\rho|+1)$ -smooth, then is $W^{1,p}(\rho)$ -smooth.
for every $p>1$ . If moreover$\nu_{\rho}$$\rho\in Cl_{p}$$\nu_{\rho}$$\nu$

Proposition 4.3. Let $\rho\in BV(E)\cap L^{q}$ for some $q\in(1, \infty)$ . Then,
is $W^{1,q/(q-1)}$ -smooth.

In Proposition 4.2, the smaller is, the stronger the claim is.
Now, we will give other examples of $D$ so thatis $D$-smooth. Let
us recall the Sobolev spaces in the context of Malliavin calculus. We
give several notations in somewhat informal way. We refer to [12] for$p$$\nu_{\rho}$

136

M. Hinoprecise definitions. Let $ L=-\nabla^{*}\nabla$ be the Ornstein-Uhlenbeck operator,,
which is regarded as a generator of. The Sobolev space$D^{\alpha,p}=(1-L)^{-\alpha/2}(L^{p})$. Each, $ 1<p<\infty$ , is given byis$||f||_{D^{\alpha,p}}:=||(1-L)^{\alpha/2}f||_{L^{p}}$. Thea separable Banach space with norm
, $q=p/(p-1)$ . When
topological dual ofis identified with, by Meyer’s equivalence,is defined as a continuous operator
from
to
and
) gives a normonwhich is equivalent to. In particular, $W^{1,p}(:=W^{1,p}(1))$
is identical withas a set and their norms are mutually equivalent.
,, $ 1<p<\infty$ , firstly
We define another Sobolev space
introduced in [24], by$D^{\alpha,p}$$\{T_{t}\}_{t>0}$$D^{\alpha,p}$$\alpha\in \mathbb{R}$$D^{-\alpha,q}$$D^{\alpha,p}$$n\in \mathbb{N}$$\nabla^{n}$$D^{n,p}$$L^{p}(E\rightarrow H^{\otimes n})$

$D^{n,p}$$||\cdot||_{L^{p}}+||\nabla^{n}\cdot||_{L^{p}(E\rightarrow H\otimes n}$$||\cdot||_{D^{n,p}}$$D^{1,p}$$E^{\alpha,p}$$D^{\alpha,p}$$\alpha\in \mathbb{R}$,if$k<\alpha<k+1$,
if$\alpha\in \mathbb{Z}$$E^{\alpha,p}=\{$$(D^{k+1,p}, D^{k,p})_{k+1-\alpha,p}$$k\in \mathbb{Z}$.

The general theory of real interpolation implies that
is identified
with, where $q=p/(p-1)$ (see also [24]). When $0<\alpha<1$ and$ 1<p<\infty$ ,satisfies conditions $(A1)-(A4)$ , $(QR1)-(QR3)$ , and (C)by virtue of Proposition 2.21, if
is equipped with a norm deduced$(W^{1,p},L^{p})_{1-\alpha,p}$. For such indices,bysince $W^{1,p}$is dense in
is dense in. For later use, following $[1, 2]$ , we introduce anotherequivalent norm on
based on the $K$ -method by$(E^{\alpha,p})^{*}$$E^{-\alpha,q}$$E^{\alpha,p}$$E^{\alpha,p}$$E^{\alpha,p}$$FC_{b}^{1}$$E^{\alpha,p}$$E^{\alpha,p}$$||f||_{E^{\alpha,p}}^{J}=(\int_{0}^{1}(\in\alpha K-(\in, f))^{p}\frac{d\in}{\in})^{1/p}$

,

where
$K(\in, f)=\inf\{||f_{1}||_{L^{p}}+\in||f_{2}||_{W^{1,p}}|f=f_{1}+f_{2}, f_{1}\in L^{p}, f_{2}\in D^{1,p}\}$

The connection between $BV(E)$ and$E^{\alpha,p}$.

is given as follows.

Theorem 4.4. Let $q>1$ . The $BV(E)\cap L^{q}\subset E^{\alpha,p}$ if $1<p<q$
and $\alpha<(1/p-1/q)/(1-1/q)$ . Also, this inclusion is continuous when
$BV(E)\cap L^{q}$ is equipped with norm $||f||_{BV(E)\cap L^{q}}=V(f)+||f||_{L^{q}}$ . In
particular, $BV(E)\cap L^{\infty-}\subset E^{\alpha,p}$ if $p>1$ and $\alpha p<1$ .

For the proof, we need the following estimates.

Lemma 4.5.

(i) When $\theta/a+(1-\theta)/b=1/p$ with $0<\theta<1$ ,, , $p\geq 1$ , we have.
For each $r\geq 0$ and $p\in(1, \infty)$ , there exists some $C$ such that$||(1-L)^{r}T_{t}f||_{L^{p}}\leq Ct^{-r}||f||_{L^{p}}$for every $t\in(0,1]$ and $f\in L^{p}$ .

$a$

(ii)

$b$

$||f||_{L^{p}}\leq||f||_{L^{a}}^{\theta}||f||_{L^{b}}^{1-\theta}$

Proof.

The claim (i) follows from a simple application of H\"older’sinequality. The claim (ii) is a consequence of Theorem 6.13 (c) of Chapter 2 in [18], since
is an analytic semigroup on
Q.E.D.
.
$\{T_{t}\}_{t>0}$$L^{p}$

137

Integral representation

Proof of Theorem$4\cdot 4$. Let $f\in BV(E)\cap L^{q}$ with $V(f)+||f||_{L^{q}}\leq 1$ .

In the following,
denotes a constant depending only on and . By
Theorem 4.1 (i), $||\nabla T_{t}f||_{L^{1}}\leq V(f)\leq 1$ for any $t>0$ . By virtue ofMeyer’s equivalence and Lemma 4.5 (ii), for $t\in(0,1]$ ,$p$$c_{\iota}$$||\nabla T_{t}f||_{L^{q}}\leq c_{1}||(1-L)^{1/2}T_{t}f||_{L^{q}}\leq c_{2}t^{-1/2}$$g$.

Applying Lemma 4.5 (i) with $a=1$ and $b=q$ , that is, $\theta=(1/p-$$1/q)/(1-1/q)$ , we havefor $t\in(0,1]$ , therefore,$||\nabla T_{t}f||_{L^{p}}\leq(c_{2}t^{-1/2})^{1-\theta}$(5)$||T_{t}f||_{W^{1,p}}\leq c_{3}t^{-(1-\theta)/2}$.

From the identity$f-T_{t}f$$=$$-\int_{0}^{t}\frac{d}{ds}T_{s}fds=-\int_{0}^{t}LT_{s}fds$$=$$\int_{0}^{t}\{((1-L)^{1/2}T_{s/2})^{2}f-T_{s}f\}ds$,

we obtain, for $t\in(0,1]$ ,(6)$||f-T_{t}f||_{L^{p}}$$\leq$$\int_{0}^{t}||((1-L)^{1/2}T_{s/2})^{2}f||_{L^{p}}ds+t||f||_{L^{p}}$$\leq$$\int_{0}^{t}c_{4}s^{-1/2}||(1-L)^{1/2}T_{s/2}f||_{L^{p}}ds+t$$\leq$$\int_{0}^{t}c_{5}s^{-1/2}||T_{s/2}f||_{W^{1,p}}ds+t$$\leq$$\int_{0}^{t}c_{6}s^{-1/2}s^{-(1-\theta)/2}ds+t\leq c_{7}t^{\theta/2}$.

Here we used Lemma 4.5 (ii) in the second line and (5) in the last line.
By combining (5) and (6), for $each\in\in(0,1]$ ,$K(\in, f)\leq||f-T_{\epsilon^{2}}f||_{L^{p}}+\in||T_{\epsilon^{2}}f||_{W^{1,p}}\leq c_{8}\in^{\theta}$

and, if$\alpha\in(0, \theta)$,,$(\int_{0}^{1}(\in\alpha K-(\in, f))^{p}\frac{d\in}{\in})^{1/p}\leq c_{8}\{p(\theta-\alpha)\}^{-1/p}<\infty$

Q.E.D.

This proves the claim.

Using Theorem 4.4, we obtain the
proposition.

.

$E^{\alpha,p}$-smoothness of$\nu_{\rho}$by the following

138

M. Hino

Proposition 4.6. Let $\rho\in BV(E)\cap L^{q}$ , $q>1$ . Then the map
in (4) extends continuously on
if $p>q/(q-1)$ and $\alpha p>q/(q-$. Therefore,is-smooth for suchand with $\alpha\in(0,1)$ . In
particular, if $\rho\in BV(E)\cap L^{\infty-}$ , then-smooth for any ,is$\alpha\in(0,1)$$\alpha
p>1$
.
with
and
$I_{\ell}$$E^{\alpha,p}$$E^{\alpha,p}$$1)$$\alpha$$p$$\nu_{\rho}$$E^{\alpha,p}$$\alpha$$p$$\nu_{\rho}$

Proof. Due to Meyer’s equivalence, the map

is continuousto, respectively. By the real interpolafrom
toand from
tion theorem, it is continuous ﬀom
tofor any $\alpha\in(0,1)$ . The
claim follows from the fact $(E^{1-\alpha,p/(p-1)})^{*}=E^{\alpha-1,p}$ and $ BV(E)\cap L^{q}\subset$$E^{1-\alpha,p/(p-1)}$
Q.E.D.
by the assumption and Theorem 4.4.
$D^{1,p}$$L^{p}$$u\mapsto\partial_{\ell}u$$D^{-1,p}$$L^{p}$$E^{\alpha,p}$$E^{\alpha-1,p}$(i) In [24], it is proved that
REMARK 4.7.
for every, $ 1<p<\infty$ and $\in>0$ . Therefore,
Theorem 4.4 and Proposition 4.6 remain valid if we replace.
by(ii) When $\rho\in BV(E)$ is an indicator function of some set $A$ ,can$A$
be regarded as a surface measure of . The smoothness ofthat is proved in the proposition above is consistent with Theorem 9 of [6] saying that the Hausdorﬀ measure of codimension
on Wiener space does not charge any set of-capacity
as long as $p>1$ and $\alpha p>n$ .
$D^{\alpha+\in,pL}\rightarrow E^{\alpha,p}\simeq\rightarrow$$D^{\alpha-\in,p}$$E^{\alpha,p}$$\alpha\in \mathbb{R}$$D^{\alpha,p}$$\nu_{\rho}$$\nu_{\rho}$$(\alpha,p)$$n$Lastly, we give a few nontrivial examples of $BV$ functions, referringto the work [2]. Note that by combining Theorem 4.8 and Theorem 4.4
we recover a part of the results in [2].

Theorem 4.8.

(i)

Let $F$ be a function such that $F\in D^{2,p}$$and||\nabla F||_{H}^{-1}\in L^{q}$
for some $p>1$ and $q>1$ with $1/p+1/q<1$ .
Let $A=\{F<x\}$ with. Then $1_{A}\in BV(E)$ .
$(E,
H,
\mu)$
Suppose thatis a classical Wiener space on $[0, 1]$ .
For $x>0$ , set $A=\{w\in E|\max_{0\leq s\leq 1}|w(s)|<x\}$ . Then$x\in \mathbb{R}$(ii)$1_{A}\in BV(E)$ .

Proof.

(i): From the assumptions, we havefor some $a>1$ . Indeed, keeping in mind the fact,$let\intend$
due to Meyer’s equivalence,
to 0 in the identity$\nabla^{*}(\nabla F/||\nabla F||_{H})\in L^{a}$$||\nabla F||_{H}$$||\nabla^{2}F||_{H\otimes H}\in$$L^{p}$$\nabla^{*}(\frac{\nabla F}{\sqrt{||\nabla F||_{H}^{2}+\in}})=-\frac{LF}{\sqrt{||\nabla F||_{H}^{2}+\epsilon}}+\frac{\langle\nabla F\otimes\nabla F,\nabla^{2}F\rangle_{H\otimes H}}{(||\nabla F||_{H}^{2}+\in)^{3/2}}$

Now, set $\psi_{n}(y)=n1_{[x-1/n,x]}(y)$ ,. Then we have$\varphi_{n}(y)=\int_{y}^{\infty}\psi_{n}(z)dz$$\varphi_{n}(F)$$||\nabla\rho_{n}||_{L^{1}(E\rightarrow H)}$$=$$\int_{E}\psi_{n}(F)||\nabla F||_{H}d\mu$.

, and $\rho_{n}=$139Integral representation,$=$$-\int_{E}\langle\nabla\rho_{n}$$=$$-\int_{E}\rho_{n}\nabla^{*}(\frac{\nabla F}{||\nabla F||_{H}})d\mu$$\leq$$||\nabla^{*}(\frac{\nabla F}{||\nabla F||_{H}})||_{L^{1}}$$\frac{\nabla F}{||\nabla F||_{H}}\rangle_{H}d\mu$,

is uniformly bounded and conwhich is bounded in . Since
pointwise, Theorem 4.1 (iii) completes the proof.
verges to(ii): Set $\rho_{n}(w)=0\vee n(1-\max_{0\leq s\leq 1}|w(s)|/x)\wedge 1$ , $w\in E$ for each
. By the calculation in the proof of Theorem 3.1 of [2], we haveis bounded in . Since
is uniformly
andpointwise, Theorem 4.1 (iii) completes thebounded and tends to
Q.E.D.
proof.
$\{\rho_{n}\}_{n\in N}$$n$$1_{A}$$n\in \mathbb{N}$$\rho_{n}\in W^{1,1}$$||\rho_{n}||_{W^{1,1}}$$n$$\{\rho_{n}\}_{n\in N}$$1_{A}$

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[3] B. Beauzamy, Espaces d’interpolation r\’eels: topologie et g\’eom\’etrie, Lecture Notes in Math. 666, Springer, Berlin, 1978.
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the French by H. G. Eggleston and S. Madan, Springer, Berlin, 1987.
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Potential Anal. 1 (1992), 177-189.
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symmetric Markov processes, Electron. J. Probab. 4 (1999), no. 18,
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M. Hino[11] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics 19, Walterde Gruyter, Berlin, 1994.
[12] N. Ikeda and S. Watanabe, Stochastic diﬀerential equations and diﬀusion
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415-444, Lecture Notes in Math. 1526, Springer, Berlin, 1992.
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Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992.
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$C_{1,p}$Graduate School of Informatics
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Japan

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 141-166

Least-Squares Approximation of
Random Variables by Stochastic Integrals
Chunli Hou and Ioannis Karatzas
Dedicated to Professor K. It\^o on the occasion of his 88th
Birthday
Abstract.
This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic
integral with respect to a given semimartingale $X$ . The criterion isminimization of $L^{2}-$ distance, or “least-squares”. This problem has
a straightforward and well-known solution when $X$ is a Brownian
motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$ . We address the general, semimartingale case by means of a duality approach; the adjoint
variables in this duality are signed measures, absolutely continuouswith respect to $P$ , under which $X$ behaves like a martingale. It isshown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it“has no gap” (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem
can be easily identified whenever it exists; and that the duality is also“strong” , in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented
here both simplifies and extends the extant work on the subject.
It has also natural connections and interpretations in terms of thetheory of “variance-optimal” and “mean-variance eﬃcient” portfolios in Mathematical Finance, pioneered by H. Markowitz and thengreatly extended by H. F\"ollmer, D. Sondermann and most notably
M. Schweizer.

Received November 21, 2002.
This work was supported in part by the National Science Foundation
under Grant NSF-DMS-00-99690.

C. Hou and I. Karatzas

142

\S 1.

Introduction

Suppose we are given a square-integrable, $d-$ dimensional process$X=\{X(t);0\leq t\leq T\}$ defined on the finite time-horizon $[0, T]$ , which
is a semimartingale on the filtered probability space $(\Omega, F, P)$ , $F=$
. How closely can we approximate in the sense of leastsquares a given, square-integrable and $F(T)-$ measurable random vari? Hereis aable $H$ , by a linear combination of the form
real number and a predictable $d-$ dimensional process for which the.
.
is well-defined and is itself
stochastic integral
a square-integrable semimartingale.
In other words, if we denote by the space of all such processes ,how do we compute$\{F(t)\}_{0\leq t\leq T}$$c+\int_{0}^{T}\theta^{l}dX$$c$$\theta$$\int_{0}$$\theta^{l}dX\equiv\sum_{i=1}^{d}\int_{0}$ $\theta_{i}dX_{i}$$\theta$$\Theta$(1.1)$V(c)=\triangle\inf_{\theta\in\ominus}E(H-c-\int_{0}^{T}\theta^{l}dX)2$

if
is given and we have the ﬀeedom to choose
as above? How do we find$c\in \mathbb{R}$(1.1)$\theta$over the class$\Theta$$V=\triangle\inf_{(c,\theta)\in \mathbb{R}\times\ominus}E(H-c-\int_{0}^{T}\theta^{J}dX)^{2}=\inf_{c\in \mathbb{R}}V(c)$when we have the freedom to select both and ? And how do wethatcharacterize, or even compute, the process
and the pairattain the infimum in (1.1) and (1.2), respectively, whenever these exist ?
To go one step further: How does one$c$$\theta$$\theta^{(c)}$(1.3)

over all(1.4)

minimize the variance$\theta\in\Theta$$(\hat{c},\hat{\theta})$

Var $(H-\int_{0}^{T}\theta^{l}dX)$

as above? Or even more interestingly, how does one$\{minimizethevarianceVar_{0}(H-\int_{=\theta^{l}dX]\mu}0\theta’TdX)OVe\Gamma\theta\in\Theta withE[\int^{T}\}$?
for some given
Questions such as (1.3) and (1.4) can be traced back to the pioneering work of H. Markowitz $(1952, 1959)$ , and have been studied
more recently by F\"ollmer&Sondermann (1986), F\"ollmer&Schweizer$\mu\in \mathbb{R}$Least-squares approximation of random variables by stochastic integrals 143(1991), Duﬃe &Richardson (1991), Sch\"al (1992), in the modern context of Mathematical Finance. Most importantly, problems (1.1)-(1.4)
have received an exhaustive and magisterial treatment in a series of
papers by Schweizer $(1992, 1994, 1995.a,b, 1996)$ and his collaborators (cf. Rheinl\"ander &Schweizer (1997), [Ph.R.S.] (1998), [DMSSS](1997), as well as Hipp (1993), [G.L.Ph.] (1996), Laurent &Pham(1999), Grandits (1999), Arai (2002) . In this context, the compo, $i=1$ ,, of the semimartingale $X$ are interpreted as thenents(discounted) stock-prices in a financial market, and $H$ as a contingent
claim, or liability, that one is trying to replicate as faithfully as possible
at time $T$ , starting with initial capital and trading in this market. Such
trading is modelled by the predictable portfolio process , whose component
represents the number of shares being held at time in the, . Thenstock, for $i=1$ ,corresponds to the (discounted) gains fiom trading accrued by the terminal
time $T$ , with which one tries to approximate the contingent claim $H$ , andone might be interested in minimizing the variance of this approximation over all admissible portfolio choices (problem of (1.3)), or just overthose portfolios that guarantee a given mean-rate-of-return (problem of(1.4) .
It turns out that solving the problem of (1.1) provides the key toattains the infimum inanswering all these questions. For instance, if
is optimal for the problem, then(1.1) andof (1.3); the pairis optimal for the problem of (1.2); and theprocesswith $c_{\mu}=(E[\pi(H)]-E(H)+\mu)/(E[\pi(1)]-1)$ is optimalfor the problem of (1.4). Here denotes the projection operator ﬀom
the Hilbert space $L^{2}(P)$ onto the orthogonal complement of its linear.
subspace$)$$d$$X_{i}(\cdot)$$\ldots$$c$$\theta$$t$$\theta_{i}(t)$$i^{th}$$\int_{0}^{T}\theta^{l}dX\equiv\sum_{i=1}^{d}\int_{0}^{T}\theta_{i}(s)dX_{i}(s)$$d$$\ldots$$)$$\theta^{(c)}$$\hat{\theta}\equiv\theta^{(\hat{c})}$$\hat{c}\equiv\arg\min_{c\in \mathbb{R}}V(c)$$(\hat{c},\hat{\theta})$$\theta^{(c_{\mu})}$$\pi$$\{\int_{0}^{T}\theta^{l}dX/\theta\in\Theta\}$

The problem of (1.1) has a very simple solution, if $X$ is a (squareintegrable) martingale; then every $H$ as above has the so-called KunitaWatanabe decomposition$H=E(H)+\int_{0}^{T}(\zeta^{H})’dX+L^{H}(T)$ ,where
orthogonal to$\zeta^{H}\in\Theta$

andis a square-integrable martingale strongly. Then the infimum in (1.1)
for every$L^{H}(\cdot)$$\int_{0}.\theta^{l}dX$$\theta\in\Theta$

is computed as $V(c)=(E[H]-c)^{2}+E(L^{H}(T))^{2}$ and is attained by, which also attains the infimum $V=E(L^{H}(T))^{2}$ in (1.2).
In order to deal with a general semimartingale $X$ we develop a simple duality approach, which in a sense tries to reduce the problem to$\zeta^{H}\in\Theta$

144

C. Hou and I. Karatzas

the “easy” martingale case just described. This approach is the main
contribution of the present paper. The dual or “adjoint” variables inthis duality are signed measures $Q$ , absolutely continuous with respect
to $P$ and with $dQ/dP$ $\in L^{2}(P)$ , under which $X$ behaves like a martingale (Definition 2.1 and Remark 2.2). A simple observation, describedin (3.1)-(3.7), leads to a dual maximization problem. The resulting duality is useful because, as it turns out, the dual problem is relatively
straightforward to solve (Proposition 3.1); its value is easily computed
as $E[\pi^{2}(H-c)]$ and coincides with the value $V(c)$ of the original problem (1.1), so there is no “duality gap”; and furthermore the duality isof (1.1)“strong”, in that one can identify the optimal integrandrather easily, under suitable conditions (Theorem 4.1 and Remark 4.1).
Several examples are presented in Section 5.
$\theta^{(c)}$We follow closely the notation and the setting of Schweizer (1996),
our great debt to which should be clear to anyone familiar with thisexcellent work. Indeed, the present paper can be considered as complementing and extending the results of this work, by means of our simple
duality approach.

\S 2.

The Problem

On a given complete probability space $(\Omega, F, P)$ equipped with afiltration $F=\{F(t) ; 0\leq t\leq T\}$ that satisfies the usual conditions,consider a process(2.1)$X(t)=X(0)+M(t)+B(t)$ ,$0\leq t\leq T$,

defined on the finite time-horizon $[0, T]$ and belonging to the space$S^{2}\equiv S^{2}(P)$ of square-integrable-dimensional semimartingales. Thisbelongs tomeans that each $X_{i}(0)$ is in $L^{2}(\Omega, F(0),$ $P)$ ; that eachthe spaceof square-integrable -martingales with $M_{i}(0)=0$ and, whereare
RCLL paths; and that we have$A_{i}^{\pm}(0)=0$
increasing, right-continuous and predictable processes with
, . We denote bytheand $ E(A_{i}^{\pm}(T))^{2}<\infty$ , for every $i=1$ ,space of “good integrands” for the square-integrable semimartingale$X=\{X(t), 0\leq t\leq T\}$ , namely, those $F-$ predictable processes whose
stochastic integrals with respect to $X$ are themselves square-integrable
semimartingales:$d$$M_{i}(\cdot)$$\mathcal{M}_{0}^{2}(P)$$F$$A_{i}^{\pm}(\cdot)$$B_{i}(\cdot)=A_{i}^{+}(\cdot)-A_{i}^{-}(\cdot)$$\cdots$$d$$\Theta$.

(2.2)$\Theta=\triangle\{\theta\in \mathcal{L}(X)/G.(\theta)\equiv\int_{0}$$\theta’dX\in S^{2}(P)\}$.

Least-squares approximation of random variables by stochastic integrals 145

Herestands for the space of all
cesses, whose stochastic integrals$\mathcal{L}(X)$(2.3)$\mathbb{R}^{d}$

Revalued and predictable pro-

$G_{t}(\theta)=\triangle\int_{0}^{t}\theta’(s)dX(s)=\sum_{i=1}^{d}\int_{0}^{t}\theta_{i}(s)dX_{i}(s)$,$o\leq t\leq T$with respect to $X$ are well-defined.
Suppose now we are given a random variable $H$ in the space $L^{2}(P)\equiv$$(\Omega, F(T)$ , $P)$ . The following problem will occupy us in this paper.
$L^{2}$

Problem 2.1. Given H $\in L^{2}(P)$ , compute(2.4)$ V=\triangle$$\inf$$E(H-c-G_{T}(\theta))^{2}$$(c,\theta)\in \mathbb{R}\times\ominus$

and try to find a pair
pair exists.

$(\hat{c},\hat{\theta})\in \mathbb{R}\times\Theta$that attains the infimum, if such$a$$\square $In other words, we are looking to find the least-squares approximation of$H$ , as the sum of a constant,and of the stochastic integral.
for some processThis problem has a rather obvious solution, if it is known that the
random variable $H$ is of the form
$c\in \mathbb{R}$$G_{T}(\theta)$$\theta\in\Theta$(2.5)$H=h+G_{T}(\zeta^{H})$,; because then we can takefor some
and
$V=0$
it
in (2.4). Now is a classical result (e.g. Karatzas
and deduce that
&Shreve (1991), pp. 181-185 for a proof) that every $H\in L^{2}(P)$ can be
is Brownian
written in the form (2.5), in fact with $h=E(H)$ , ifgenerated bymotion and if is the (augmentation of the) filtrationin terms of theitself. One can then also describe the integrandfamous Clark (1970) formula, under suitable conditions on the randomvariable $H\in L^{2}(P)\equiv L^{2}(\Omega, F^{X}(T),$ $P)$ . Thus, in this special case, we
can take, and have $V=0$ in (2.4).
is a squareA little more generally, suppose that
integrable martingale (i.e., $X(0)=0$ andin (2.1)). Then again$H\inL^{2}(P)$admits the so-called Kunitait is well-known that every
Watanabe (1967) decomposition$h\in \mathbb{R}$$\hat{c}\equiv h,\hat{\theta}\equiv\zeta^{H}$$\zeta^{H}\in\Theta$$X(\cdot)$$F^{X}$$F$$\zeta^{H}$$X(\cdot)$$\hat{c}=E(H),\hat{\theta}=\zeta^{H}$$X(\cdot)\in \mathcal{M}_{0}^{2}(P)$$A(\cdot)\equiv 0$(2.6)$H=h+G_{T}(\zeta^{H})+L^{H}(T)$, and some square-integrable martingale
for $h=E(H)$ , some;which is strongly orthogonal tofor every$\zeta^{H}\in\Theta$$L^{H}(\cdot)\in \mathcal{M}_{0}^{2}(P)$$G.(\theta)$$\theta\in\Theta$

C. Hou and I. Karatzas

146

in particular,(2.7)$E[L^{H}(T)\cdot G_{T}(\theta)]=0$,$\forall\theta\in\Theta$.

Then$E(H-c-G_{T}(\theta))^{2}=(h-c)^{2}+E(G_{T}(\zeta^{H}-\theta))^{2}+E(L^{H}(T))^{2}$ ,and it is clear that Problem 2.1 admits again the solution $\hat{c}=E(H)$ ,, but now with $V=E(L^{H}(T))^{2}$ in (2.4).
What happens for a general, square-integrable semimartingale$S^{2}(P)$ ? In view of the above discussion it is tempting to try and “reduce” this general problem to the case whereis a martingale. Thiscan be accomplished by absolutely continuous change of the probability
measure $P$ . We formalize this idea as in Schweizer (1996).
$\hat{\theta}=\zeta^{H}$$ X(\cdot)\in$$X(\cdot)$

Definition 2.1. A signed measure $Q$ on
martingole measure, if $Q(\Omega)=1$ , $Q<<P$ withis called a signed$(dQ/dP)\in L^{2}(P)$ , and$\Theta-$$(\Omega, \mathcal{F})$(2.8)$E[\frac{dQ}{dP}\cdot G_{T}(\theta)]$$=0$ ,$\forall\theta\in\Theta$

We shall denote by
the set of all such signed
sures, and introduce the closed, convex set$P_{s}(\Theta)$(2.8)$=\triangle$$D$.

$\Theta$-martingale mea-

{ $D\in L^{2}(P)/D=(dQ/dP)$ , some}
{ $D\in L^{2}(P)/E(D)=1$ and $E(DG_{T}(\theta))=0$ ,$Q\in P_{s}(\Theta)$$=$$\forall\theta\in\Theta$}.
$\square $We shall assume from now onwards, that
(2.10)$P_{s}(\Theta)\neq\emptyset$(equivalently$D$$\neq\emptyset$).

Remark 2.1 : The linear subspace $ G_{T}(\Theta)=\triangle\{\int_{0}^{T}\theta’(s)dX(s)/\theta\in$of the Hilbert space $L^{2}(P)$ is not necessarily closed for a general(it is, if $B(\cdot)\equiv 0$ in (2.1), or equivalently if
semimartingale
is a square-integrable martingale, since then the stochastic-integral of(2.3) is an isometry). For a semimartingale
with continuous paths,
necessary and suﬃcient conditions for the closedness of
havebeen provided by Delbaen, Monat, Schachermayer, Schweizer&Strieker[DMSSS] (1997); see also Theorem 2 in Rheinl\"ander&Schweizer (1997),
as well as Corollary 4 in Pham, Rheinl\"ander&Schweizer [Ph.R. S.] (1998)
and section 5, equation (5.11) of the present paper, for suﬃcient conditions. As noted by W. Schachermayer (see Schweizer (1996), p. 210;$\ominus\}$$X(\cdot)$$X(\cdot)$$X(\cdot)$$G_{T}(\Theta)$

Least-squares approximation of random variables by stochastic integrals 147

Lemma 4.1 in Schweizer (2001) , the assumption (2.10) is equivalent to
the requirement that$)$

(2.10)’$\left\{\begin{array}{lll}the & L^{2}(P)closureinof & G_{T}(\ominus)\\does & notcontainthe & constant1\end{array}\right\}$.

On the other hand, the orthogonal complement(2.11)$(G_{T}(O-))^{\perp}=\triangle\{D\in L^{2}(P)/E(DG_{T}(\theta))=0, \forall\theta\in\Theta\}$ofis a closed linear subspace of $L^{2}(P)$ , and its orthogonal complement
is the smallest closed, linear subspace of $L^{2}(P)$ that
. Clearly,
includes the set $V$ of (2.9), and thecontains
requirement $(2.10)’$ amounts to$G_{T}(\Theta)$$(G_{T}(\Theta))^{\perp\perp}$$(G_{T}(\Theta))^{\perp}$$G_{T}(\Theta)$(2.10)”$1\not\in(G_{T}(\Theta))^{\perp\perp}$

Remark 2.2 : The notion of signed -martingale measure in Defitself, as well as on the definiinition 2.1 depends on the space. In many cases of in,tion of the stochastic integralofterest, though, every $Q\in P_{s}(\Theta)$ belongs also to the space
$X$
$Q$
on
signed martingale measures for , namely those signed measures$\inL^{2}(P)$$Q(\Omega)=1$,
with $Q\ll P$, $dQ/dP$
and$\Theta$$\Theta$$G_{T}(\theta)$$\theta\in\Theta$$P_{s}^{2}(X)$$(\Omega, F)$(2.12)$E[\frac{dQ}{dP}.$$(X(t)-X(s))|F(s)]$ $=0$ ,

$a.s$

.

for any $0\leq s\leq t\leq T$ . (If $Q$ is a true probability measure, as opposed toa signed measure with $Q(\Omega)=1$ , then (2.12) amounts to the martingale
property of $X$ under $Q.$ ) See M\"uller (1985), Lemma 12(b) in Schweizer(1996), as well as conditions (5.1)-(5.4) and the paragraph immediately
following them in the present paper.
In addition to Problem 2.1, it is useful to consider also the following
question, which is interesting in its own right.
Problem 2.2. Given H $\in L^{2}(P)$ and c(2.13)$\in \mathbb{R}$, compute$V(c)=\triangle\inf_{\theta\in O-}E(H-c-G_{T}(\theta))^{2}$and try to find$\theta^{(c)}\in\Theta$that attains the infimum in (2.13), if such exists.

coincides with the quantity $V$ of (2.4); and if thisClearly,, then the pairlast infimum is attained by someattains theinfimum in (2.4). In the next Sections we shall try to solve Problems 2.2$\inf_{c\in \mathbb{R}}V(c)$$\hat{c}\in \mathbb{R}$$(\hat{c}, \theta^{(\hat{c})})$

C. Hou and I. Karatzas

148

and 2.1 using very elementary duality ideas. In this eﬀort, the elementsof the set $D$ in (2.9) will play the role of adjoint or dual variables. For
the duality methodology to work in any generality it is critical to allow,
as we did in Definition 2.1, for signed measures $Q$ with $Q(\Omega)=1$ , as
opposed to just standard probability measures.

\S 3.

The Duality

The duality approach to Problem 2.2 is simple, and is based on theelementary observation(3.1)$\min_{x\in \mathbb{R}}[(H-x)^{2}+yx]$$=$$(H-(H-y/2))^{2}+y(H-y/2)$$=$$yH-y^{2}/4$ ,$\forall y\in \mathbb{R}$.

The key idea now, is to read (3.1) with $x=c+G_{T}(\theta)$ , $y=2kD$ for
given
and arbitrary, $D\in D$ as in (2.2) and (2.9), and with, to obtainarbitrary$\theta\in O-$$c\in \mathbb{R}$$k\in \mathbb{R}$$(H-c -G_{T}(\theta))^{2}+2kD(c+G_{T}(\theta))\geq 2kDH-k^{2}D^{2}$ .

(3.2)

Note also that (3.2) holds as equality for some
$D$ and, if and only if$\theta\equiv\theta^{(c)}\in\Theta$, $ D\equiv D^{(c)}\in$$k\equiv k^{(c)}\in \mathbb{R}$(3.3)$c+G_{T}(\theta^{(c)})=H-k^{(c)}D^{(c)}$.

Now let us take expectations in (3.2) to obtain, in conjunction with theproperties of (2.9):(3.4)

for every(3.5)$E(H-c-G_{T}(\theta))^{2}\geq-k^{2}E(D^{2})+2k[E(DH)-c]$ ,$k\in \mathbb{R}$, $D\in D$ and$\theta\in\Theta$. Clearly,$E(D^{2})-1=$ Var(D)$\geq 0$,$\forall D\in D$

and the mapping $k\mapsto-k^{2}E(D^{2})+2k[E(DH)-c]$ attains overmaximal value $(E(DH)-c)^{2}/E(D^{2})$ at the point(3.6)$k_{D,c}=\triangle\frac{E(DH)-c}{E(D^{2})}$$\mathbb{R}$itsLeast-squares approximation of random variables by stochastic integrals 149

Thus, we obtain from (3.4) the inequality$=\triangle$$V(c)$(3.7)$\inf_{\theta\in\Theta}E(H-c-G_{T}(\theta))^{2}$$\geq$$\sup_{D\in D}\sup_{k\in \mathbb{R}}[-k^{2}E(D^{2})+2k(E(DH)-c)]$$\sup_{D\in D}\frac{(E(DH)-c)^{2}}{E(D^{2})}=:\tilde{V}(c)$$=$,which is the basis of our duality approach. Here $V(c)$ is the value of ouris the value
original $(‘‘ primal’’ )$ optimization Problem 2.2, whereas$(‘‘
dual’’)$of an auxiliary
optimization problem.
This kind of duality is useful, only if the dual problem is easier tosolve than the primal Problem 2.2 and if there is no “duality gap” (i.e.,equality holds in (3.7) , so that by computing the value of the dual
problem we also compute the value of the primal. Both these features
hold for our setting, as we are about to show. Furthermore, the duality
for theis “strong”, in the sense that we can identify an optimal
dual problem, namely$\overline{V}(c)$$)$$\overline{D}_{c}\in D$$\overline{V}(c)=\frac{(E(\tilde{D}_{c}H)-c)^{2}}{E(\tilde{D}_{c}^{2})}$(3.8)

for all but a critical value of the parameter , and then obtain ﬀom thisan optimal process
for the primal problem via (3.3).
In order to make headway with this program, let us start by in:with thetroducing the projection operator
property$c$$\theta^{(c)}$

$\pi$

(3.9)

$E[(H-\pi(H))\cdot D]=0$ ,$L^{2}(P)\rightarrow(G_{T}(\Theta))^{\perp}$$\forall H\in L^{2}(P)$,$\forall D\in(G_{T}(\ominus))^{\perp}$

In particular,(3.9)’$E[(H_{1}-\pi(H_{1}))\cdot\pi(H_{2})]=0$ ,$\forall H_{1}$, $H_{2}\in L^{2}(P)$ ,and ﬀom (3.9)’ and $(2.10)’$ we have
(3.10)$E[\pi(1)]=E[\pi^{2}(1)]>0$ .

Proposition 3.1. The value of the dual problem in (3.7), namely(3.10)$\overline{V}(c)=\triangle\sup_{D\in D}\frac{(E(DH)-c)^{2}}{E(D^{2})}$

,

150

C. Hou and I. Karatzas

can be computed as(3.12)$\tilde{V}(c)=E[\pi^{2}(H-c)]$ ,$\forall c\in \mathbb{R}$.

The supremum in (3.11) is attained by(3.13)$\tilde{D}_{c}=\triangle\frac{\pi(H-c)}{E[\pi(H-c)]}$it is not attained for$c=\hat{c}$,$\forall c\neq\hat{c}=\triangle\frac{E[\pi(H)]}{E[\pi(1)]}$;.

, we shall call the random variable
For everyof (3.13),
the “density of the dual-optimal signed martingale measure” innamely$\tilde{D}_{c}\in D$$c\neq\hat{c}$$P_{s}(\Theta)$(3.14),$\tilde{Q}_{c}(A)=\triangle\int_{A}\tilde{D}_{c}dP$

Remark 3.1 : Suppose that for some
(3.15)$E(DH)=h$ ,$A\in F$ .

$h\in \mathbb{R}$

we have$\forall D\in D$.

(For instance, this is the case when $H$ is of the form (2.5).) Then the
dual value function of (3.11) becomes(3.16)$\tilde{V}(c)=\frac{(h-c)^{2}}{\inf_{D\in D}E(D^{2})}$

and, for $c\neq h$ , the dual-optimal(3.17)$\tilde{D}_{c}$of (3.13) coincides with$\tilde{D}=\triangle\arg\min_{D\in D}E(D^{2})=\frac{\pi(1)}{E[\pi(1)]}=E(\tilde{D}^{2})+R\in D$, and $E(\tilde{D}^{2})=1/E[\pi(1)]\geq 1$ , as we shallfor some
establish below. Following Schweizer (1996), we shall call the “densityof the variance-optimal signed martingale measure”$R\in(G_{T}(\ominus))^{\perp\perp}$$\overline{D}$(3.18)$\tilde{Q}(A)=\triangle\int_{A}\tilde{D}dP$,$A\in F$in
. This terminology should be clear ﬀom (3.5) and the definitionin (3.17).
$P_{s}(\ominus)$$\blacksquare$

Proof of Proposition 3.1 : For every $D\in D$ , we have$E(DH)-c=E[D(H-c)]=E[D\cdot\pi(H-c)]$

Least-squares approximation of random variables by stochastic integrals 151

thanks to (3.9). Thus, from the Cauchy-Schwarz inequality,$(E(DH)-c)^{2}=(E[D\cdot\pi(H-c)])^{2}\leq E(D^{2})\cdot E[\pi^{2}(H-c)]$,which implies$\tilde{V}(c)\leq E[\pi^{2}(H-c)]$.

Now these last two inequalities are valid as equalities, if and only if weof the form
can find$\tilde{D}_{c}\in D$(3.13)’$\overline{D}_{c}=$const . $\pi(H-c)$ ,. This is imposwhere the constant has to be chosen so that$E[\pi(H-c)]=E[\pi(H)-c\cdot\pi(1)]$equal
to zero, i.e., if
issible to do if$c=\hat{c}$as in (3.13); in other words, the supremum of (3.11) cannot be, the normalizing constant in (3.13)’
attained in this case. But for
can be taken as $1/E[\pi(H-c)]$ , leading to the expression of (3.13) andto (3.12) as well.
$E(\overline{D}_{c})=1$$c\neq\hat{c}$It remains to show that (3.12) holds even forand$c-1/n$ ,$c=\hat{c}$. For this, let$ c_{n}=\triangle$$n\in \mathbb{N}$$\varphi_{n}=\triangle\pi(H-c_{n})$,$\varphi=\triangle\pi(H-c)$,$\tilde{D}_{c_{n}}=\frac{\varphi_{n}}{E(\varphi_{n})}\in D$

so that$\frac{(E(\tilde{D}_{c_{n}}H)-c)^{2}}{E(\overline{D}_{c_{n}}^{2})}=\frac{(E(\tilde{D}_{c_{n}}H)-c_{n}-1/n)^{2}}{E(\overline{D}_{c_{n}}^{2})}$$=\frac{(E(\tilde{D}_{c_{n}}H)-c_{n})^{2}}{E(\tilde{D}_{c_{n}}^{2})}+\frac{1/n^{2}}{E(\tilde{D}_{c_{n}}^{2})}-\frac{2}{n}.\frac{E[\tilde{D}_{c_{n}}(H-c_{n})]}{E(\tilde{D}_{c_{n}}^{2})}$$=E[\pi^{2}(H-c_{7\iota})]+\frac{1/n^{2}}{E(\tilde{D}_{c_{n}}^{2})}-\frac{2}{n}E[\pi(H-c_{n})]$$=E(\varphi_{n}^{2})+\frac{1/n^{2}}{E(\tilde{D}_{c_{n}}^{2})}-\frac{2}{n}E(\varphi_{n})$$\rightarrow E(\varphi^{2})=E[\pi^{2}(H-c)]$

as

$n$

$\rightarrow\infty$. We have used the inequality $o<1/E(\overline{D}_{c_{n}}^{2})\leq 1$ ; the facts$\varphi_{7l}-\varphi=\pi(1)/n\rightarrow 0a.s.$, $|\varphi_{n}|\leq|\varphi|+\pi(1)\in L^{2}(P)$ ; the Dominated152

C. Hou and I. Karatzas

Convergence Theorem; and the observation that, for$c\neq\hat{c}$, we have(3.19)$\underline{E[\tilde{D}_{c}\cdot(H-c)]}\underline{E[\tilde{D}_{c}\cdot\pi(H-c)]}=$$E(\tilde{D}_{c}^{2})$$E(\tilde{D}_{c}^{2})$$=\frac{E[\pi^{2}(H-c)]}{E[\pi(H-c)]}$. $\frac{(E[\pi(H-c)])^{2}}{E[\pi^{2}(H-c)]}=E[\pi(H-c)]$from (3.9)’, (3.13).
Proof of (3.17) : For any $D\in D$ , we have$\square $$\blacksquare$1$=(E(D\cdot 1))^{2}=(E(D\cdot\pi(1)))^{2}\leq E(D^{2})\cdot E[\pi^{2}(1)]$ ,from (3.9) and the Cauchy-Schwarz inequality. Equality holds if andonly ifconst . $\pi(1)$ ,$ D=\tilde{D}=\triangle$

and the normalizing constant has to be chosen so that $E(\tilde{D})=1$ , namely,
equal to $1/E[\pi(1)]$ . We conclude that $\overline{D}=\pi(1)/E[\pi(1)]$ satisfies(3.20)$E(D^{2})\geq\frac{1}{E[\pi^{2}(1)]}=\frac{1}{E[\pi(1)]}=E(\tilde{D}^{2})$,$\forall D\in D$.

On the other hand, since(3.21)$ 1=\pi(1)+\eta$for some$\eta\in(G_{T}(\ominus))^{\perp\perp}$,

we have $\tilde{D}=(1-\eta)/E[\pi(1)]=E(\tilde{D}^{2})+R$ , with $R=-\eta/E[\pi(1)]$ .

Remark 3.2 : If
1

(3.21)’$G_{T}(\ominus)$$\square $is closed in $L^{2}(P)$ , then (3.21) becomes$=\pi(1)+G_{T}(\xi^{1})$,for some$\xi^{1}\in\Theta$and (3.17), (3.20) give
(3.22)\S 4.

$\tilde{D}=E(\tilde{D}^{2})+G_{T}(\tilde{\xi})$

,

with

$\tilde{\xi}=\triangle-E(\tilde{D}^{2})\cdot\xi^{1}\in\ominus$.

Results

We are now in a position to use the duality developed in the previous
section in order to provide solutions to Problems 2.1 and 2.2. First, alemma from Schweizer (1996), pp. 230-231; we provide the proof forcompleteness.

Least-squares approximation of random variables by stochastic integrals 153

Lemma 4.1. Suppose that the infimum in (2.13) is attained by some
. Then this process satisfies$\theta^{(c)}\in\Theta$

(4.1)

(4.2)

$E\{$ $H-c-G_{T}(\theta^{(c)})]$

and$=\frac{E(\tilde{D}H)-c}{E(\tilde{D}^{2})}$$E[H-c-G_{T}(\theta^{(c)})]^{2}=\frac{c^{2}-2c\cdot E(DH)}{E(\tilde{D}^{2})}+E[\pi^{2}(H)]$,in the notation of (3.9) and (3.17).

Proof of (4.1) : The assumption implies that, for any given
the function$\xi\in\ominus$,$f(\epsilon)=\triangle E[H-c-G_{T}(\theta^{(c)}+\in\xi)]^{2}$$=\epsilon^{2}\cdot E[G_{T}^{2}(\xi)]-2\epsilon\cdot E[($ $H-c-G_{T}(\theta^{(c)}))\cdot

G_{T}(\xi)]+V(c)$attains its minimum over $\mathbb{R}at\in=0$ . This gives $f’(0)=0$ , or equivalent(4.3)$E[($ $H-c$ $-G_{T}(\theta^{(c)}))\cdot G_{T}(\xi)]$$=0$ ,$\forall\xi\in\Theta$.

Let us also notice that the mapping(4.4)$D\mapsto E(\tilde{D}D)$

is constant on $D$ ,since we have$E(\tilde{D}^{2})-E(\tilde{D}D)=E[\tilde{D}(\tilde{D}-D)]=E[\pi(1)(\tilde{D}-D)]/E[\pi(1)]$$=E(\tilde{D}-D)/E[\pi(1)]=0$

thanks to (3.9).
Now denote $\gamma=\triangle E[H-c-G_{T}(\theta^{(c)})]$ . If $\gamma=0$ , then the randomvariable$D_{1}=\triangle\tilde{D}+(H-c-G_{T}(\theta^{(c)}))$

belongs to $D$ by virtue of (4.3), and (4.4) implies$0=E[\tilde{D}(D_{1}-\tilde{D})]=E[\tilde{D}(H-c-G_{T}(\theta^{(c)}))]=E(\tilde{D}H)-c$ ,C. Hou and I. Karatzas

154

so that (4.1) holds. If
and by (4.4) once again:$\gamma\neq 0$, then $ D_{2}=\triangle[H-c-G_{T}(\theta^{(c)})]/\gamma$ is in $D$ ,$E(\tilde{D}^{2})=E(\tilde{D}D_{2})=\frac{1}{\gamma}(E(\tilde{D}H)-c)$,and so (4.1) holds in this case too.
Proof of (4.2) : From (4.3), the random variable $H-c-G_{T}(\theta^{(c)})$ belongs to the closed subspaceof (2.11), so we have$\square $$\blacksquare$$(G_{T}(\ominus))^{\perp}$$E[H-c-G_{T}(\theta^{(c)})]^{2}$$=$$E([H-c-G_{T}(\theta^{(c)})]\cdot[$ $H-\pi(H)+\pi(H)-c-G_{T}(\theta^{(c)})])$$=$$E(\{$ $H-c-G_{T}(\theta^{(c)})]\cdot[\pi(H)-c])$$=$$E[\pi^{2}(H)]-cE[\pi(H)]-c\frac{E(\tilde{D}H)-c}{E(\tilde{D}^{2})}$

thanks to (4.3), (3.9) and (4.1). The equation (4.2) now follows from
(4.5)

$E(\tilde{D}H)=E(\tilde{D}^{2})\cdot E[\pi(H)]$.

In order to check (4.5), recall (3.17), (3.20) and use (3.9)’ repeatedly, to
wit:
(4.6)

$\frac{E(\tilde{D}H)}{E(\tilde{D}^{2})}=E[H\pi(1)]=E[\pi(H)\pi(1)]=E[\pi(H)]$ .

$\square $Theorem 4.1. (i) Suppose that there exists some $\theta^{(c)}\in\ominus which$attains the infimum in (2.13). Then thissatisfies$\theta^{(c)}$(4.7)$H-c-G_{T}(\theta^{(c)})$$=\pi(H-c)$ ,and there is no duality gap in (3.7), namely(4.8)$V(c)=\tilde{V}(c)$$=$$E[\pi^{2}(H-c)]$$(E(\tilde{D}H)-c)^{2}$$=$.

$\overline{E(\tilde{D}^{2})}+E[\pi^{2}(H)]-\frac{(E\pi(H))^{2}}{E[\pi(1)]}$.

Least-squares approximation of random variables by stochastic integrals 155(ii) Conversely, suppose there exists somesabisfies (4-7);
then thisattains the infimum in (2.13), and the equalities of (4-8)
hold.
$\theta^{(c)}\in\ominus that$$\theta^{(c)}$$\blacksquare$

Proof of (4.8), Part (i) : Under the assumption of (i), we claim that

$V(c)$(4.9)$E(H-c-G_{T}(\theta^{(c)}))^{2}$$=$$\frac{(E(\tilde{D}H)-c)^{2}}{E(\overline{D}^{2})}+E[\pi^{2}(H)]-(E\pi(H))^{2}\cdot E(\tilde{D}^{2})$$=$$E[\pi^{2}(H-c)]=\tilde{V}(c)$ ,$=$which clearly proves (4.8) in light of the last equality in (3.20). Indeed,
the first equality in (4.9) holds by assumption, whereas the second is aconsequence of (4.2), (4.5). The third equality is a consequence of (4.6),(3.20) and (4.2), thanks to the simple computation$E[\pi^{2}(H-c)]$$=$$E(\pi(H)-c\cdot\pi(1))^{2}$$=$$E[\pi^{2}(H)]+c^{2}\cdot E[\pi^{2}(1)]-2c\cdot E[\pi(1)\pi(H)]$$=$$E[\pi^{2}(H)]+\frac{c^{2}-2c\cdot E(\tilde{D}H)}{E(\tilde{D}^{2})}$.

Finally, the last equality in (4.9) is just (3.12).
$\blacksquare$

Proof of (4.7), Part (i);as in (3.13), and$c\neq\hat{c}$: Let us write (3.2) with$\theta\equiv\theta^{(c)}$$D\equiv\tilde{D}_{c}$$k\equiv k_{c}=\triangle k_{\overline{D}_{c},c}=\frac{E(\tilde{D}_{c}H)-c}{E(\tilde{D}_{c}^{2})}$as in (3.6): namely,(4.10)$(H-c$ $-G_{T}(\theta^{(c)}))^{2}+2k_{c}\tilde{D}_{c}(c+G_{T}(\theta^{(c)}))$$\geq 2k_{c}\tilde{D}_{c}H-(k_{c}\tilde{D}_{c})^{2}$, $a.s$ .

,156

C. Hou and I. Karatzas

Taking expectations in (4.10), and recalling the optimality ofas Proposition 3.1, we obtain$V(c)$(4.11)$=$$\geq$$\theta^{(c)}$as well$E(H-c-G_{T}(\theta^{(c)}))^{2}$$-k_{c}^{2}E(\tilde{D}_{c}^{2})+2k_{c}(E(\tilde{D}_{c}H)-c)$$\frac{(E(\tilde{D}_{c}H)-c)^{2}}{E(\tilde{D}_{c}^{2})}=\tilde{V}(c)$$=$.

But ﬀom (4.8) we know that (4.11) actually holds as equality, whichmeans that the left-hand side and the right-hand side of (4.10) have thesame expectation. In other words, (4.10) must hold as equality, which
we know happens only if (3.3) holds, namely only if$H-c-G_{T}(\theta^{(c)})=k_{c}\tilde{D}_{c}=\frac{E(\tilde{D}_{c}H)-c}{E(\tilde{D}_{c}^{2})}\frac{\pi(H-c)}{E[\pi(H-c)]}=\pi(H-c)$holds $a.s.$ , thanks to (3.19).
Proof of (4.7), part (i);of duality, namely with

$c=\hat{c}$$\blacksquare$(4.12),: In this case we shall need a new kind$\mathcal{L}=\triangle\{L\in(G_{T}(\theta))^{\perp}/E(L)=0\}$

replacing the space $D$ of (2.9); the elements of will be the dual (adjoint)
variables in this new duality. We begin by writing (3.1) with $x=c+$:, $y=2L$ for arbitrary,$\mathcal{L}$$\theta\in\Theta$$G_{T}(\theta)$(4.13)$L\in \mathcal{L}$$(H-c-G_{T}(\theta))^{2}+2L(c+G_{T}(\theta))\geq 2LH-L^{2}$ ,with equality if and only if(4.14)$H-c-G_{T}(\theta)=L$

holds $a.s$ . Taking expectations in (4.13), we obtain$E$ $(H-c -G_{T}(\theta))^{2}$(4.15)$\geq$$E[2L(H-c)-L^{2}]$$=$$E[2L\cdot\pi(H-c)-L^{2}]$$=$$E[\pi^{2}(H-c)]-E(L-\pi(H-c))^{2}$, the elThis suggests that we should read (4.13)-(4.15) withement ofthat attains the infimum in (2.13) and is thus optimal for$\theta\equiv\theta^{(c)}$$\Theta$Least-squares approximation of random variables by stochastic integrals 157

Problem 2.2, and $L\equiv\tilde{L}=\triangle\pi(H-c)$ , noting that $E(\overline{L})=0$ since$c=\hat{c}=E[\pi(H)]/E[\pi(1)]$ . With these choices, the le -most memberof (4.15) becomes$ft$$E(H-c-G_{T}(\theta^{(c)}))^{2}=V(c)$ ,whilst its right-most member is $E[\pi^{2}(H-c)]=\tilde{V}(c)$ . From (4.8) weknow that these two quantities are equal, so the two sides of (4.13) havethe same expectation. This means that (4.13) must holds as equality,
which happens only if (4.14) is valid, namely$H-c-G_{T}(\theta^{(c)})=\pi(H-c)$ ,Proof of Part (ii) : Suppose there exists some
(4.7); then$\blacksquare$

$a.s$

.

that satisfies$\theta^{(c)}\in\Theta$$E(H-c-G_{T}(\theta^{(c)}))^{2}=E[\pi^{2}(H-c)]=\tilde{V}(c)$from Proposition 3.1. But we also have$\tilde{V}(c)\leq V(c)\leq E(H-c-G_{T}(\theta^{(c)}))^{2}$,

thanks to (3.7) and (2.13). In other words, these last two inequalities arevalid as equalities,
attains the infimum in (2.13), and (4.8) holds.
$\theta^{(c)}$$\square $closed.
Remark 4-1 : The case of$L^{2}(P)$, then the infimum in (2.13) is attained,If
is closed inas was assumed in Lemma 4.1 and in Theorem 4.1(i). In this case we, and every $H\in L^{2}(P)$ admits ahave of course
decomposition of the form
$G_{T}(\ominus)$$G_{T}(\ominus)$$G_{T}(\ominus)=(G_{T}(\ominus))^{\perp\perp}$(4.16)$H=\pi(H)+G_{T}(\xi^{H})$In particular, there exists$\xi^{1}\in\ominus so$for some$\xi^{H}\in\Theta$.

that (3.21) holds, and thus

$H-c-\pi(H-c)=[H-\pi(H)]-c[1-\pi(1)]=G_{T}(\xi^{H}-c\xi^{1})$ .

Comparing this expression with (4.7), we conclude that (4.7) is satisfied
with the choice(4.17)$\theta^{(c)}=\xi^{H}-c\xi^{1}$.

According to Theorem 4.1(ii), the process $\theta^{(c)}\in\ominus of(4\cdot 17)$ is thenholds.
optimal for Problem 2.2, and$(4\cdot 8)$

C. Hou and I. Karatzas

158

The asExample 4.1. F\"ollmer-Schweizer decomposition.
is notsertion at the end of Remark 4.1 remains valid even if$L^{2}(P)$, provided that (3.21) and (4.16) still hold. Consider,closed infor example, the case of a semimartingale $X(\cdot)\in S^{2}(P)$ and of a random variable $H\in L^{2}(P)$ which admits the so-called “F\"ollmer-Schweizer
decomposition” ; this means that $H$ can be written in the form $H=$
$h+G_{T}(\zeta^{H})+L^{H}(T)$ of (2.6), for $h=E(H)$ , some, and somemartingale
that satisfies the property (2.7).
Suppose that (3.21) is also satisfied; then $\pi(H)=h\pi(1)+L^{H}(T)$and we have $H-\pi(H)=h(1-\pi(1))+G_{T}(\zeta^{H})=G_{T}(h\xi^{1}+\zeta^{H})$ , so
we may takein (4.16) and$G_{T}(\Theta)$$\zeta^{H}\in\Theta$$L^{H}(\cdot)\in \mathcal{M}_{0}^{2}(P)$$\xi^{H}\equiv h\xi^{1}+\zeta^{H}$(4.17)’$\theta^{(c)}\equiv(h-c)\cdot\xi^{1}+\zeta^{H}$in (4.7). The processof (4.17)’ is then optimal for the Problem 2.2,
i.e., attains the infimum in (2.13), which can be readily computed as$\theta^{(c)}$$V(c)=(h-c)^{2}\cdot E[\pi(1)]+E(L^{H}(T))^{2}$This simple result may be compared with Theorem 3 and Proposition
18 in Schweizer (1994).

We are now in a position to discuss the solution of Problem 2.1 as
well.
Theorem 4.2. Suppose thatvalue of Problem 2.1 is given as(4.18)$G_{T}(\Theta)$

is closed in $L^{2}(P)$ . Then the$V=V(\hat{c})=E[\pi^{2}(H)]-\frac{(E[\pi(H)])^{2}}{E[\pi(1)]}$

with the notation(4.19)$\hat{c}=\frac{E[\pi(H)]}{E[\pi(1)]}=E(\tilde{D}H)=E(\frac{d\tilde{Q}}{dP}H)$

Furthermore, the infimum in (2.4) is attained by the pairwith as in (4 19) and withof (3.13).
$(\hat{c},\hat{\theta})$$\hat{c}$$\cdot$(4.20)$\hat{\theta}=\triangle\theta^{(\hat{c})}=\xi^{H}-\hat{c}\xi^{1}$

Proof : Immediate from Theorem 4.1 and Remark 4.1, when it is$overc\in \mathbb{R}observedt.h$

at the number$\hat{c}$of (4.19) minimizes the expression of$(4.8)\square $Least-squares approximation of random variables by stochastic integrals 159of (3.8) is a probability meaNote that when the signed measure$P[\pi(1)\geq0]=1$), the quantity of (4.19) is just thesure (i.e., when
expectation of the random variable $H$ under the dual-optimal measure. Suﬃcient conditions are spelled out in the next section.
$\overline{Q}$$\tilde{Q}$: Variance Minimization. If
Remarkthen the process of (4.20) also$4\cdot 2$$G_{T}(\Theta)$

is closed in $L^{2}(P)$ ,$\hat{\theta}$(4.21)

minimizes

This is because for any

Var $(H-G_{T}(\theta))$

Var $(H-G_{T}(\theta))$ ,$\theta\in\ominus$

over all$\theta\in\ominus$.

, and with $c_{\theta}=\triangle E[H-G_{T}(\theta)]$ , we have:$=$$E(H-c_{\theta}-G_{T}(\theta))^{2}$$\geq$$E(H-\hat{c}-G_{T}(\hat{\theta}))^{2}=Var(H-G_{T}(\hat{\theta}))$,from Theorem 4.2.
, the processof (4.17)
More generally, for any given
has the “mean-variance eﬃciency” property(4.2 2)
Var $(H-G_{T}(\theta^{(c)}))\leq$ Var $(H-G_{T}(\theta))$ , for any$E[H-G_{T}(\theta)]=E[H-G_{T}(\theta^{(c)})]\}$ .
that satisfies$\theta^{(c)}\in O-$$c\in \mathbb{R}$$\{$$\theta\in\ominus$

Indeed, let $\mu_{c}=\triangle E[H-G_{T}(\theta^{(c)})]$ and observe that, for any
with $E[H-G_{T}(\theta)]=\mu_{c}$ , we have$\theta\in\Theta$

Var $(H-G_{T}(\theta))$$=$$Var(H-c-G_{T}(\theta))=E(H-c-G_{T}(\theta))^{2}-(\mu_{c}-c)^{2}$$\geq$$E(H-c-G_{T}(\theta^{(c)}))^{2}-(E(H-c-G_{T}(\theta^{(c)})))^{2}$$=$

Var $(H-c-G_{T}(\theta^{(c)}))=Var(H-G_{T}(\theta^{(c)}))$ .

is closed: Mean-Variance Frontier. Suppose that
Remark$E(\tilde{D}^{2})>1$
in $L^{2}(P)$ , that we have $P[\pi(1)\neq E(\pi(1))]>0$ ; this implies, consider thein (3.5), hence $E[\pi(1)]<1$ . For some given
following problem:$G_{T}(\ominus)$$4\cdot 3$$m\in \mathbb{R}$(4.20)$\{TominimizethevarianceVar(H-G_{T}(\theta))over\theta\in\ominus withE[H-G_{T}(\theta)]=m.’\}$

In view of the property (4.22), it suﬃces to show that we can findsuch that(4.24)$E[H-G_{T}(\theta^{(c)})]$$=m$ .

$c\equiv c_{m}$160

C. Hou and I. Karatzas

asThen the solution of the problem (4.23) will be given byin (4.17), with
. Now ﬀom (4.1) we have $E[H-G_{T}(\theta^{(c)})]=$$\theta^{(c)}\in\Theta$$c\equiv c_{m}$$c+\frac{E(\overline{D}H)-c}{E(\overline{D}^{2})}$, so that (4.24) amounts to$c=c_{m}=\triangle\frac{m\cdot E(\tilde{D}^{2})-E(\tilde{D}H)}{E(\tilde{D}^{2})-1}=\frac{m-E[\pi(H)]}{1-E[\pi(1)]}$,

thanks to (4.6) and (3.20). We take these two Remarks 4.2, 4.3 ﬀom
Schweizer $(1994, 1996)$ .

\S 5.

A Mathematical Finance Interpretation

The Problems 2.1, 2.2 have an interesting interpretation in the context of Mathematical Finance, when one interprets the componentsof the semimartingale in (2.1) as the (discounted) prices of several risky
represents the numassets in a financial market. In this context,
asset,
held by an investor atber of shares in the correspondingtime $\in[0, T]$ . The resulting process $\theta\in\ominus stands$ then for the investor’s (self-financing) trading strategy, and $G_{t}(\theta)=\int_{0}^{t}\theta’(s)dX(s)=$for the (discounted) gains-from-trade associatedwith the strategy by time .
Suppose now that the investor faces a contingent claim (liability)$H$ at the end $T$ of the time-horizon $[0, T]$ .
Starting with a given ini, the investor seeks totial capital , and using a trading strategy
replicate this contingent claim $H$ as faithfully as possible, in the sense ofminimizing the expected squared-error loss $E(H-c-G_{T}(\theta))^{2}$ This
leads us to Problem 2.2. When the determination of the “right” initial
capital is also part of the problem, one is led naturally to the formulation of Problem 2.1. Similarly, one may consider minimizing thevariance of the discrepancy $H-c$ $-G_{T}(\theta)$ over all trading strategies(problem of (4.21)), or just over those strategies that guarantee
a given “mean-gains-ﬀom-trade” level $E[G_{T}(\theta)]=E(H)-m$ (problemof (4.23) .
If one decides to stick with this interpretation, it makes sense to
ask whether the model for the financial assets represented by (2.1) admits arbitrage opportunities; these are trading strategies $\theta\in\ominus with$$P[G_{T}(\theta)\geq 0]=1$ and $P[G_{T}(\theta)>0]>0$ .
To this eﬀect, let $A$
be an increasing, predictable and RCLL process with values in, , and suppose that the proand $A(0)=0$ ,for $i=1$ ,cesses $M$ $=(M_{1}, \ldots, M_{d})$ and $B=(B_{1}, \ldots, B_{d})$ , in the decomposition$X_{i}(\cdot)$$\theta_{i}(t)$$i^{th}$$t$$\sum_{i=1}^{d}\int_{0}^{t}\theta_{i}(s)dX_{i}(s)$$\theta$$t$$\theta\in\Theta$

$c$

$c$

$\theta\in\ominus$$)$$[0, \infty)$$\langle M\rangle_{i}<<A$$\cdots$$d$Least-squares approximation of random variables by stochastic integrals 161$X=X(0)+M+B$ of (2.1) for the semimartingale $X=(X_{1, }\ldots, X_{d})$ ,satisfy(5.1)$B_{i}(\cdot)$$<<$$\langle M\rangle_{i}(\cdot)$(5.2)$B_{i}(t)$$=$$\int_{0}^{t}\gamma_{i}(s)$(5.3)$\langle M_{i}, M_{j}\rangle(t)$$=$$\int_{0}^{t}\sigma_{ij}(s)$for $i=1$ ,$\cdots$,$d$dA(s),$o\leq t\leq T$

dA(s),$o\leq t\leq T$

and, . Hereand $j=1$ ,are suitable predictable processes that satisfy$\cdots$$d$$\gamma(\cdot)=(\gamma_{1}(\cdot), \cdots, \gamma_{d}(\cdot))’$$\sigma(\cdot)=\{\sigma_{ij}(\cdot)\}_{1\leq i,j\leq d}$(5.4)$\sigma(t)\lambda(t)=\gamma(t)$almost surely, for some

,

$a.e$

.

$t$

$\in[0, T]$$\lambda(\cdot)=(\lambda_{1}(\cdot), \cdots, \lambda_{d}(\cdot))’$in the space(5.5)$\mathcal{L}^{2}(M)=\triangle\{\theta$: $[0, T]\rightarrow \mathbb{R}^{d}predictable/$$E\int_{0}^{T}\theta’(s)\sigma(s)\theta(s)$dA(s)$=E(\int_{0}^{T}\theta’(s)dM(s))^{2}<\infty\}$ .

Following Schweizer (1996), we shall refer to (5.1)-(5.4) as the structure.
conditions on the semimartingale
Under these conditions, it can be shown that the semimartingale
does not admit arbitrage opportunities, and that we have equality$P_{s}(\ominus)=P_{s}^{2}(X)$ in Remark 2.2 (cf. Ansel&Strieker (1992); Schweizer
has con(1995); and Schweizer (1996), Lemma 12). If, in addition,, then it can be shown that the variance-optimal martingale
tinuous$Q$
measure of (3.18) is nonnegative, namely a probability measure:$X(\cdot)$$X(\cdot)$$X(\cdot)$$p\underline{a}ths$(5.6)$P[\tilde{D}\geq 0]=1$

and$\tilde{Q}(\Omega)=E(\tilde{D})=1$

in (3.17), (3.18). This is in fact equivalent to $P$ (i.e., $P[\pi(1)>0]=1$ ),under the extra assumption$\tilde{Q}$(5.7)

is a $Q-$ local martingale under some$(dQ/dP)\in L^{2}(P)\}$
probability measure $Q\sim P$ with
$X(\cdot)$$\{$(cf. Schweizer (1996), Theorem 13; Delbaen &Schachermayer (1996),
Theorem 1.3). This condition (5.7) also implies that

(5.8)

the mapping$\theta\mapsto G_{T}(\theta)$

is injective;

162

C. Hou and I. Karatzas

cf. [DMSSS], Lemma 3.5.
Always under the structure conditions (5.1)-(5.4) on the semimartingaleof (2.1), consider now the so-called mean-variance tradeoﬀ
process$X(\cdot)$$K(t)=\triangle\int_{0}^{t}\lambda’(s)dB(s)=\int_{0}^{t}\lambda’(s)\sigma(s)\lambda(s)dA(s)=\langle\int\lambda’dM\rangle(t)$,$0\leq t\leq T$.

If this process is P-a.s. bounded, then
(5.9)$\Theta=\mathcal{L}^{2}(M)$in the notation of (2.2), (5.5) and(5.10)

is closed in$G_{T}(\ominus)$$L^{2}(P)$([Ph.R.S.], Corollary 4 and below; Schweizer (1996), Lemma 12). See
also [DMSSS] where conditions both necessary and suﬃcient for (5.10)
are presented.
Remark 5.1 : In the one-dimensional case $d=1$ , the structure conditions (5.1)-(5.4) are satisfied if there exists a process
with$\lambda\in \mathcal{L}^{2}(M)$(5.11)$X(t)=X(0)+M(t)+\int_{0}^{t}\lambda(s)d\langle M\rangle(s)$,$o\leq t\leq T$.

In this case, the mean-variance tradeoﬀ process of (5.9) takes the form

(5.12)$K(t)=\int_{0}^{t}\lambda^{2}(s)d\langle M\rangle(s)$.

Remark 5.2 : Suppose thathas continuous paths, and that
(5.11) and (5.7) hold. If the random variable $H\in L^{2}(P)$ is of the form
$H=h+G_{T}(\zeta^{H})$ in (2.5), then$X(\cdot)$$H-\pi(H)=G_{T}(h\xi^{1}+\zeta^{H})$and the injectivity property (5.8) allows us to make the identifications(5.13)$\xi^{H}=h\xi^{1}+\zeta^{H}$,$\theta^{(c)}=(h-c)\cdot\xi^{1}+\zeta^{H}$in (4.16) and (4.17), respectively.
More generally, under the assumptions of Remark 5.2 but now for
any $H\in L^{2}(P)$ , we have the Kunita-Watanabe decomposition underthe variance-optimal probability measureof (3.18), namely$\tilde{Q}$(5.14)$\tilde{E}[H|\mathcal{F}(t)]=\overline{h}+G_{t}(\overline{\zeta}^{H})+\tilde{L}^{H}(t)$,$o\leq t\leq T$

Least-squares approximation of random variables by stochastic integrals 163(cf. Ansel&Strieker (1993), or Theorem 3 in Rheinl\"ander&Schweizer(1997) . Heredenotes expectation with respect to the probability, and $\tilde{L}^{H}\in S^{2}(P)$ is ameasure-martingale withand$\tilde{E}$$)$$\overline{Q},\tilde{h}=\triangle\overline{E}(H)=E(\tilde{D}H),\tilde{\zeta}^{H}\in\Theta$$\overline{L}^{H}(0)=0$$\tilde{Q}$(5.15)$\langle\tilde{L}^{H}$,$X_{i}\rangle(\cdot)\equiv 0$

On the other hand, sincehave its decomposition(5.16)$\tilde{L}^{H}(T)$,$\forall i=1$

,

, .
$d$

$\cdots$

belongs to the space $L^{2}(P)$ , we also$\tilde{L}^{H}(T)=G_{T}(\tilde{\theta}^{H})+\pi(\overline{L}^{H}(T))$

from the closedness of(5.17)$G_{T}(\Theta)$, where$\overline{\theta}^{H}\in\Theta$

is such that

$E[(\tilde{L}^{H}(T)-G_{T}(\tilde{\theta}^{H}))\cdot G_{T}(\theta)]=0$

Back in (5.14), this gives$(H-c)-\pi(H-c)$,$\forall\theta\in\ominus$.

, so that$H=\tilde{h}+G_{T}(\tilde{\zeta}^{H}+\tilde{\theta}^{H})+\pi(\tilde{L}^{H}(T))$$=$$(\tilde{h}-c)(1-\pi(1))+G_{T}(\overline{\zeta}^{H}+\tilde{\theta}^{H})$$=$$G_{T}((\tilde{h}-c)\xi^{1}+\tilde{\zeta}^{H}+\tilde{\theta}^{H})$from (3.21)’. Again, the injectivity property (5.8) allows us to make theidentification(5.18)$\theta^{(c)}=(\tilde{h}-c)\xi^{1}+\tilde{\zeta}^{H}+\tilde{\theta}^{H}$in (4.17). Finally, let us consider the positive(5.19)$\tilde{D}(t)$$=\triangle\tilde{E}[\tilde{D}|\mathcal{F}(t)]$$\overline{Q}$-martingale$=$$E(\tilde{D}^{2})+G_{t}(\tilde{\xi})$$=$$E(\tilde{D}^{2})[1-G_{t}(\xi^{1})]$,$0\leq t\leq T$

obtained by taking conditional expectations in (3.22) under$\tilde{Q}$.

appearing in
We are now in a position to identify the process(5.16), (5.18) and state the following result, which simplifies and generalizes Theorems 5, 6 of Rheinl\"ander&Schweizer (1997).
$\tilde{\theta}^{H}$

Theorem 5.1. Suppose that the semimartingale $X(\cdot)\in S^{2}(P)$ has
continuous paths, and that (5.7), (5.11) hold. Then the optimal process
for Problem 2.2 takes the form$\theta^{(c)}\in\Theta$(5.20)$\theta^{(c)}=\tilde{\zeta}^{H}+[(E(\tilde{D}H)-c)+E(\tilde{D}^{2})\int_{0}.$in the notation of (5.14), (5.19).

$\frac{d\tilde{L}^{H}(s)}{\tilde{D}(s)}]\cdot\xi^{1}$

C. Hou and I. Karatzas

164

Sketch of Proof: The -martingales$0\leq t\leq T$ andof (5.19) are orthogonal, since$\tilde{N}(t)=\triangle\int_{0}^{t}(1/\tilde{D}(s))d\tilde{L}^{H}(s)$$\tilde{Q}$,$\tilde{D}(\cdot)$.
$\langle\tilde{N},\tilde{D}\rangle(\cdot)=\sum_{i=1}^{d}\int_{0}$$\frac{\tilde{\xi}_{i}(s)}{\tilde{D}(s)}d\langle\overline{L}^{H}, X_{i}\rangle(s)\equiv 0$

from (5.15) and (5.19). Thus, integration by parts gives(5.21)$\tilde{L}^{H}(T)$$=$$\int_{0}^{T}\tilde{D}(s)d\tilde{N}(s)=\tilde{D}(T)\tilde{N}(T)-\int_{0}^{T}\tilde{N}(s)\tilde{\xi}’(s)dX(s)$$=$$\tilde{D}\tilde{N}(T)-G_{T}(\tilde{N}\tilde{\xi})$,

and transforms (5.17) into
(5.22)$E[G_{T}(\overline{\theta}^{H}+\tilde{N}\tilde{\xi})\cdot G_{T}(\theta)]=\tilde{E}[\tilde{N}(T)G_{T}(\theta)]$,$\forall\theta\in\ominus$.

But the right-hand-side of (5.22) vanishes, since$\tilde{E}[\tilde{N}(T)G_{T}(\theta)]$$=$$\tilde{E}[\int_{0}^{T}\frac{d\tilde{L}^{H}(s)}{\tilde{D}(s)}\cdot\sum_{i=1}^{d}\int_{0}^{T}\theta_{i}(s)dX_{i}(s)]$$=$$\tilde{E}\sum_{i=1}^{d}\int_{0}^{T}\frac{\theta_{i}(s)}{\tilde{D}(s)}d\langle\tilde{L}^{H}, X_{i}\rangle(s)=0$

thanks to (5.15). Thus the left-hand-side of (5.22) also vanishes for every, which suggests$\theta\in\ominus$.
$\tilde{\theta}^{H}=-\tilde{N}\tilde{\xi}=E(\tilde{D}^{2})\xi^{1}\int_{0}$$\frac{d\tilde{L}^{H}(s)}{\overline{D}(s)}$and leads to (5.20) after substitution into (5.18).

$\square $In order to justify the legitimacy of the above argument, particularly
the steps leading to (5.22), one needs to show that the random variablebelongs to $L^{2}(P)$ ; this is carried out
on pp. 1820-1823 of Rheinl\"ander&Schweizer (1997).
$\sup_{0\leq t\leq T}|\overline{D}(t)\int_{0}^{t}(1/\tilde{D}(s))d\tilde{L}^{H}(s)|$Least-squares approximation of random variables by stochastic integrals 165

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$L^{2}$

Chunli Hou
Nomura Securities International
2 World Financial Center, Building B
New York, NY 10281
chou@us.nomura.com

Ioannis Karatzas
Departments of Mathematics and Statistics
619 Mathematics Building
Columbia University
New York, NY 10027
ik@math.columbia.edu

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 167-187

Quadratic Wiener Functionals, Kalman-Bucy
Filters, and the KdV Equation
Nobuyuki Ikeda and Setsuo Taniguchi

Dedicated to Professor Kiyosi It\^o on his 88th birthday
Abstract.

Soliton solutions and the $tau$ function of the $KdV$ equation arestudied within the stochastic analytic ﬀamework. A key role is playedby the It\^o formula and the Cameron-Martin transformation.

\S Introduction
In this paper, we investigate the Korteweg-de Vries $(KdV)$ equation
within the ﬀamework of stochastic analysis. We shall study soliton solutions with the help of the It\^o formula, whose original form was achievedin 1942 ([9]). The Cameron-Martin transformation, which was established in the early 1940’s([2, 3]), also plays a key role.
Let $x>0$ and-valued continuous functions
be the space of
$P$
$[0,
x]$
.
starting at the origin, and let be the Wiener measure on
on
Following the idea of Cameron-Martin [3], we can show that$\mathcal{W}^{n}$$R^{n}$$\mathcal{W}^{n}$(1)$I(x, t):=\int_{\mathcal{W}^{1}}\exp[-\frac{a^{2}}{2}\int_{0}^{x}w(y)^{2}dy-\frac{a}{2}\tanh(a^{3}t)w(x)^{2}]P(dw)$$=(\cosh(a^{3}t))^{1/2}(\cosh(ax+a^{3}t))^{-1/2}$

where $w(y)\in R$ denotes the position of $w$[8] . Then $u(x, t)=-4\partial_{x}^{2}\log I(x, t)$ , where
solution of the $KdV$ equation$\in \mathcal{W}^{1}$

$)$

(2)

for any $a>0$ ,at time$\partial_{x}=\partial/\partial x$$\frac{\partial u}{\partial t}=\frac{3}{2}u\frac{\partial u}{\partial x}+\frac{1}{4}\frac{\partial^{3}u}{\partial x^{3}}$(see \S 4 and, is a 1-soliton$y$.

Received February 27, 2003.
Revised June 16, 2003.
The second author is partially supported by Grant-in-Aid for Scientific
Research (A) (1) 14204010.

N. Ikeda and S. Taniguchi

168

A reflectionless potential with scattering data
is by definition a function(3)$q(x)=-2\frac{d^{2}}{dx^{2}}$ $\log$$\eta_{j}$, $m_{j}>0,1\leq j\leq n$ ,$\det(I+A(x )$

where
(4)

$A(x)=(\frac{\sqrt{m_{i}m_{j}}}{\eta_{i}+\eta_{j}}e^{-(\eta_{i}+\eta_{j})x})_{1\leq i,j\leq n}$

We denote by
the totality of all reflectionless potentials with scatbe the set of alltering data consisting of $2n$ positive numbers. Let$\sigma\pm
on(-\infty,
0]$ such thatof non-negative measures
pairs
. For
, set
for any$Q_{n}$$\Sigma$$\sigma=(\sigma_{+}, \sigma_{-})$$\int_{(-\infty,0]}e^{\lambda\sqrt{-z}}\sigma_{\pm}(dz)<\infty$$\lambda>0$$\sigma\in\Sigma$$G(u, v; \sigma)=\frac{1}{4}\int_{-\infty}^{0}\frac{1}{\sqrt{-z}}(e^{\sqrt{-z}(u+v)}-e^{\sqrt{-z}|u-v|})\sigma_{+}(dz)$$+\frac{1}{4}\int_{-\infty}^{0}\frac{1}{\sqrt{-z}}(e^{-\sqrt{-z}|u-v|}-e^{-\sqrt{-z}(u+v)})\sigma_{-}(dz)$.

We consider a family of all Gaussian processes
with mean 0 and$G(u,v;\sigma)$. We also consider the totality $Q$,
covariance function, defined byof all functions ,$X^{\sigma}$$\mathcal{G}$$\sigma\in\Sigma$$q^{\sigma}$$\sigma\in\Sigma$,$q^{\sigma}(x)=-4\frac{d^{2}}{dx^{2}}\log E[\exp(-\frac{1}{2}\int_{0}^{x}|X^{\sigma}(y)|^{2}dy)]$$X^{\sigma}\in \mathcal{G}$,where $E$ stands for the expectation with respect to the underlying prob$=$,ability measure. In [12], Kotani showed that $Q$ includes all, and any element of $Q$ is obtained as a limit of reflectionless, 2,
potentials in the topology of uniform convergence on compacts.
Furthermore, it is well known that $q(x, t)$ defined by (3) and (4), $1\leq j\leq n$ , gives a rise of anwith $m_{j}(t)=m_{j}\exp[-2\eta_{j}^{3}t]$ instead of-soliton solution $u(x, t)=-q(x, t)$ of the $KdV$ equation (2).
The facts mentioned above indicate that soliton solutions of the$KdV$ equation may be represented in terms of Gaussian processes. In
this paper, we shall establish such an expression of soliton solutions
and the $tau$ function, which plays a fundamental role in the study of the$KdV$ hierarchy (see [14, 16, 17]), in the Wiener space.
are discrete measures, then theIf both components, whereis a setcorresponding Gaussian process belongs tosuperpositions
independentasof
1processesof Gaussian
obtained, seedimensional Ornstein-Uhlenbeck processes (for the definition of$Q_{n}$$1$$\ldots$$m_{j}$$n$$n$$\sigma\pm of\sigma\in\Sigma$$\mathcal{G}_{n}$$\bigcup_{n\in N}\mathcal{G}_{n}$$n$$\mathcal{G}_{n}$

$n$

169

Quadratic Wiener functionals

1.2). In this case, the correspondence between and $Q$ is given con. See \S 2.
tocretely; for every $\in N$ , we shall give a mapping ﬀom
Moreover, not only reflectionless potentials but also -soliton solutions
and the $tau$ function of the $KdV$ equation can be represented in terms. See \S 4. These expressions show thatof Gaussian processes in“a superposition” to make an -soliton solution out of 1-soliton ones
can be realized in the Wiener space. Further, we can explicitly see how
speed parameters of 1-solitons reflect on those of the -solitons obtained
as superpositions. See \S 4.
An exact expression of Wiener integrals of Wiener functionals of theplays a basic role
form $\exp[-(a^{2}/2)\int_{0}^{x}X(y)^{2}dy+R(x)]$ ofin this paper, where $R(x)$ is a Wiener functional which varies according
as we deal with reflectionless potentials, -soliton solutions, and the $tau$
function. Such exact expressions are achieved with the help of the It\^oformula and the Cameron-Martin transformation. The Cameron-Martin
transformation we deal with is determined by a second order ordinary
diﬀerential equation. When a 1-dimensional Wiener process, which is in, is considered, the equation is the Sturm-Liouville one employed in[3]. The ordinary diﬀerential equations in this paper appear in diﬀerent
features ﬀom place to place, while they correspond to the same Wiener
integral. Namely, we encounter several types of $n\times n$ -matrix Riccati
equations and second order $n\times n$ -matrix and first order $2n\times n$ -matrix
linear ordinary diﬀerential equations. These diﬀerent features are unified
in terms of Grassmannians. See 1.1. It should be also mentioned thatthe above Riccati equations play an important role in the theory of thelinear filtering problem by Kalman-Bucy. See \S 3.
of Gaussian
Before closing this section, we note that the classprocesses is closely related to the one studied by Hida-Streit [5] andOkabe [15].
The authors are grateful to Professor S. Kotani for his helpful comments and informing us of his recent achievement privately. They also
thank Professor M. Yor for stimulating discussions with him. The subject of the paper grew out when the first author was visiting the Mathematics Research Centre, the University of Warwick in summer in 2001
at the invitation of Professor K.D. Elworthy, and discussed with Professor M. Kleptsyna. The second author had a chance to talk about the
prototype of this paper with Professor Elworthy while he visited Japan
in 2001. It is a great pleasure of the authors to thank both of them.
$\mathcal{G}$$n$$Q_{n}$$\mathcal{G}_{n}$$n$$\bigcup_{n\in N}\mathcal{G}_{n}$

$n$

$n$

$X\in\bigcup_{n\in N}\mathcal{G}_{n}$$n$$\mathcal{G}_{1}$$\mathcal{G}$

170

\S 1.

N. Ikeda and S. Taniguchi

Cameron-Martin transformation –Ornstein-Uhlenbeck
process

1.1. Ordinary diﬀerential equations
We first recall several known facts about ordinary diﬀerential equa, wheretions. For $n\in N$ , we set$A_{n}=\prime \mathcal{P}_{n}\times C_{n}$$P_{n}=$$C_{n}=${ $p=$ ( ,{ $c=$ ( ,$p_{1}$$c_{1}$$\ldots,p_{n})\in R^{n}$,$\ldots$$c_{n})\in R^{r\iota}$:$p_{i}\neq p_{j}$for $i\neq j$ },: $c_{i}>0$ for $1\leq i\leq n$ }.

For $(p, c)\in A_{n}$ and $a>0$ , we define $n\times n$-matrices$D_{p}=diag[p_{1}, \ldots,p_{n}]$and$E_{p,c}(a)=D_{p}^{2}+a^{2}c\otimes c$.

and $E_{p,c}(a)$ , respectively.
We shall often write simply $D$ and $E(a)$ forLet
be a solution of a first order $2n\times n$ -matrix ordinary differential equation$D_{p}$$\Phi_{a}(y)$

(5)

where

$\Phi’+M_{p,c,a}\Phi=0$ ,$M_{p,c,a}=\left(\begin{array}{ll}D_{p} & I\\a^{2}c\otimes c & -D_{p}\end{array}\right)$-matrices $A$ and $B$ ,andstands for the derivative of . For
we often write $\Phi_{a}(y;A, B)$ to emphasize the initial condition $\Phi_{a}(0)=$$f$$f^{/}$$\left(\begin{array}{l}A\\B\end{array}\right)$. Denote bysubmatrices of$\phi_{a}(y)$$\Phi_{a}(y)$and$\psi_{a}(y)$$n$the upper and the lower half $n\times n-$, respectively;.

$\Phi_{a}(y)=\left(\begin{array}{l}\phi_{a}(y)\\\psi_{a}(y)\end{array}\right)$

Then
(6)

$\phi_{a}(y)$$\times n$obeys a second order ordinary diﬀerential equation$\phi^{JJ}-E(a)\phi=0$.

In the cases we deal with in this paper,is always shown to be$\Phi_{a}(y)=\Phi_{a}(y;I,0)$ , which is the$y\geq0$. Moreover, if
invertible for any
case investigated in \S 2 and \S 3, thenis also invertible for $z>0$(see a paragraph after Theorem 2.1). Hence, in the remainder of thissubsection, we assume thatandare both invertible for any$y\geq 0$ and $z>0$ . Then
determines an -ﬀame in a $2n$ -dimensional
vector space $V(2n)$ over $R$ , and hence gives a rise of a dynamics on
the Grassmannian $GM(n, V(2n))$ consisting of all -dimensional vector
subspaces of $V(2n)$ . Moreover,is identified in $GM(n, V(2n))$ with$\phi_{a}(y)$$\psi_{a}(z)$$\phi_{a}(y)$$\Phi_{a}(y)$$\psi_{a}(z)$

$n$

$n$

$\Phi_{a}(y)$$\left(\begin{array}{l}I\\\psi_{a}(y)\phi_{a}^{-1}(y)\end{array}\right)=\left(\begin{array}{l}I\\\gamma_{a}(y)\end{array}\right)$

,

171

Quadratic Wiener functionals

where $\phi_{a}^{-1}(y)=(\phi_{a}(y))^{-1}$ and $\gamma_{a}(y)=\psi_{a}(y)\phi_{a}^{-1}(y)=-\phi’(y)\phi_{a}^{-1}(y)-$$D$ .
obeys the-matrix
Due to the Cole-Hopf transformation,
Riccati equation$n$$\gamma_{a}$$\times n$$\gamma’-\gamma D-D\gamma-\gamma^{2}+a^{2}c\otimes c=0$). We next consider an -ﬀame obtained by reversing the time of. Set $\mu(y)=\gamma_{a}(x-y)$ and $\iota/(y)=\mu(y)^{-1}$ , $y<x$ .
Then, for $y<x$ ,determines a point in $GM(n, V(2n))$ identified
(see

$n$

$[1\underline{8]}$$\Phi_{a}(\cdot);\Phi_{a}(y)=\Phi_{a}\underline{(}x-y)$$\Phi_{a}(y)$

with

and

$\left(\begin{array}{l}I\\\mu(y)\end{array}\right)$$(_{I}^{\nu(y)})$.

Thus a dynamics of$\overline{\Phi}_{a}(y)$, $y<x$ , in the

Grassmannian is expressed in two diﬀerent ways by Riccati equations$\mu’+\mu D+D\mu+\mu^{2}-a^{2}c\otimes c=0$ ,$lJ’-l/D-D\iota/+\iota/(a^{2}c\otimes c)_{l/}-I=0$ .

The second equation is a Riccati equation which an error matrix appearing in the linear filtering theory obeys (see \S 3 and [1]).
, $ 0\leq i\leq$
be the $(i,j)$ -component of the -frameLet$2n$ $-1,0\leq j\leq n-1$ . The Pl\"ucker coordinate ofis given by$\alpha_{ij}(y)$$\Phi_{a}(y)$$n$$\Phi_{a}(y)$$\alpha_{I}(y)=\det[(\alpha_{i_{k}j}(y))_{0\leq k,j\leq n-1}]$

where I is the totality of$i_{n-1}\leq 2n-1$ .

We set

$I$

,

$I$

$=(i_{0, }\ldots, i_{n-1})\in I$

with $0\leq i_{0}<\cdots<$$=(i_{0}, \ldots, i_{n-1})\in Z^{n}$$F=\left(\begin{array}{ll}-D & -I\\-a^{2}c\otimes c & D\end{array}\right)$,

and define a$\left(\begin{array}{l}2n\\n\end{array}\right)$$\times$$\left(\begin{array}{l}2n\\n\end{array}\right)-$matrix $G$ by$ G\alpha_{I}=\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}F_{i_{k}j}\alpha_{i_{0},,i_{k-1},j,i_{k+1},,i_{n-1}}\ldots\ldots$,$I$$=(i_{0}, \ldots, i_{n-1})\in I$,

is defined in the same,
where, for, $k_{n-1}\leq 2n-1$ ,.
manner as . We then have a dynamics on the Grassmannian in termsof the Pl\"ucker coordinate;$0\leq k_{0}$$\alpha_{k_{0}},$$\ldots$’$k_{n-1}$$\alpha_{I}$$\frac{d}{dy}(\alpha_{I}(y))_{I\in \mathcal{T}}=G(\alpha_{I}(y))_{I\in I}$.

It should be mentioned that$\alpha_{I}(y)=\det\phi_{a}(y)$For related results, see also [4].

for

$I$

$=(0,1, \ldots, n -1)$ .

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N. Ikeda and S. Taniguchi

1.2. Cameron-Martin transformation
be the unique solution
For $p\in P_{n}$ , letof the-valued stochastic diﬀerential equation$\xi_{p}(y)={}^{t}(\xi_{p}^{1}(y), \ldots, \xi_{p}^{n}(y))$$R^{n}$$d\xi(y)=dw(y)+D\xi(y)dy$ ,(7)$\xi(0)=0$ .

. For $(p, c)\in A_{n}$ , we define a superposition
We setby,,of 1-dimensional Ornstein-Uhlenbeck processes$\mathcal{O}\mathcal{U}_{n}=\{\xi_{p} :p\in P_{n}\}$$\xi_{p}^{1}(y)$$ X_{p,c}(y)=\langle c, \xi_{p}(y)\rangle$$\xi_{p}^{n}(y)$$\ldots$,. $X_{p,c}(y)$ , $0\leq y\leq x$ , is awheredenotes the inner product oncontinuous Gaussian process with mean 0 and covariance function$R^{n}$$\langle\cdot, \cdot\rangle$(8)$R(u, v)=\sum_{j=1}^{n}\frac{c_{j}^{2}}{2p_{j}}(e^{p_{j}(u+v)}-e^{p_{j}|u-v|})$,$o\leq u$,$v$$\leq x$.

We set$\mathcal{G}_{n}=\{X_{p,c} :(p, c)\in A_{n}\}=\{\langle c, \xi_{p}\rangle :(p, c)\in A_{n}\}$.

is closed under
Obviously, $aX_{p,c}=X_{p,ac}$ for $a>0$ , and hence
is invarithe multiplication by positive numbers. Moreover,ant under permutation in the sense that $X_{p’,c’}=X_{p,c}$ if $(p, c)\in A_{n}$ ,is a permutation of $\{1, \ldots, n\}$ , and $p’={}^{t}(p_{\sigma(1)}, \ldots,p_{\sigma(n)})$ , $c’=$$\mathcal{G}_{n}$$X_{p,c}\in \mathcal{G}_{n}$$\sigma$${}^{t}(c_{\sigma(1)}, \ldots, c_{\sigma(n)})$.

function on $[0, x]$ with $\det\kappa(y)\neq$
Given an-matrix valued
for each $y\in[0, x]$ , we define two Cameron-Martin transformations $K$byand on$C^{1}$$n\times n$$\kappa$$0$$L$$\mathcal{W}^{n}$(9)$K[w](y)=w(y)-\int_{0}^{y}\kappa’(u)\kappa^{-1}(u)w(u)du$ ,(10)$L[w](y)=w(y)-\kappa(y)\int_{0}^{y}(\kappa^{-1})’(u)w(u)du$,$w$$\in \mathcal{W}^{n}$.

By a change of variables formula on $[0, x]$ , we see that
(11)

$K[L[w]]=L[K[w]]=w$

and let be an
Setfunction on $[0, x]$ . We then have$\overline{\theta}(y)=\kappa’(y)\kappa^{-1}(y)$for any $w$$\hat{\theta}$$n\times n$$\in \mathcal{W}^{n}$.

-matrix valued continuous

173

Quadratic Wiener functionals

Lemma 1.1. For any measurable
(12)

$f$

:

$\mathcal{W}^{n}\rightarrow[0, \infty)$, it holds that$\int_{\mathcal{W}^{n}}f(\xi_{p})\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy$$+\frac{1}{2}\langle(\overline{\theta}(x)-D)\xi_{p}(x), \xi_{p}(x)\rangle-\frac{1}{2}\int_{0}^{x}|\hat{\theta}(y)\xi_{p}(y)|^{2}dy]dP$$=\int_{\mathcal{W}^{n}}f(w)\exp[-\frac{1}{2}\int_{0}^{x}\langle E(a)w(y), w(y)\rangle dy$$+\frac{1}{2}\langle\overline{\theta}(x)w(x), w(x)\rangle-\frac{1}{2}\int_{0}^{x}|\hat{\theta}(y)w(y)|^{2}dy]dPe^{-(x/2)trD}$.

By using the Maruyama-Girsanov transformation ([8, 13,19]), we obtain that

Proof.

the leﬃ hand side of (12)$=\int_{\mathcal{W}^{n}}f(w)\exp[-\frac{a^{2}}{2}\int_{0}^{x}\langle c, w(y)\rangle^{2}dy$$+\frac{1}{2}\langle(\overline{\theta}(x)-D)w(x), w(x)\rangle-\frac{1}{2}\int_{0}^{x}|\hat{\theta}(y)w(y)|^{2}dy$$+\int_{0}^{x}\langle Dw(y), dw(y)\rangle-\frac{1}{2}\int_{0}^{x}|Dw(y)|^{2}dy]P(dw)$,where the identity may hold as $\infty=\infty$ . Applying the It\^o formula ([8,Q.E.D.
13]) to $\langle Dw(x), w(x)\rangle$ , it is easily seen that this implies (12).
, $y\in[0, x]$ , of the ordinary diﬀerential
Suppose that a solution
equation (6) satisfies the condition that $\det\phi_{a}(y)\neq 0,0\leq y\leq x$ , where
the initial condition is not specified. We set$\phi_{a}(y)$(13)$\beta_{a,x}(y)=-(\phi_{a}’\phi_{a}^{-1})(x-y)$and denote byandparts, respectively. Lettion$\overline{\beta}_{a,x}(y)$its symmetric and skew-symmetricbe the solution of the diﬀerential equa-$\hat{\beta}_{a,x}(y)$$\kappa_{a,x}(y)$$\kappa’(y)=\overline{\beta}_{a,x}(y)\kappa(y)$and define linear transformations(10) with.
$\kappa=\kappa_{a,x}$,,$K_{a,x}$$\kappa(x)=I$,$L_{a,x}$:,$\mathcal{W}^{n}\rightarrow \mathcal{W}^{n}$by (9) and174

N. Ikeda and S. Taniguchi

Proposition 1.1. For any bounded measurable
it holds that
(14)

$f$

:

$\mathcal{W}^{n}\rightarrow[0, \infty)$,$\int_{\mathcal{W}^{n}}f(\xi_{p})\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy$$+\frac{1}{2}\langle(\overline{\beta}_{a,x}(x)-D)\xi_{p}(x), \xi_{p}(x)\rangle-\frac{1}{2}\int_{0}^{x}|\hat{\beta}_{a,x}(y)\xi_{p}(y)|^{2}dy]dP$$=(\det\phi_{a}(0))^{1/2}(e^{xtrD}\det\phi_{a}(x))^{-1/2}\int_{\mathcal{W}^{n}}foL_{a,x}dP$

Proof.
$\overline{\beta}_{a,x}$

, and

(15)

Since

For the sake of simplicity, we write, respectively. It follows ﬀom (6) that

, and

and

.

$\beta,\overline{\beta}$$\hat{\beta}$for.

$\beta_{a,x}$$\hat{\beta}_{a,x}$$\beta’=E(a)-\beta^{2}$$tr\overline{\beta}=tr\beta$$\overline{\beta}’=E(a)-\overline{\beta}^{2}-\hat{\beta}^{2}$, by virtue of the It\^o formula, we then have$\frac{1}{2}\langle\overline{\beta}(x)w(x), w(x)\rangle$$=\frac{1}{2}\int_{0}^{x}\langle E(a)w(y), w(y)\rangle dy+\frac{1}{2}\int_{0}^{x}|\hat{\beta}(y)w(y)|^{2}dy$$+\int_{0}^{x}\langle\overline{\beta}(y)w(y), dw(y)\rangle-\frac{1}{2}\int_{0}^{x}|\overline{\beta}(y)w(y)|^{2}dy+\frac{1}{2}\int_{0}^{x}tr\beta(y)dy$.

Combining this with Lemma 1.1, we have(16)$\int_{\mathcal{W}^{n}}f(\xi_{p})\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy$$+\frac{1}{2}\langle(\overline{\beta}(x)-D)\xi_{p}(x), \xi_{p}(x)\rangle-\frac{1}{2}\int_{0}^{x}|\hat{\beta}(y)\xi_{p}(y)|^{2}dy]dP$$=\int_{\mathcal{W}^{n}}f(w)\exp[\int_{0}^{x}\langle\overline{\beta}(y)w(y), dw(y)\rangle-\frac{1}{2}\int_{0}^{x}|\overline{\beta}(y)w(y)|^{2}dy]P(dw)$$\times\exp[-\frac{x}{2}trD+\frac{1}{2}\int_{0}^{x}tr\beta(y)dy]$

Applying the Maruyama-Girsanov transformation to the equation$dz(y)=dw(y)+\overline{\beta}(y)z(y)dy$

,

.

,

175

Quadratic Wiener functionals

and noting that(17)$\overline{\beta}=\kappa_{a,x}^{J}\kappa_{a,x}^{-1}$, we obtain that$\int_{\mathcal{W}^{n}}(g\circ K_{a,x})(w)\exp[\int_{0}^{x}\langle\overline{\beta}(y)w(y), dw(y)\rangle$$-\frac{1}{2}\int_{0}^{x}|\overline{\beta}(y)w(y)|^{2}dy]P(dw)=\int_{\mathcal{W}^{n}}gdP$

for any bounded measurable
we have
(18)

$g$

:

$\mathcal{W}^{n}\rightarrow[0, \infty)$. By the definition of$\beta$,$\exp[\int_{0}^{x}tr\beta(y)dy]=\det\phi_{a}(0)(\det\phi_{a}(x))^{-1}$By (11), combining (17) and (18) with (16), we obtain (14).

Q.E.D.

1.3. Eigenvalues of $E(a)$Let $(p, c)\in A_{n}$ and $a>0$ . We shall specify the eigenvalues of$E(a)=E_{p,c}(a)$ . As for $p={}^{t}(p_{1}, \ldots,p_{n})$ , rearranging if necessary, we
may assume that there exist $m$ and $1\leq j(1)<\cdots<j(m)\leq n$ suchthat$|p_{j}|\leq|p_{j+1}|$ for $j=1,2$ ,, $n-1$ , $p_{j(\ell)}>0$ and $p_{j(\ell)+1}=$$(H)_{m}$$\ell=1$,, $m$ , and $\#\{|p_{1}|, \ldots, |p_{n}|\}=n-m$ .
for$\ldots$$-p_{j(\ell)}$$\ldots$

When $m=0$ , this conditions means that $|p_{1}|<|p_{2}|<\cdots<|p_{n}|$ .
$=1$ , thenIfonholds. We define a Herglotz function$C^{+}=\{z\in C:Imz>0\}$ by$h_{p,c,a}$$(H)_{0}$$n$$h_{p,c,a}(z)=\frac{1}{2}\int_{0}^{\infty}\frac{1}{u-z}\{\sigma_{+}+\sigma_{-}\}(-du)$

,

where
(19)

$\sigma_{+}(du)=2a^{2}.\sum_{i\cdot p_{i}\geq 0}c_{i}^{2}\delta_{-p_{i}^{2}}$(du),$\sigma_{-}(du)$$=2a^{2}.\sum_{i\cdot p_{i}<0}c_{i}^{2}\delta_{-p_{i}^{2}}$(du)(cf. [11]). Then $h_{p,c,a}(\lambda+t\sqrt{-1})$ converges to$h_{p,c,a}(\lambda+0\sqrt{-1})=a^{2}\sum_{j=1}^{n}\frac{c_{j}^{2}}{p_{j}^{2}-\lambda}$

as$t\downarrow 0$.

Under $(H)_{m}$ , the equation $h_{p,c,a}(r+0\sqrt{-1})=-1$ possesses$-m$roots $0<r_{1}<\cdots<r_{n-m}$ such that $r_{j}^{1/2}\not\in\{p_{j(1)}, \ldots,p_{j(m)}\}$ , $j=$,
, $-m$ . Take ,,so that$n$$1$$\ldots$$n$$\eta_{1}$$\ldots$$\eta_{n}\in R$$\{|\eta_{1}|, \ldots, |\eta_{n}|\}=\{p_{j(1)}, \ldots, p_{j(m)}, r_{1}^{1/2}, \ldots, r_{n-m}^{1/2}\}$

.

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N. Ikeda and S. Taniguchi

Define an $n\times n$ matrix $U=(U_{ij})_{1\leq i,j\leq n}$ by$\frac{c_{i}}{|(D^{2}-r_{k}I)^{-1}c|(p_{i}^{2}-r_{k})}$

(20)

,

,

if

$\eta_{j}^{2}=r_{k}$if$\eta_{j}^{2}=p_{j(\ell)}^{2}$$U_{ij}=\{$$\frac{\delta_{i,j(\ell)+1}c_{j(\ell)}-\delta_{i,j(\ell)^{C}j(\ell)+1}}{(c_{j(\ell)}^{2}+c_{j(\ell)+1}^{2})^{1/2}}$,.

Lemma 1.2. $U\in O(n)$ and it holds that

Proof.

$E(a)=UR^{2}U^{-1}$ ,where $R=diag[\eta_{1}, \ldots, \eta_{n}]$ .

It is easily checked that$p_{j(\ell)}^{2}$is an eigenvalue of $E(a)$ witheigenvector.
$u={}^{t}(0,\ldots,0\tilde{j(\ell)-1}, -c_{j(\ell)+1}, c_{j(\ell)}, 0, \ldots, 0)$

is an eigenvalue with
Noting that $D^{2}-r_{j}I$ is invertible, we see thateigenvector $u=(D^{2}-r_{j}I)^{-1}c$ . Thus we have obtained distinct eigenvalues of $E(a)$ and the associated eigenvectors. In conjunction with the
Q.E.D.
symmetry of $E(a)$ , we obtain the desired assertion.
$r_{j}$

$n$

\S 2.

Reflectionless potentialby (19). Then
For $(p, c)\in A_{n}$ and $a>0$ , define$G(u, v;\sigma)$ coincides with $a^{2}R(u, v)$ , the covariance function described in. We shall spell(8). Hence we can identify
with $aX_{p,c}$ , and.
andout a correspondence between$(H)_{m}$, we define $0<r_{1}<\cdots<r_{n-m}$ as described before
Assuming$0<\eta_{1}<\cdots<\eta_{n}$,, $m_{n}>0$ by
and
1.2.
Lemma
Define$\sigma=(\sigma_{+}, \sigma_{-})\in\Sigma$$X^{\sigma}$$\mathcal{G}_{n}\subset \mathcal{G}$$Q_{n}$$\mathcal{G}_{n}$$m_{1}$$\ldots$$\{\eta_{1}, \ldots, \eta_{n}\}=\{p_{j(1)}, \ldots,p_{j(m)}, r_{1}^{1/2}, \ldots, r_{n-m}^{1/2}\}$.

$2\eta_{i}\frac{c_{j(\ell)+1}^{2}}{c_{j(\ell)}^{2}}\prod_{k\neq i}\frac{\eta_{k}+\eta_{i}}{\eta_{k}-\eta_{i}}\prod_{k\neq j(\ell),j(\ell)+1}\frac{p_{k}+\eta_{i}}{p_{k}-\eta_{i}}$,if $i=j(\ell)$ ,$m_{i}=\{$$-2\eta_{i}\prod_{k\neq i}\frac{\eta_{k}+\eta_{i}}{\eta_{k}-\eta_{i}}\prod_{k=1}^{n}\frac{p_{k}+\eta_{i}}{p_{k}-\eta_{i}}$Mention that(21)$\eta_{j(\ell)}=p_{j(\ell)}$, $\ell=1$ ,$\ldots$,otherwise.

, $m$ . We set$I_{p,c,a}(x)=\int_{\mathcal{W}^{n}}\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy]dP$

.

177

Quadratic Wiener functionals$\eta_{i}$Theorem 2.1. Define $A(x)$ by (4) with the above scattering data
, $m_{i}>0$ , $i=1$ ,, . Then it holds that$\ldots$

(22)

$n$

$\log(I_{p,c,a}(x))=-\frac{1}{2}\log\det(I+A(x))$$+\frac{1}{2}\log\det(I+A(0))-\frac{x}{2}\sum_{i=1}^{n}(p_{i}+\eta_{i})$.

In particular,$4\frac{d^{2}}{dx^{2}}\log(I_{p,c,a}(x))=-2\frac{d^{2}}{dx^{2}}\log\det(I+A(x))$.

Note that $I_{p,c,a}(x)$ is invariant under the permutation of parameters$(p, c)$ in the sense as stated after (8).
Thus, so is the reflectionlesspotential associated with $I_{p,c,a}(x)$ .
The proof of the theorem is divided into two steps, each step being
and letbea lemma. In the sequel, we write $R$ for$\Phi_{a}(y;I,0)$. Thenthe upper half of$diag[\eta_{1}, \ldots, \eta_{n}]$$\phi_{a}(y)$$\phi_{a}(y)=U\{\cosh(yR)-\sinh(yR)R^{-1}U^{-1}DU\}U^{-1}$ ,(23)

where $U\in O(n)$ is defined by (20), and, for $n\times n$-matrix $A$ ,$\cosh(A)=(e^{A}+e^{-A})/2$

and$\sinh(A)=(e^{A}-e^{-A})/2$ .

, we see thatare
By computing the product matrix
and$y\geq0$
by (13) with, $z>0$ . Then we can defineboth invertible for$\beta_{a,x}(x)=D$. Obviously. Sinceobeys the Riccati equation
this(15), it turns out to be symmetric. Applying Proposition 1.1, we obtain$\psi_{a}(z)$$\phi_{a}(y)$$\phi_{a}\psi_{a}$$\beta_{a,x}$$\phi_{a}$(24)$\beta_{a,x}$$I_{p,c,a}(x)=(\det\phi_{a}(0))^{1/2}(\det\phi_{a}(x))^{-1/2}e^{-(x/2)trD}$.

In this expression, for the latter use, we left $\det\phi_{a}(0)$ while it is equalholds, then we set
to one. If$(H)_{0}$$X_{ij}=(p_{j}+r_{i}^{1/2})^{-1}$,$Y_{ij}=(p_{j}-r_{i}^{1/2})^{-1}$$X=(X_{ij})_{1\leq i,j\leq n}$,,$1\leq i$$Y=(Y_{ij})_{1\leq i,j\leq n}$, $j\leq n$ ,,$V(c)=diag[|(D^{2}-r_{1}I)^{-1}c|^{-1}, \ldots, |(D^{2}-r_{n}I)^{-1}c|^{-1}]$$\sigma(i)=sgn[\prod_{\beta=1}^{n}(p_{\beta}-\eta_{i})]$,and $B=diag[b(1), \ldots, b(n)]$ .

,$b(i)=\sigma(i)\{-2\eta_{i}\frac{\prod_{\alpha\neq i}(\eta_{\alpha}^{2}-\eta_{i}^{2})}{\prod_{\beta=1}^{n}(p_{\beta}^{2}-\eta_{i}^{2})}\}^{1/2}$

,

178

N. Ikeda and S. Taniguchi

Lemma 2.1. Suppose that(25)

holds. Then it holds that$(H)_{0}$$\phi_{a}(y)=-\frac{1}{2}UV(c)R^{-1}B(I+A(y))e^{yR}B^{-1}XC(c)$ ,where $C(c)=diag[c_{1}, \ldots, c_{n}]$ . Moreover, the identity (22) holds.

Proof.

Due to (23), we have$\phi_{a}(y)=\frac{1}{2}UR^{-1}\{e^{yR}(RU^{-1}-U^{-1}D)+e^{-yR}(RU^{-1}+U^{-1}D)\}$ .

Set $Z=(Z_{ij})_{1\leq i,j\leq n}$ , where$Z_{ij}=(p_{i}^{2}-r_{j})^{-1}$,$1\leq i,j\leq n$.

Then it holds that

$U=C(c)ZV(c)$ ,$U^{-1}=V(c){}^{t}ZC(c)$ .

Since $R$ , $D$ , $C(c)$ , and $V(c)$ are all diagonal matrices, we have that
$RU^{-1}-U^{-1}D=-V(c)XC(c)$ ,$RU^{-1}+U^{-1}D=V(c)YC(c)$ .

Hence we obtain(26)$\phi_{a}(y)=-\frac{1}{2}UR^{-1}V(c)\{I-e^{-yR}YX^{-1}e^{-yR}\}e^{yR}XC(c)$ .

We now compute $YX^{-1}$ . Applying Cauchy’s identity (cf. [14])(27)$\det((\frac{1}{\alpha_{i}+\beta_{j}})_{1\leq i,j\leq n})=\frac{\prod_{1\leq i<j\leq n}(\alpha_{i}-\alpha_{j})(\beta_{i}-\beta_{j})}{\prod_{i,j=1}^{n}(\alpha_{i}+\beta_{j})}$for$\alpha_{1}$,$\ldots$,$\alpha_{n}$,$\beta_{1}$,$\ldots$,$\beta_{n}\in C$,

to the cofactor matrix of $X$ , we obtain$(X^{-1})_{k\ell}=\frac{\prod_{\alpha\neq\ell}(p_{k}+\eta_{\alpha})\prod_{\beta=1}^{n}(p_{\beta}+\eta\ell)}{\prod_{\beta\neq k}(p_{\beta}-p_{k})\prod_{\alpha\neq\ell}(\eta_{\alpha}-\eta_{\ell})}$,$1\leq k$,$\ell\leq n$.

Using Lagrange’s interpolation formula$\sum_{k=1}^{n}\frac{\prod_{j=1}^{n-1}(a_{k}+b_{j})\prod_{\beta\neq k}(a_{\alpha}-z)}{\prod_{\beta\neq k}(a_{\alpha}-a_{k})}=\prod_{j=1}^{n-1}(z+b_{j})$,$z\in C$,

we have$(YX^{-1})_{ij}=\frac{\prod_{\alpha\neq i}(\eta_{\alpha}+\eta_{i})}{\prod_{\beta=1}^{n}(p_{\beta}-\eta_{i})}\frac{2\eta_{i}}{\eta_{i}+\eta_{j}}\frac{\prod_{\beta=1}^{n}(p_{\beta}+\eta_{j})}{\prod_{\alpha\neq j}(\eta_{\alpha}-\eta_{j})}$,$1\leq i,j\leq n$

.

179

Quadratic Wiener functionals

Since$sgn[(\prod_{\beta=1}^{n}(p_{\beta}+\eta_{i}))/(\prod_{\alpha\neq i}(\eta_{\alpha}-\eta_{i}))]=-\sigma(i)$$b(i)\sqrt{m_{i}}=2\eta_{i}\frac{\prod_{\alpha\neq i}(\eta_{\alpha}+\eta_{i})}{\prod_{\beta=1}^{n}(p_{\beta}-\eta_{i})}$,, it holds that$\frac{\sqrt{m_{i}}}{b(i)}=-\frac{\prod_{\beta=1}^{n}(p_{\beta}+\eta_{i})}{\prod_{\alpha\neq i}(\eta_{\alpha}-\eta_{i})}$,$1\leq i\leq n$

.

Hence
$(YX^{-1})_{ij}=-\frac{b(i)\sqrt{m_{i}}\sqrt{m_{j}}(b(j))^{-1}}{\eta_{i}+\eta_{j}}$,$1\leq i,j\leq n$.

Combining this with (26), we obtain (25).
The identity (25) implies$\det\phi_{a}(0)(\det\phi_{a}(x))^{-1}=\det(I+A(0))(e^{xtrR}\det(I+A(x)))^{-1}$Thus the second assertion follows from this and (24).

Q.E.D.

Lemma 2.2. Let $m\geq 1$ and suppose that $(H)_{m}$ is satisfied. Then
(22) holds$Proo/$.

$For\in>0$ , set$ p_{i}-\in$,$p_{i}^{\in}=\{$$p_{i}$,

if $i=j(\ell)+1$ for some ,otherwise,$\ell$$1\leq i\leq n$.

Choosing a suﬃciently $sma\square \in>0$ , we may assume that$|p_{1}^{\Xi}|<|p_{2}^{\Xi}|<\cdots<|p_{n}^{\mathcal{E}}|$.

Let $0<r_{1}^{\in}<\cdots<r_{n}^{\Xi}$ be roots of $a^{2}\sum_{i=1}^{n}c_{i}^{2}/\{(p_{i}^{\Xi})^{2}-r\}=-1$ . Then it
holds that(28)$(p_{i}^{\Xi})^{2}<r_{i}^{\in}<(p_{i+1}^{g})^{2}<r_{i+1}^{\epsilon}$,$i=1,2$ ,$\ldots$, $n-1$ .

, , with these ’s,, $i=1$ ,Define scattering data ,’s as described before Theorem 2.1. By Lemma 2.1, we have$\eta_{i}^{\Xi}$$m_{i}^{\Xi}>0$$\ldots$$n$$p_{i}^{\Xi}$$r_{i}^{\Xi}$’s and$c_{i}$$\log(I_{p^{\epsilon},c,a}(x))=-\frac{1}{2}\log\det(I+A^{\Xi}(x))$$+\frac{1}{2}\log\det(I+A^{\Xi}(0))-\frac{x}{2}\sum_{i=1}^{n}(p_{i}^{\epsilon}+\eta_{i}^{\Xi})$

where, . Since,$p^{\Xi}={}^{t}(p_{1}^{\epsilon}, \ldots,p_{n}^{\Xi})$$1$$\ldots$$n$$p_{i}^{\Xi}\rightarrow p_{i}$

as

and

is defined by (4) with, , we have that
, $i=1$ ,$A^{\in}(x)$$\in\downarrow 0$$\ldots$$n$$\log(I_{p^{\Xi},c,a}(x))\rightarrow\log(I_{p,c,a}(x))$$as\in\downarrow 0$.

$\eta_{i}^{\Xi}$,$m_{i}^{\Xi}$, $i=$

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N. Ikeda and S. Taniguchi

Moreover, recalling that
, $i=1$ ,that, $i=1$ ,, . Hence$\eta_{i}^{\in}=(r_{i}^{\in})^{1/2}$$\eta_{i}^{\epsilon}\rightarrow\eta_{i}as\in\downarrow 0$$\ldots$$\ldots$, , it follows ﬀom (28)
$n$

$n$

$\sum_{i=1}^{n}(p_{i}^{\Xi}+\eta^{\Xi})\rightarrow\sum_{i=1}^{n}(p_{i}+\eta_{i})$as,$\in\downarrow 0$.

Thus the proof completes once we have shown the convergence ofto $A(y)as\in\downarrow 0$ .
, it suﬃces to show thatTo see the convergence oftends to, . If $i\neq j(\ell)$ for any , then it is easily
asfor every $i=1$ ,. We now consider the case that $i=j(\ell)$ .
seen that
,
Since
as
$A^{\in}(y)$$A^{\Xi}(y)$$m_{i}$$\in\downarrow 0$$\ldots$$m_{i}^{\Xi}$$\ell$$n$$m_{i}^{\in}\rightarrow m_{i}as\in\downarrow 0$$\eta_{j(\ell)}^{\epsilon}\rightarrow\eta_{j(\ell)}=p_{j(\ell)}$(29)$\in\downarrow 0$$\frac{p_{j(\ell)}^{\epsilon}+\eta_{j(\ell)}^{\epsilon}}{p_{j(\ell)+1}^{\in}-\eta_{j(\ell)}^{\in}}\rightarrow-1$

It follows ﬀom (28) and the identitythat,$as\in\downarrow 0$.

$a^{2}\sum_{j=1}^{n}c_{j}^{2}/\{(p_{j}^{\Xi})^{2}-r_{j(\ell)}^{\Xi}\}=-1$$c_{j(\ell)}^{2}\{(p_{j(\ell)}+\in)^{2}-r_{j(\ell)}^{\Xi}\}+c_{j(\ell)+1}^{2}\{p_{j(\ell)}^{2}-r_{j(\ell)}^{\in}\}=O(\in^{2})$,$as\in\downarrow 0$.

This yields that(30)$(p_{j(\ell)}^{\epsilon})^{2}-(\eta_{j(\ell)}^{\Xi})^{2}=\frac{-2p_{j(\ell)}c_{j(\ell)}^{2}}{c_{j(\ell)}^{2}+c_{j(\ell)+1}^{2}}\in+O(\in^{2})$(31)$(p_{j(\ell)+1}^{\in})^{2}-(\eta_{j(\ell)}^{\Xi})^{2}=\frac{2p_{j(\ell)}c_{j(\ell)+1}^{2}}{c_{j(\ell)}^{2}+c_{j(\ell)+1}^{2}}\in+O(\in^{2})$,,$as\in\downarrow 0$

.

Hence
$\frac{p_{j(\ell)+1}^{\in}+\eta_{j(\ell)}^{\in}}{p_{j(\ell)}^{\in}-\eta_{j(\ell)}^{\Xi}}\rightarrow c_{j(\ell)+1}^{2}c_{j(\ell)}^{-2}$

Combining this with (29) and the definition of$m_{j(\ell)}as\in\downarrow 0$,$as\in\downarrow 0$$m_{j(\ell)}^{g}$.

, we see that

$ m_{j(\ell)}^{\in}\rightarrow$

Q.E.D.

.

Remark 2.1. (i) The exponent of the integrand in the right handside of (21) is a sum of a quadratic Wiener functional and a constant.
Hence the right hand side of (24) can be expressed in terms of theCarleman-Fredholm determinant of the symmetric Hilbert-Schmidt operator determining the quadratic Wiener functional. Moreover,is asolution of the Jacobi equation associated with the Lagrangian related
to the Wiener functional (cf. [6, 7]).
(ii) In Theorem 2.1, for each $\in N$ , a mapping fromto the space$\phi_{a}$$n$$A_{n}$

181

Quadratic Wiener functionals$\{(\eta_{1}\ldots, \eta_{n}, m_{1}\ldots, m_{n}) :of scat-0<\eta_{1}<\cdots<\eta_{n}, m_{1}\ldots, m_{n}>0\}$tering data was established. If $n=2$ , the mapping is invertible.

Filtering theory\S 3.

In this section, we
see that the change of variables formula (24)relates to the filtering theory. On the $(n+1)$ -dimensional Wiener space, consider the following filtering problem.
$sha\square $$\mathcal{W}^{n+1}$$d\xi_{p}(y)=dw(y)+D\xi_{p}(y)dy$ ,$dY(y)=db(y)+a\langle c, \xi_{p}(y)\rangle dy$,$\xi(0)=0$ ,(system),$Y(0)=0$ ,(observation),

. Let
be
where $(p, c)\in A_{n}$ , $a>0$ andto the filteringthe -field generated by $Y(u)$ , $u\leq y$ . A solution
problem with respect to $Y(u)$ , $u\leq y$ , is realized as a function whose-valuederror matrix is minimal in the space of error matrices of-measurable functions, where the order is the one inherited from thecoincides
non-negative definiteness ([1, Theorem 4.1]). In our case,, whichgiven
ofwith the conditional expectation
is called the Kalman-Bucy filter. The corresponding error matrix$(w, b)\in \mathcal{W}^{7\iota}\times \mathcal{W}^{1}=\mathcal{W}^{n+1}$$\mathcal{F}_{y}^{Y}$$\hat{\xi}_{p}(y)$$\sigma$$R^{n}$$F_{y}^{Y}$$\hat{\xi}_{p}(y)$$\xi_{p}(y)$$E[\xi_{p}(y)|\mathcal{F}_{y}^{Y}]$$F_{y}^{Y}$$P_{a}(y)=\int_{\mathcal{W}^{n+1}}(\xi_{p}(y)-\hat{\xi}_{p}(y))^{t}(\xi_{p}(y)-\hat{\xi}_{p}(y))dP$

obeys the $n\times n$-matrix Riccati equation$P(0)=0$ ,$P’=DP+PD-P(a^{2}c\otimes c)P+I$ ,which we have already seen as an equation forbe the error variance of $X_{p,c}(y)$ ;$\nu(y)$in 1.1. Let$\rho_{p,c,a}(y)=\int_{\mathcal{W}^{n+1}}|X_{p,c}(y)-E[X_{p,c}(y)|F_{y}^{Y}]|^{2}dP$$\rho_{p,c,a}(y)$.

It then holds that$\rho_{p,c,a}(y)=tr[(c\otimes c)P_{a}(y)]$Let $\Phi_{a}(y;I, 0)=$$y\geq 0$$\left(\begin{array}{l}\phi_{a}(y)\\\psi_{a}(y)\end{array}\right)$.

. As was seen before (24), $\det\phi_{a}(y)\neq 0$ ,. We set $\gamma_{a}(y)=\psi_{a}(y)\phi_{a}^{-1}(y)$ and $\gamma_{a,x}(y)=\gamma_{a}(x-y)$ . Note that$\gamma_{a,x}’=-D\gamma_{a,x}-\gamma_{a,x}D-\gamma_{a,x}^{2}+a^{2}(c\otimes c)$

on $[0, x]$ ,$\gamma_{a,x}(x)=0$

.

182

N. Ikeda and S. Taniguchi

Then it holds that$\{\det(I-\gamma_{a},{}_{xa}P)\}’=(-tr[a^{2}(c\otimes c)P_{a}+\gamma_{a,x}])\det(I-\gamma_{a},{}_{xa}P)$.

Since $P(0)=\gamma_{a,x}(x)=0$ ,$1=\det(I-\gamma_{a,x}(x)P_{a}(x))=\exp[-\int_{0}^{x}tr[a^{2}(c\otimes c)P_{a}(y)+\gamma_{a,t}(y)]dy]$.

Combined with (18), this implies$\exp[-\int_{0}^{x}tr[a^{2}(c\otimes c)P_{a}(y)]dy]=\frac{e^{-xtrD}}{\det\phi_{a}(x)}$.

By (24), we obtain$\int_{\mathcal{W}^{n}}\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy]dP=\exp[-\frac{a^{2}}{2}\int_{0}^{x}\rho_{p,c,a}(y)dy]$,

which can be also shown by applying the result due to $M.L$ . Kleptsyna
and A. Le Breton [10].
Letbe the unique solution of the integral equation$\delta(y, u)$(32)$\delta(y, u)=a^{2}R(y, u)-\int_{0}^{y}\delta(y, v)\delta(u, v)dv$

In [10], they also have shown that\S 4.

$a^{2}\rho_{p,c,a}(y)$,$o\leq y$, $u\leq x$ .

coincides with$\delta(y, y)$.

$KdV$ equation

Throughout this section, we assume $(H)_{m}$ . Let scattering data $0<$$\eta_{1}<\cdots<\eta_{n}$ ,, $m_{n}>0$ and,-matrices $U$ and $R$ be the ones
stated before and after Theorem 2.1. Set$\ldots$$n\times n$$T=\{t=(t_{1}, t_{2}, \ldots, ) :t_{j}\in R, \neq\{j :$m_{1}$

t_{j}\neq 0\}<\infty\}$.

For $x\in R$ , $t\in T$ , define$\zeta_{j}(x, t)=x\eta_{j}+\sum_{\alpha=1}^{\infty}t_{\alpha}\eta_{j}^{2\alpha+1}$,$\zeta(x, t)=diag[\zeta_{1}(x, t), \ldots, \zeta_{n}(x, t)]$.

The $tau$ function $\tau(x, t)$ of the $KdV$ equation is of the form
(33)$\tau(x, t)=1+\sum_{p=1}^{n}\sum_{1\leq i_{1}<<i_{p}\leq n}\cdots\prod_{j=1}^{p}\frac{m_{i_{j}}}{2\eta_{i_{j}}}\prod_{1\leq j<k\leq p}(\frac{\eta_{i_{j}}-\eta_{i_{k}}}{\eta_{i_{j}}+\eta_{i_{k}}})^{2}$$\times$$\exp[-2\sum_{j=1}^{p}\zeta_{i_{j}}(x, t)]$

.

183

Quadratic Wiener functionals

For details, see $[14, 16]$ . If we set$A(x, t)=(\frac{\sqrt{m_{i}m_{j}}}{\eta_{i}+\eta_{j}}e^{-\{\zeta_{i}(x,t)+\zeta_{j}(x,t)\})}1\leq i,j\leq n$,then, with the help of Cauchy’s identity (27), we can show that(34)$\tau(x, t)=\det(I+A(x, t))$ ,$x\in R$ , $t\in T$ .

We shall show that $\tau(x, t)$ can be expressed in terms of Wiener
integral. To do this, let(35)$\phi_{a}(x, t)=U\{\cosh(\zeta(x, t))-\sinh(\zeta(x, t))R^{-1}U^{-1}DU\}U^{-1}$ .

Then, for each $t\in T$ ,initial condition$\phi_{a}(\cdot, t)$obeys the diﬀerential equation (6) with$\phi_{a}(0)=U\{\cosh(\zeta(0, t))-\sinh(\zeta(0, t))R^{-1}U^{-1}DU\}U^{-1}$,$\phi_{a}’(0)=U\{R\sinh(\zeta(0, t))-\cosh(\zeta(0, t))U^{-1}DU\}U^{-1}$ .

As we shaU see in Lemma 4.1 below, $\det\phi_{a}(x, t)\neq 0$ , and then we candefine$\beta_{a,x,t}(y)=-((\partial_{x}\phi_{a})\phi_{a}^{-1})(x-y, t)$.

We shall show thatis symmetric. To do this, write, the partial derivative. For each $k=1,2$ ,ofsatisfies that
respect to$\beta_{a,x,t}(y)$

for
with

$\beta(y, t)$$\beta_{a,x,t}(y)$$\partial_{t_{k}}\phi_{a}$$\phi_{a}$$\ldots$$t_{k}$$\partial_{t_{k}}\phi_{a}(x, t)=U\{R^{2k+1}\sinh(\zeta(x, t))-R^{2k}\cosh(\zeta(x, t))U^{-1}DU\}U^{-1}$.

This implies that$\partial_{t_{k}}\phi_{a}(x, t)=E(a)^{k}\partial_{x}\phi_{a}(x, t)$

for any $k=1,2$ ,(36)$\ldots$

and$\partial_{t_{k}}^{2}\phi_{a}(x, t)=E(a)^{2k+1}\phi_{a}(x, t)$

We then obtain that$\partial_{t_{k}}\beta(y, t)=-E(a)^{k+1}+\beta(y, t)E(a)^{k}\beta(y, t)$,$k=1,2$ ,$\ldots$ofSince $E(a)$ is symmetric, the transpose
satisfies theis symmetric if and only if so issame identities in (36). Hence$\beta(y, t[k])$ for some $k=1,2$ ,$t[k]$
is obtained ﬀom by replacing, whereby 0. As was seen in the paragraph before (24), $\beta(y, 0)$ is symmetric,
where $0=(0, 0, \ldots)\in T$ . In conjunction with the above observation,
this implies thatis symmetric for $t=(t_{1},0,0, \ldots)$ , $t_{1}\in R$ .
is symmetric
Apply the above observation again, it follows that${}^{t}\beta(y, t)$$\beta(y, t)$$\beta(y, t)$$t$$\ldots$$t_{k}$$\beta(y, t)$$\beta(y, t)$184

N. Ikeda and S. Taniguchi, $t_{2}\in R$ . Repeating this argument successively,$\beta(y, t)$ is symmetric for any $(y, t)\in R\times T$ .

for $t=(t_{1}, t_{2},0,0, \ldots)$ ,we can conclude that
We set$t_{1}$$I_{p,c,a}(x, t)=\int_{\mathcal{W}^{n}}\exp[-\frac{a^{2}}{2}\int_{0}^{x}X_{p,c}(y)^{2}dy$$+\frac{1}{2}\langle(\beta_{a,x,t}(x)-D)\xi_{p}(x), \xi_{p}(x)\rangle]dP$.

To state our result, we introduce a set $J=\{(j(\ell)+1, j(\ell))$ : $\ell=$,, $m\}$ and a quantity$1$$\ldots$$Z_{m}(p, c)=(-1)^{m}\prod_{i=1}^{n}\frac{c_{i}}{2\eta_{i}}\prod_{1\leq i<j\leq 7b}(p_{i}-p_{j})(\eta_{i}-\eta_{j})\{\prod_{(i,j)\not\in J}(p_{i}+\eta_{j})$$\times\prod_{k=1}^{n-m}|(D_{p}^{2}-r_{k}I)^{-1}c|\prod_{\ell=1}^{m}\frac{c_{j(\ell)+1}(c_{j(\ell)}^{2}+c_{j(\ell)+1}^{2})^{1/2}}{2p_{j(\ell)^{C}j(\ell)}}\}^{-1}$where, if $m=0$ , then$\prod_{\ell=1}^{0}(\cdots)=1$$ J=\emptyset$,“$(i,j)\not\in J$”,

means $‘‘ 1\leq i,j\leq n’’$ , and.

Theorem 4.1. (i) It holds that(37)$\det\phi_{a}(x, t)=\tau(x, t)e^{tr\zeta(x,t)}Z_{m}(p, c)$(38)$\log(I_{p,c,a}(x, t))=-\frac{1}{2}\log\tau(x, t)+\frac{1}{2}\log\tau(0, t)-\frac{x}{2}\sum_{i=1}^{n}(p_{i}+\eta_{i})$(ii) Let $t=(t, 0, \ldots)$ . We write(a) Set

$t$

,

.

for .
$t$

$q_{p,c,a}(x, t)=-4\partial_{x}^{2}\log(I_{p,c,a}(x, t))$.

Then $q_{p,c,a}(x, t)$ solves the $KdV$ equation (2).
(b) Both $\det\phi_{a}(x, t)$ and $(I_{p,c,a}(x, t))^{-2}$ solve the Hirota equation:(38)$(4D_{t}D_{x}-D_{x}^{4})u\cdot u=0$,

where $(D_{x}, D_{t})$ denotes the Hirota derivatives with respect to the variables $(x, t)([14])$ .
While $q_{p,c,a}(x, t)$ is defined only on
in our framework, by$R\timesR$
virtue of (37), it extends toso that the extension $ako$ solves the$KdV$ equation (2).
To prove Theorem 4.1, we first show a relation between
and$[0, \infty)\times R$$\phi_{a}(x, t)$$A(x, t)$ .

185

Quadratic Wiener functionals

Lemma 4.1. It holds that(40)$\det\phi_{a}(y, t)=\det(I+A(y, t))e^{tr\zeta(y,t)}Z_{m}(p, c)$.

In particular, (37) holds and $\det\phi_{a}(y, t)\neq 0$ , $y\geq 0$ .

as in the proof of
Lemma 2.2. For suﬃciently small , we see thatsatisfies the condition. In the sequel, as in the proof of Lemma 2.2, we use the superscript
indicate the dependence on ; given a quantity defined with ,we writefor the same quantity defined withinstead of . Then, in
repetition of the argument employed to prove Lemma 2.1, we can showthat

Proof.

$For\in>0$ , define$p^{\Xi}={}^{t}(p_{1}^{\in}, \ldots,p_{n}^{\epsilon})\in P_{n}$$p^{\Xi}$$\in$$(H)_{0}$$\in to$$f$$p^{\Xi}$$p$$p^{\in}$$f^{\Xi}$$p$$\phi_{a}^{\in}(y, t)=-\frac{1}{2}U^{\Xi}(R^{\Xi})^{-1}V^{\epsilon}(c)B^{\in}\{I+A^{\Xi}(y, t)\}(B^{\in})^{-1}e^{\zeta^{\epsilon}(y,t)}X^{\Xi}C(c)$Applying Cauchy’s identity (27) to computing
have
(41)

$\det X^{\in}$and$\det\phi_{a}^{\Xi}(y, t)=\det(I+A^{\Xi}(y, t))e^{tr\zeta^{\epsilon}(y,t)}Z_{0}(p^{\Xi}, c)$$\det U^{\Xi}$

.

, we

.

,
As we have already seen in the proof of Lemma 2.2, as, . Moreover, taking the advantage of (30)andfor $i=1$ ,.
,andand (31), we can show that,
Hence, lettingin (41), we obtain (40).
(37) is an immediate consequence of (34) and (40). The non-singulaQ.E.D.
rity offollows ﬀom that of $I+A(y, t)$ and (40).
$\in\downarrow 0$$m_{i}^{\Xi}\rightarrow m_{i}$$\ldots$$\eta_{i}^{\epsilon}\rightarrow\eta_{i}$$n$$as\in\downarrow 0$$U^{\Xi}\rightarrow U$$Z_{0}(p^{\in}, c)\rightarrow Z_{m}(p, c)$$\in\downarrow 0$$\phi_{a}(y, t)$

Proof of Theorem 4 1. (i) We have already seen (37) in Lemma 4.1.
By the same lemma, $\phi_{a}(y, t)\neq 0$ , $y\geq 0$ . Applying Proposition 1.1 with$\phi_{a}(y)=\phi_{a}(y, t)$ , we have$\cdot$$I_{p,c,a}(x, t)=(\det\phi_{a}(0, t))^{1/2}(e^{xtrD}\det\phi_{a}(x, t))^{-1/2}$

By (37), it holds that$\det\phi_{a}(0, t)(\det\phi_{a}(x, t))^{-1}=\tau(0, t)\tau(x, t)^{-1}\exp[tr\zeta(0, t)-tr\zeta(x, t)]$.

Since $\zeta(0, t)-\zeta(x, t)=-xR$ , we obtain (38).
(i) (a) The assertion follows form (34) and (38).
(b) It is well known ([14]) that $\tau(x, t)$ solves the Hirota equation (38).
Since $\det\phi_{a}(x, t)$ and $(I_{p,c,a}(x, t))^{-2}$ are both of the form $k(t)e^{cx}\tau(x, t)$with a constant and a function :, they also obey the same$\tau(x,t)$
Q.E.D.
Hirota equation (39) that
does.
$c$

$k$

$R\rightarrow R$

186

N. Ikeda and S. Taniguchi

References[1] R. Bucy and P. Joseph, Filtering for stochastic processes with applications
to guidance, Intersci. Publ., 1968.
[2] R.H. Cameron and W.T. Martin, The Wiener measure of Hilbert neighbourhoods in the space of real continuous functions, J. Math. Phys.
M.I.T., 23 (1944), 195-209.
[3] R.H. Cameron and W.T. Martin, Evaluation of various Wiener integrals
by use of certain Sturm-Liouville diﬀerential equations, Bull. A.M.S.,
51 (1945), 73-90.
[4] K. Hara and N. Ikeda, Quadratic Wiener functionals and dynamics on
Grassmannians, Bull. Sci. math., 125 (2001), 481-528.
[5] T. Hida and L. Streit, On quantum theory in terms of white noise, Nagoya
Math. J., 68 (1977), 21-34.
[6] N. Ikeda and S. Manabe, Asymptotic formulae for stochastic oscillatory
integrals, in “Asymptotic problems in probability theory: Wiener functionals and asymptotics” , ed. by K.D. Elworthy and N. Ikeda, Pitman
Research Notes Math. 284, Longman, 1993, 136-155.
[7] N. Ikeda and S. Manabe, Van Vleck-Pauli formula for Wiener integrals
and Jacobi fields, in “It\^o’s stochastic calculus and probability theory”,
Springer-Verlag, 1996, 141-156.
[8] N. Ikeda and S. Watanabe, Stochastic Diﬀerential Equations and Diﬀusion Processes, 2nd edition, North-Holland/Kodansha, 1989.
[9] K. It\^o, Diﬀerential equations determining a Markoﬀ process (in Japanese), Journ. Pan-Japan Math. Coll. No.1077 (1942). English transl. in“Kiyosi It\^o Selected papers” ed. by D. Stroock and S.R.S. Varadhan,
Springer, 1987, 42-75.
[10] M.L. Kleptsyna and A. Le Breton, A Cameron-Martin type formula forgeneral Gaussian processes–A filtering approach, Stoch. Stoch. Rep.,
72 (2002), 229-250.
[11] S. Kotani, One-dimensional random Schr\"odinger operators and Herglotz functions, in “Probabilistic methods in mathematical physics(Katata/Kyoto, 1985)” ed. by K. It\^o and N. Ikeda, Academic Press,
1987, 219-250.
[12] S. Kotani, Probabilistic approach to reflectionless potentials (in Japanese), Symposium on random matrices and related topics at Tsukuba
Univ., 2000.
[13] H. McKean, Stochastic integrals, Academic Press, New York, 1969.
[14] T. Miwa, M. Jimbo and E. Date, Solitons, Cambridge Univ. Press, Cambridge, 2000.
[15] Y. Okabe, On the stationary Gaussian process with $T$-positivity and its
associated Langevin equation and -matrix, Jour. Fac. Sci. Univ. Tokyo
Sect. IA Math.,26 (1979), 115-165.
[16] M. Sato, “Lectures of Mikio Sato (1984-1985)” noted by T. Umeda (inJapanese), RIMS Lecture Notes No.5, RIMS, Kyoto Univ., 1989.
$ $

$S$

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Quadratic Wiener functionals[17] M. Sato and Y. Sato, Soliton equations as dynamical systems on infinitedimensional Grassmann manifold, in “Nonlinear PDE in Applied Sci.
US-Japan Seminar, Tokyo 1982” , ed. by H. Fujita, P.D. Lax and G.
Strang, North-Holland, 1983, 259-271.
[18] K. Takasaki, Integrable systems as deformations of -modules, Proc.
Sympo. Pure Math., 49 (1989), 143-168.
[19] A.I. Yashin An extension of the Cameron-Martin result, J. Appl. Prob.,
30 (1993), 247-251.
$\mathcal{D}$Nobuyuki Ikeda
Suzukakedai 2-40-3, Sanda 669-1322, Japan

Setsuo Taniguchi
Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japane-mail: taniguch@math.kyushu-u.ac.jp

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 189-207

Homogenization on Finitely Ramified Fractals
Takashi Kumagai
Abstract.

Let
beacontinuous time Markov chain on some finitely ramified fractal graph given by putting $i.i.d$ . random resistors on eachcell. We prove that under an assumption thata renormalization map
converges inof resistors has a non-degenerate fixed point,,
law toanon-degenerate diﬀusion process on the fractal as
where
is a spatial scale and is a time scale of the fractal. Especially, when the fixed point of the renormalization map is unique,the diﬀusion is a constant time change of Brownian motion on thefractal. These results improve and extend our former results in[10].
$X_{t}$$\alpha^{-n}X_{\tau^{n}t}$$n$$\alpha$\S 1.

$\rightarrow\infty$$\tau$

Introduction

In this paper, we consider the Homogenization problem on uniform
finitely ramified ﬀactals, which is a class of finitely ramified Fractals
witha unique spatial scaling rate. We put random resistors on each cellof the 丘 actal graph and set
be the corresponding continuous time
converges in law to a
Markov chain. Our aim is to show that
. Here is thenon-degenerate 垣 fusion process on the fractal as
spatial scale and isatime scale of the fractal.
Homogenization ofadiﬀusion process is interpreted asalimit theocase, it is discussed
rem ofarandom process for changing scales. For
a
time
of
Brownianmotion aschangethatconverges to constant
has random diﬀusion coeﬃcients or it
under a condition thatmoves in some random environment such as random scatterers. (See [8]for general references on Homogenization of diﬀerential operators.) The
martingale method has been well developed for this problem.
For the case of fractals, typical diﬀusions are sub-diﬀusive, in thefor some $d_{w}>2$ . Those diﬀusions
sense thatasare not semi-martingales, thus we need a diﬀerent approach. In[10], we$X_{t}$$\alpha^{-7l}X_{\tau^{n}t}$

$d$

$n$

$\rightarrow\infty$$\alpha$$\tau$$R^{d}$$\epsilon X_{t/\epsilon^{2}}$$\epsilon\rightarrow 0$$X_{t}$$E[|B_{t}|]\approx t^{1/d_{w}}$$t$$\rightarrow\infty$

Received February 17,2003.
Partially supported by Grant-in-Aid for Scientific Research (C) (2)14540113.

190

T. Kumagai

developed atheory to be applied for homogenization problem on nested
ﬀactals, a class of finitely ramified fractals with good symmetries. In
this paper, we inherits the basic approach of[10], which we now explain.
We first consider a Dirichlet form corresponding to the continuous time Markov chain with random resistors which are almost surely
bounded from above and below by some non-degenerate resistor (thiscorresponds to the uniform ellipticity condition for the operator)andthe distributions are $i.i.d$ . for each ce 垣. As a solution of a variation
problem of the Dirichlet form, we induce a renormalization map $F$ on
the space of matrices, by which we can produce a new form that is arenormalization of the original one. We assume that there is a nondegenerate fixed point of the map. Then, what we should prove are thefollowing:1) Convergence of the iteration of the renormalization map to afixed
point and convergence of the forms.
2) Convergence of finite dimensional distributions and tightness.
In[10], we prove1) under certain condition for an adjoint of a
Fr\’echet derivative of the renormalization map at afixed point (see Remark 3.61) for details). Unfortunately, the condition is not easy tocheck in general and there are examples(even for nested fractals) thatthe condition does not hold. On the other hand, the dynamics of theiteration of the renormalization map has been well studied recently forthe finitely ramified fractals (see [16, 13, 15] etc.). In this paper, weapply the results to improve our former results in [10]. In Section4, we
will prove 1) under a very mild condition on the renormalization map(see Assumption2.3). We note that the map $F$ we study is onainfinite
dimensional space since we have infinite number of resistors, whereas therenormalization map studied in[16, 13, 15] is onafinite dimensional
space. We thus make some eﬀﬀ 化 $rts$ to show the stability of the map $F$from that of .
In general, non-degenerate fixed points of the renormalization map
are not necessarily unique. When the uniqueness (up to constant multiples) is guaranteed, we can further show that the diﬀusion obtained as
the limit is a constant time change of some special diﬀusion (which canbe called Brownian motion)for the fractal. Especially, when we consider random resistors which are invariant under all reflection maps on
a nested fractal, then the diﬀusion obtained is a constant time change
of Brownian motion on the fractal.
For the proof of 2), uniform Harnack inequality and uniform heat
kernel estimates of the Markov chains play important roles. Here we canadopt stability results of parabolic Harnack inequalities and heat kernel
estimates which are actively studied recently for the fractal graph cases$\hat{F}$$ F\wedge$

Homogenization on fractals

191(see [2,3,4,5]). In this paper, we skip the proof of 2)since we canapply the same argument as in [10], but note that we can shorten the
proof by applying the results in[5].
The organization of the paper is as follows. In Section 2, we define uniform finitely ramified ﬀactals(graphs), renormalization maps ofresistors on them and briefly mention about Dirichlet forms and theirheat kernel estimates on the fractals. In Section3, we give our main
theorem on homogenization. Section 4 is for the proof of 1) above. In
Section 5, we state main propositions concerning 2) above. In Appendix, we give the proof of the stability results of the(finite dimensional)
renormalization map studied in[16,13,15].
$\hat{F}$\S 2.

Uniform finitely ramified graphs and their Dirichlet forms

Uniform finitely ramified graphs
be afamily of
For $\alpha>1$ and $=\{1, 2, \cdots, N\}$ , let 重$\Psi_{i}x=\alpha^{-1}U_{i}x+\gammai$. An -similitude is a map,
similitudes on. We will impose several
isaunitary map andwhereassumption on this family. First, we assume(H-O)
satisfies the open set condition,i.e., there is a non-empty, bounded open set $W$ such thatisafamily of contraction. As
are disjoint and$=$maps, there existsa unique non-void compact set
such that. We assume(H-1)
is connected.
Let Fix be the set of fixed points of the 重 ’s, $i\in I$ . Apoint $x\in Fix$
is called an essential fifixed point if there exist , $j\in I$ , $i\neq j$ and $y\in Fix$
be the set of $i\in I$ for which the fixedsuch that $\Psi_{i}(x)=\Psi_{j}(y)$ . Let
for the set of essential
point ofis an essential fixed point. We write. We further assume thefixed points. Denotefollowing finitely ramified property.
,
are distinct sequences, then(H-2) If
2.1.

$I$$R^{D}$$R^{D}$$\{$$ i\}_{i\in I}$$\alpha-$$x$$\alpha$$\in$$\gamma_{i}\in R^{D}$$U_{i}$$\{\Psi_{i}\}_{i\in I}$$\{\Psi_{i}(W)\}_{i\in I}$$\bigcup_{i\in I}\Psi_{i}(W)\subset W$$\{\Psi_{i}\}_{i\in I}$$ K\wedge$$ K\wedge$$\bigcup_{i\in I}\Psi_{i}(\hat{K})$$\hat{K}$

$i$

$i$

$I_{F}$$V\wedge 0$$\Psi_{i}$$\Psi_{i_{1},\ldots,i_{n}}=\Psi_{i_{1}}\circ\cdots\circ\Psi_{i_{n}}$$\{i_{1}, \ldots, i_{n}\}$$\{j_{1}, \ldots, j_{n}\}$$\Psi_{i_{1},\ldots,i_{n}}(\hat{K})\cap\Psi_{j_{1},\ldots,j_{n}}(\hat{K})=\Psi_{i_{1},\ldots,i_{n}}(\hat{V}_{0})\cap\Psi_{j_{1},\ldots,jn}(\hat{V}_{0})$.

(compact) uniform fifinitely ramifified fracDefinition 2.1. ([5])
set determined bysimilitudes
tal ($u.f.r$ . fractal for short)is.
satisfying the assumption(H-0), (H-1), (H-2) and that$A$$\hat{K}$$a$$\{\Psi_{i}\}_{i\in I}$$\alpha$$Q\hat{V}_{0}\geq 2$

If we further assume the following symmetry condition, thencalled a (compact) nested fractals introduced in [12].

$ K\wedge$

is

192

T. Kumagai(SYM) If ,, then the reflection in the hyperplane $ H_{xy}=\{z\in$: $|z-x|=|z-y|$ } maps
to itself, where$y\in\hat{V}_{0}$$x$$R^{D}$$V\wedge n$(2.1)$\hat{V}_{n}=\bigcup_{i_{1},\cdots,i_{n}\in I}\Psi_{i_{1},\ldots,i_{n}}(\hat{V}_{0})$.

Thus, $u.f.r$ . fractals form a class of fractals which is wider than nested
丘 actals, and is included in the class of p.c.f. self-similar sets ([9]).
, we calla set of the form,
For eachand ,an -complex. For , $y\in\hat{K}$ ,an -cell and$\{x_{0}, \cdots, x_{m}\}$is called a -chain 丘 $omx$ to if $x0=x$ , $x_{m}=y$ ,are in the same -complex for 0
for 1 $\leq j\leq m-1$ and ,$i\leq m-1$ . We then have the following topological properties of $u.f.r$ .
fract als.
$n$$\Psi_{i_{1},\cdots,i_{n}}(\hat{V}_{0})$$\geq$$0$$\cdots$$i_{1}$$i_{n}$

$\in$

$I$

$\Psi_{i_{1},\cdots,i_{n}}(\hat{K})$

$n$

$n$

$n$

$x$

$xj\in\hat{V}_{n}$$y$$x_{i}$$n$$x_{i+1}$$\leq$

Lemma2.2.
1)Each element inbelongs to only one -cell for each2) Any1-cell contains at most one element of .
, there exists-chain3) For eachandto such that ,.
,$V\wedge 0$

$n$

$n$

$\geq 0$.

$V\wedge 0$$x\in\hat{V}_{1}$

$x$

$y$

$x_{1}$$y\in\hat{V}_{0}$$\cdots$$a$$1$$\{x_{0}, \cdots, x_{m}\}$from$x_{m-1}\not\in\hat{V}_{0}$1) and 2) can be proved in the same way as [11] (Lemma 2.8
and Proposition2.9)and [12] (Proposition $IV.13$ and Corollary IVA4).
(They discuss for nested fractals, but the symmetry assumption is not
used there.) For3), we first note that any 1-junction is not an element ofdue to1), whereis calleda1-junction if there exist $i\neq j\in I$重口重. Using $(H_{-}1)$ and (H-2), we can choosesuch that, $x_{m-1}$ are 1-junctions. Sincea 1-chain $\{x_{0}, \cdots, x_{m}\}$ such that ,$Q.E.D$ .
1-junction is not an element of , we obtain the result.

Proof.

$x\in\hat{V}_{1}$$\hat{V}_{0}$$ x\in$$i(\hat{V}_{0})$$j(\hat{V}_{0})$$x_{1}$$\cdots$$V\wedge 0$

Next we define unbounded $u.f.r$ . fractals. We assume without loss of.
generality that 重 1 $(x)=\alpha_{1}^{-1}x$ and 0 belongs to . Set
Then, clearly 重 1 $(K)=K$ . We call $K$ an unbounded uniform finitely
andfor $\in Z$ .
ramﬃed fractal. Let(Note that this labelling is the opposite to the one given in [5]. Asgets smaller
gets bigger, the graph distance between each vertex of$n\inZ$, we define n-cells
and。.)Then, $K=Cl(\bigcup_{n\in Z}V_{n})$ . For
and -complexes similarly as the compact fractals.
We now introduce uniform finitely ramified graphs. These will begraphs with vertices $V$ and a collection of edges $B$ . In order to define
the edges, we first define $\hat{B}_{0}:=\{\{x, y\} : x\neq y\in\hat{V}_{0}\}$ . Then insideand we denote by $B$ the set of all
each0-cell we placea copy ofthe edges determined in this way. We call the graph $(V, B)$ auniform
finitely ramified $(u.f.r.)$ graph. If we construct the graph starting ﬀom$V\wedge 0$$V=V_{0}=\bigcup_{n=0}^{\infty}\alpha^{n}\hat{V}_{r\iota}$$K=\bigcup_{n=1}^{\infty}\alpha^{7\iota}\hat{K}$$V_{n}=\alpha^{-r\iota}V$

$n$

$n$

$V_{n}$$V_{7\iota-1}\subset V$$n$$B\wedge 0$

193

Homogenization on fractals

a nested fractal, then it will be called anested fractal graph. Let$\Omega$$\Omega_{+}$

{
{

$=$

$=$

: there is an $n\in Z$ such that$\omega\in I^{Z}$: there is an $n\in N$ such that$\omega\in I^{N}$

Then, there is a continuous map$\pi$$=1$ , $k\geq n$$\omega k$$\omega k$},$=1$ , $k\geq n$}.

$=$such that. It is easy to see $K=\pi(\Omega)$ .
denotes an element of given by:$\Omega\rightarrow R^{D}$$\pi(\omega)$$\lim_{n\rightarrow\infty}\alpha^{n}\Psi_{\omega_{n}}(\Psi_{\omega_{n-1}}(\cdots(\Psi_{\omega_{-n}}(0))\cdots))$

For any$\omega\in\Omega_{+}$

and$i\in I_{F}$, let$[\omega, i]$$\Omega$$\omega_{k}$$[\omega, i](k)=\{$
$i$

,
,

$k\geq 1$$k\leq 0$.

Then, $V=\{\pi([\omega, i]) : \omega\in\Omega_{+}, i\in I_{F}\}$ .

2.2. Renormalization maps
Let $Q$ be the set of $Q=\{Q_{ij}\in R : i, j\in I_{F}xI_{F}\}$ such that
$Q_{ij}=Q_{ji}$ for any, $j\in I_{F}$ and that $\sum_{j\in I_{F}}Qij=0$ , $i\in I_{F}$ . $Q$ isgiven byvector space with an inner product
$i$

$a$

$(\cdot, \cdot)Q$$(Q, Q’)_{Q}=\sum_{j,k\in I_{F}}Q_{jk}Q_{jk}’=\prime baceQ^{t}Q’$,

Fora set $A$ , we denote $l(A)=\{f : A\rightarrow R\}$ . Let$(\xi, \xi)\geq 0$ for any $\xi\in l(I_{F})$ , wheresuch that$Q$$Q_{+}$, $Q’\in Q$ .

be the set of $Q\in Q$$\hat{S}_{Q}$$\hat{S}_{Q}$$(\xi, \xi)=-\sum_{i,j\in I_{F}}Q_{ij}\xi_{i}\xi_{j}=\frac{1}{2}\sum_{i,j\in I_{F}}Q_{ij}(\xi_{i}-\xi_{j})^{2}$.

,. Note that there exist$c_{2.2}>0$ such thatfor all $ Q\in Q+\cdot$ We$(Q, Q)Q$. Let
be the set of $Q$ $\in Q$ suchsometimes denoteas
be the set ofthat $Q_{ij}\geq 0$ for any , $j\in I_{F}$ with $i\neq j$ . Also, let$(\xi, \xi)=0$ if and an$Q\in Q_{M}$ such that
if is constant. Note that

Set$||Q||^{2}=\sup_{\xi\in l(I_{F})}\hat{S}_{Q}$ $(\xi, \xi)/(\sum_{i\in I_{F}}\xi_{i}^{2})$$\leq c_{2.2}||Q||^{2}$$ c_{2.1}||Q||^{2}\leq$$S\wedge Q$$(\xi, \xi)$$c_{2.1}$$S\wedge Q(\xi)$$Q_{M}$$Q_{irr}$$i$$S\wedge Q$$y$$Q_{irr}\subset Q_{M}\subset Q_{+}$

$\xi$

.

Take $Q_{*}\in Int(Q_{M}):=\{Q\in Q_{M}$ : $Q_{ij}>0$ for any $ i\neq j\in$and let $\mathcal{X}_{+}=\{X\in C(\Omega_{+}, Q_{+})$ : there exists $C_{0}>0$ such that
. Also, letandfor any
口 $C(\Omega_{+}, Q_{irr})$ and
,
$C$ (
,
are convex cones. For any
and, Int $(Q_{M})$ ). Then
given
let
denote a non-negative definite bilinear form on
by$I_{F}\}$$\hat{S}_{X(\omega)}$$(\xi, \xi)\leq C_{0}\hat{S}_{Q_{*}}(\xi, \xi)$$\mathcal{X}_{M}=\mathcal{X}_{+}\cap C(\Omega_{+}, Q_{M})$$\Omega_{+}$$\omega\in\Omega_{+}$$\xi=(\xi_{j})_{j\in I_{F}}\}$$\mathcal{X}_{irr}=\mathcal{X}_{+}$$\mathcal{X}_{+}$$Int(\mathcal{X}_{M})=\mathcal{X}+\cap$$\mathcal{X}_{M}$$S_{X}$$S_{X}$$(u, u)=\frac{1}{2}\sum_{\omega\in\Omega_{+}}\hat{S}_{X(\omega)}(u(\pi([\omega, \cdot])), u(\pi([\omega, \cdot])))$$X\in \mathcal{X}_{+}$$L^{2}$$(V, d\nu_{0})$, $u\in L^{2}(V, d\nu_{0})$ .

194

T. Kumagai

isameasure on $V$ so that $\nu_{0}(\{x\})=1/N$ for all $ x\in$ $V$ . If$(V, d\nu 0)$ .
, thenis a Dirichlet form on
So there is a
Markov process which we denote by $\{P_{X}^{x} : x\in V\}$ . We introduce an
as follows.
order relationHere$\nu_{0}$$X\in \mathcal{X}_{M}$$L^{2}$$S_{X}$$\leq in\mathcal{X}_{+}$$X\leq Y$

The norm on

For any$\mathcal{X}_{+}$

if$S_{X}$

is given by$X\in \mathcal{X}_{+}$

for all $u\in L^{2}(V, d\nu_{0})$ .

$(u, u)\leq S_{Y}(u, u)$, let$||X||^{2}=\sup_{u\in L_{(V,d\nu_{0})}^{2}}S_{X}$ $(u, u)/||u||_{L^{2}(V,d\nu_{0})}^{2}$$S_{X}^{\overline{F}}$:$L^{2}$$S_{X}^{\overline{F}}(u)=\inf\{S_{X}(v, v) :$(V, d\nu_{0})\rightarrow[0, \infty)$.

be given by

v\in L^{2}(V, d\nu_{0}), v(\alpha x)=u(x), x\in V\}$.

Let
, , $v\in L^{2}(V, d\nu_{0})$ . Then
we see thatis a Dirichlet form on. Moreover, by the self$K$
similarity of , we see that there isarenormalization map :such thatfor all $X\in\chi_{+}$ and $u\in L^{2}(V, d\nu_{0})$ . Let:be such that $\iota(Q)(\omega)=Q$ for all
and $Q\in Q_{+}$ .
We definea renormalization map
as(it is independent of the choice offor). Note thatis in general a non-linear map. By Schauder’sfixed point theorem, weknow that there exists $Q_{*}\in Q_{M}$ (with $(Q_{*})_{ij}>0$ for some $i\neq j$ ) and$\beta Q_{*}>0$ such that
. Throughout this paper, we assume
the following.
$S_{X}^{\overline{F}}$$(u, v)=\frac{1}{2}(S_{X}^{\overline{F}}(u+v)-S_{X}^{\overline{F}}(u)-S_{X}^{\overline{F}}(v))$$L^{2}$$S_{X}^{\overline{F}}$$u$$(V, d\nu_{0})$$F-$$\mathcal{X}_{+}\rightarrow \mathcal{X}_{+}$$S_{X}^{\overline{F}}(u)=S_{\overline{F}(X)}(u, u)$$\iota$$Q_{+}\rightarrow\chi_{+}$$\omega\in\Omega_{+}$$F\sim:$$Q_{+}\rightarrow Q_{+}$$\omega\in\Omega_{+}$$\tilde{F}(Q)=\overline{F}(\iota(Q))(\omega)$$F-,\tilde{F}$$\omega\in\Omega_{+}$$F\sim(Q_{*})=\rho_{Q_{*}}^{-1}Q_{*}$$Assumption-$ $2.3$. 1) For each

$Q\in Q_{irr}$, there exists

such that.
for all$Q_{0}\in
Int(Q_{M})$2) There exists
and $\rho Q_{o}>0$ such that$F^{n}(Q)\in Int(Q_{M})$$l$$=l(Q)\in N$$n\geq l$$F\sim(Q_{0})=\rho_{Qo}^{-1}Q_{0}$.

$>0$ is uniquely
Remark 2.4. 1) By Corollary 6.20 of [1],determined,, if, $Q_{2}\in Q_{irr}$ satisfies $\tilde{F}(Q_{j})=\rho_{Q_{j}}^{-1}Q_{j}(j=1,2)$,
with, then. In the class of fractal graphs
we consider, we can prove $\rho Q_{0}>1$ (see [9] etc.).
2)
suﬃcient condition for Assumption2.31) is the following.
(H-3) There exists $\in N$ such that for each ,, there is $al$ -chain
from to such that for each1 $\leq i\leq m-2$ , there is-cell containing
andthat does not contain any element of .
Indeed, if (H-3) holds, it is easy to show $F-n(Q)\in Int(Q_{M})$ for,$Q\in Q_{irr}$ by observing the corresponding Markov chain on.
3) Every nested fractals satisfy Assumption2.31) and2). Indeed, (H-3)
can be shown for nested fractals using(SYM) and[11] Lemma2.10([12]Proposition $IV$. 垣刀 so that1) holds. 2) is proved in [11] Theorem3.10
and in[12] Theorem $V5$ .
$\rho_{Q_{o}}$$i.e.$$\rho_{Q_{1}}$$Q_{1}$$\rho_{Q_{2}}>0$$\rho_{Q_{1}}=\rho_{Q_{2}}=\rho_{Q_{0}}$

$A$

$l$

$\{x\circ, \cdots, x_{m}\}$

$x$

$x$

$y\in\hat{V}_{0}$

$a$

$y$

$\hat{V}_{0}$$l$$x_{i}$$x_{i+1}$$n$$V\wedge n$$\geq l$

Homogenization on fractals

Set$L^{2}$:$F=\rho_{Qo}\overline{F}$$(V, d\nu_{0})$

. Set

and
in the same way.

for $ u\in$$S_{X}^{F}(u)=\rho_{Q_{0}}S_{X}^{\overline{F}}(u)$$\mathcal{X}_{+}\rightarrow \mathcal{X}_{+}$$F\wedge=\rho_{Qo}\tilde{F}$195

2.3. Dirichlet forms and heat kernel estimates, define
For ,
$u$

(2.2)

$v\in l(\hat{V}_{n})$$\hat{\mathcal{E}}_{Q_{0}}^{n}(u, v)=\rho_{Q_{0}}^{n}\sum_{i_{1},\cdots,i_{n}\in I}\hat{S}_{Q_{0}}(u\circ\Psi_{i_{1},\ldots,i_{n}}, v\circ\Psi_{i_{1},\ldots,i_{n}})$.

Let be a normalized Hausdorﬀ measure on $K$ . Then, the following is
known(see for example, [6, 9]).
$\nu\wedge$

Theorem 2.5. Let $Q_{0}\in Int(Q_{M})$ be as in Assumption2.32),$\hat{F}(Q_{0})=Q_{0}$ .
Then, there islocal regular Dirichlet form
satisfying the following,$(\hat{\mathcal{E}},\hat{\mathcal{F}})$

$a$

$i.e$

.

in

$L^{2}$ $(\hat{K}, d\hat{\nu})$ナ$\hat{\mathcal{E}}(u, v)$, R) :$=${ $u\in C$ ($=$$\lim_{n\rightarrow\infty}\hat{\mathcal{E}}_{Q_{o}}^{n}(u, v)$$\hat{K}$$\sup_{n}\hat{\mathcal{E}}_{Q_{o}}^{n}(u,$

for

$u$

,

$ u)<\infty$$v\in\hat{\mathcal{F}}$},.

: $C$ (, R)
For each $m\in N$ , let $K_{m}=\alpha^{m}\hat{K}$ and define$C$ (,, R) by $\sigma_{m}u(x)=u(\alpha^{m}x)$ for $ x\in$ $K$ . Set. Let beaHausdorﬀ
for ,measure on $K$ such thatand $N\nu=\nu o$ 重 . Now let$K_{m}$$\sigma_{m}$$\hat{K}$$\rightarrow$$\mathcal{F}_{<m>}=\sigma_{-m}\hat{\mathcal{F}}$$\mathcal{E}_{<m>}$$(u, v)=\rho_{Qo}^{-m}\hat{\mathcal{E}}(\sigma_{m}u, \sigma_{m}v)$$u$$v\in \mathcal{F}_{<m>}$$1-1$$\nu|_{\hat{K}}=\hat{\nu}$$\mathcal{F}$$=${ $u\in l(K)$ :$\lim_{m\rightarrow\infty}\mathcal{E}$$\mathcal{E}(u, v)$$=$

イー $>$$\nu$$u|_{K_{m}}\in \mathcal{F}_{<m>}$for all$(u|_{K_{m}}, u|_{K_{m}})<\infty\}$

口 $L^{2}(K, d\nu)$for$\lim_{m\rightarrow\infty}\mathcal{E}_{<m>}(u|_{K_{m}}, v|_{K_{m}})$$m\in N$ ,$u$,$v\in \mathcal{F}$.

Then the following holds.
Theorem 2.6.
is local regular Dirichlet form on $L^{2}(K, d\nu)$ .
( $K$ , R) and this form has the following scaling property,$(\mathcal{E}, \mathcal{F})$$a$$\mathcal{F}\subset C$$\mathcal{E}(u, v)=\rho_{Qo}\mathcal{E}(u\circ\Psi_{1}, v\circ\Psi_{1})$$u.f.r$

for

$u$

,

$v\in \mathcal{F}$.

Finally, we $w$ 徂 mention heat kernel estimates for Markov chains on. graphs. For $X\in Int(\mathcal{X}_{M})$ and $x\neq y\in V$ , define$R_{X}$$(x, y)=(\inf\{S_{X} (u, u) :u\in l(V), u(x)=1, u(y)=0\})^{-1}$ .

$(x, x)=0$ for $x\in V$ . Then,
isametric which is called aLetresistance metric. By simple modifications of the proof of Corollary4.12
in [5], the following holds (note that as we will mention later in Remark
3.64), Assumption2.3of[5] always holds under our Assumption2.3).
$R_{X}$$R_{X}(\cdot, \cdot)$

196

T. Kumagai, letTheorem 2.7. For eachbe the heat kernelof the discrete time Markov chain which is induced from the continuous
time Markov chain corresponding to $(S_{X}, L^{2} (V, dI/_{0}))$ . Then, there exists$C2.3$ ,, $C2.6>0$ (which depend on $X$ ) andsuch that foreach , $y\in V$ and $k\geq d(x, y)$ ,$X\in Int(\mathcal{X}_{M})$$p_{k}^{X}(\cdot, \cdot)$$0<\gamma_{1}\leq\gamma_{2}$$\cdots$$x$$(x, y)$$\leq$$c_{2.3}k^{-\frac{s}{S+1}}\exp(-c_{2.4}(\frac{R_{X}(x,y)^{S+1}}{k})^{\gamma_{1}})$,$(x, y)+p_{k+1}^{X}(x, y)$$\geq$$c_{2.5}k^{-\frac{s}{S+1}}\exp(-c_{2.6}(\frac{R_{X}(x,y)^{S+1}}{k})^{\gamma 2})$,$p_{k}^{X}$$p_{k}^{X}$

where $S=\log N/\log\rho Q_{0}$ and$d(\cdot, \cdot)$

is

$a$

graph distance.

We note that similar heat kernel estimates forand(given in Theorem 2.5 and 2.6) can be also obtained (cf. [6]).
Let $\beta>0$ . We say $(S_{X}, L^{2} (V, d\nu_{0}))$ satisfies $(PHI(\beta))$ , aparabolic
Harnack inequality of order
if whenever $u(n, x)$
is defined on$[0, 4N]$
and satisfies$(\hat{\mathcal{E}},\hat{\mathcal{F}})$$(\mathcal{E}, \mathcal{F})$$\geq 0$$\beta$$\times\overline{B}(y, 2r)$$u(n+1, x)-u(n, x)=\mathcal{L}u(n, x)$($\mathcal{L}$$(n, x)\in[0,4N]\times B(y, 2r)$ ,is the corresponding diﬀerence operator), then$N\leq n\leq 2N\max_{x\in B(y,r)}u(n, x)\leq c_{2.7}3$$x\in B(y,r)\min_{N\leq n\leq 4N},(u(n, x)+u(n+1, x))$,(cf. [2, 3, 4, 5]). By Theorem
where $N\geq 2r$ and
2.7 and a standard argument, we can deduce the following.
$c_{2.8}r^{\beta}\leq N\leq c_{2.9}r^{\beta}$

Proposition2.8.
satisfies $(PHI(S+1))w.r.t$ . the resistance metric.

$(S_{X}, L^{2} (V, d\nu_{0}))$Homogenization\S 3.

In this section, we will state our main theorem. First, we give somedefinition for later use. Let $\hat{V}_{0}=\{a_{i} : i\in I_{F}\}$ . For
Int $(Q_{M})$ , we
, $k\in I$ by
define a matrix$ Q^{*}\in$$A_{k,Q^{*}}\in l(\hat{V}_{0})$(3.1)$(A_{k,Q^{*}})_{ij}=P_{Q^{*}}^{\Psi_{k}(a_{i})}(w^{1}(\tau_{\hat{V}_{0}})=a_{j})$フ

whereisaMarkov chain on
whose transition probability is determined byand. Then, by Lemma2.23),
the following clearly holds for $u.f.r$ . graphs.
$w^{1}$$V\wedge 1$$Q^{*}$$\tau_{\hat{V}_{0}}=\inf\{n\geq 0 :w_{n}^{1}\in\hat{V}_{0}\}$

Lemma 3.1. $0<(A_{k,Q^{*}})_{ij}<1$ if $k\neq i$ and$(A_{k,Q^{*}})_{kj}=\delta_{kj}$

.

197

Homogenization on fractals

For any$S_{X}^{H_{Q_{*}}}$:$L^{2}$$X\in \mathcal{X}_{+}$

and any
Int $(Q_{M})$ with $F\wedge(Q*)=Q*$ ’ let
be given by$ Q_{*}\in$$(V, d\nu_{0})\rightarrow[0, \infty)$

for$S_{X}^{H_{Q_{*}}}(u)=\rho_{Qo}S_{X}(v, v)$

where $v\in L^{2}$
harmonic on$(V, d\nu_{0})$$u\in L^{2}(V, d\nu_{0})$satisfies $v(\alpha x)=u(x)$ ,,,$V\backslash (\alpha^{-1}V)$,$x\in V$ ,

and

for

,

$v$

is

$Q_{*}-$$i.e.$$v(\pi([\omega\cdot i, j]))=\sum_{k\in I_{F}}(A_{i,Q_{*}})_{jk}u(\pi([\omega, k]))$$i\in I$ $j\in I_{F}$.

.
, $n\geq 2$ andHereis given by, we can define a Dirichlet form
. In the same way as we did for
on $L^{2}(V, d\nu_{0})$ . It is easy to see that $S_{X}^{H_{Q_{*}}}(u)=S_{H_{Q_{*}}}(X)(u, u)$where $H_{Q_{*}}(X)(\omega)=\rho_{Qo}\sum_{k\in I}{}^{t}A_{k,Q_{*}}X(\omega\cdot k)A_{k,Q_{*}}$ for all. We definea map
as $\hat{H}Q_{*}(Q)=$and $u\in L^{2}$
。:
).
for
(it is independent of the choice of$(\omega.

$i\in\Omega_{+}$$\omega$i)_{7\iota}=\omega_{n-1}$$(\omega\cdot i)_{1}=$$S_{X}^{\overline{F}}$$i$$S_{X}^{H_{Q_{*}}}$$(\cdot, \cdot)$$X\in \mathcal{X}_{+}$$H\wedge Q$$(V, d\nu_{0})$$H_{Q_{*}}(\iota(Q))(\omega)$$Q_{+}\rightarrow Q_{+}$$\omega\in\Omega_{+}$$\omega\in\Omega_{+}$

Definition 3.2.
satisfying the following.
Let be probability measure on
1)
are independently identically distributed $Q_{M}$ -valued
random variables under .
) $=1$ for some
: $X(\omega)\in Q_{C_{1}Q_{0},C_{2}Q_{0}}$ for all2) (, $C_{2}>0$ , where $Q_{C_{1}Q_{0},C_{2}Q_{0}}:=\{Q\in Q_{M} : C_{1}Qo\leq Q\leq C_{2}Q_{0}\}$ .
$\mathcal{X}_{M}$

$a$

$\mu$

$\{X(\omega) :\omega\in\Omega_{+}\}$

$\mu$

$\mu$

$\omega\in\Omega_{+}\}$$\{X\in \mathcal{X}_{M}$$C_{1}$

The following properties are easy, but important.
, $Q_{*}\in Int(Q_{M})$ be as above.
Proposition 3.3. Let:1):
are continuous maps.
and2) $F(\iota(Qo))=\iota(Qo)$ , $F(\iota(Q_{*}))=H_{Q_{*}}(\iota(Q_{*}))=\iota(Q_{*})$ .
3)If $X$ ,and $X\leq Y$ , then $F(X)\leq F(Y)$ and $ HQ_{*}(X)\leq$$Q_{0}$$F$$\mathcal{X}_{M}\rightarrow \mathcal{X}_{M}$$H_{Q_{*}}$$\mathcal{X}_{M}\rightarrow \mathcal{X}_{M}$$Y\in \mathcal{X}_{+}$$H_{Q_{*}}(Y)$.

.
4) $F(X)\leq H_{Q_{*}}(X)$ for all5) For any $X$ , $Y\in \mathcal{X}+anda$ , $b\geq 0$ , $F(aX+bY)\geq aF(X)+bF(Y)$ and$X\in \mathcal{X}_{+}$$H_{Q_{*}}(aX+bY)=aH_{Q_{*}}(X)+bH_{Q_{*}}(Y)$ .
$\theta)E_{\mu}[F(X)]\leq F(E^{\mu}[X])$for all$X\in \mathcal{X}_{+}$.

.
Note that the same results hold for and
$F$ .
Let
be the -th iteration ofThen we have the following key
theorem.
$\hat{F}$$F^{n}$$\hat{H}Q_{*}$$n$

Theorem 3.4. Under Assumption2.3, there exists $Q_{\mu}\in Int(Q_{M})$,such that for all$\omega\in\Omega+$(3.2)$Q_{\mu}=\lim_{n\rightarrow\infty}F^{n}(X)(\omega)$in$L^{1}(Q_{M}, \mu)$.

T. Kumagai

198

Since $\hat{F}(Q_{\mu})=Q_{\mu}\in Int(Q_{M})$ , we can construct a local regular(see Theorem 2.6). We denote the correDirichlet form on $K$ using
sponding diﬀusion as. We now state our main theorem.
$Q_{\mu}$$(X_{\mu}, \{P_{\mu}^{x}\}_{x\in K})$

Theorem 3.5. Let
inition3.2 and let
holds.

$\mu$

be the probability measure on
as in Def. Under Assumption2.3; the following$\mathcal{X}_{M}$$\tau Q_{0}:=\rho_{Q_{0}}N$$E^{P_{X}^{x_{n}}}[f(\alpha^{-n}w(\tau_{Q_{0}}^{n}\cdot))]\rightarrow E^{P_{\mu}^{x}\infty}[f(w(\cdot))]$

as

$n$

$\rightarrow\infty$, for any $f\in C_{b}(D([0, \infty), K)\rightarrow R)$ and any
sequencewithas. Here theexpectations are taken over $\omega\in D([0, \infty)$ , $K)$ .
Further, if there is convex conesuch that the following
holds; )there exists,
unique (up to constant口
multiples)
which satisfies $F(Q)=Q$ ,the support. Thenisconstant time change of the diﬀusionof is in$K$
$Q$
constructed fromon .

in probability under$\mu$$\alpha^{-n}x_{n}\rightarrow x_{\infty}\in K$$\{x_{n}\}_{n=1}^{\infty}\subset V$$\mathcal{X}_{sub}\subset \mathcal{X}_{irr}$

$a$

$a$

$F(\mathcal{X}_{sub})\subset \mathcal{X}_{sub}$$Q\in \mathcal{X}_{sub}$$\mathcal{X}_{sub}$$\mu$$ n\rightarrow\infty$

$b)$

$a$

$c)$

$Int(\mathcal{X}_{M})$$P_{\mu}$$a$Remark 3.6. 1) In [10]; similar statement is given under the assumption that there exists $Q_{1}\in Q_{irr}$ such that $H\wedge Q_{o}*(Q_{1})=Q_{1}$ whereis an adjoint operator of
in $Q$ $f[10]$ Assumption3.1). In general, it is not easy to check this assumption and there is example innested fractals that this does not hold.
be the set of all bijective maps on such that $\sigma(I_{F})=I_{F}$ ,2) Letand let $G$ be subgroup of . Then, as in[10] Section7, we can obtain
similar results for random resistors on,
subcone of
which
$G$
consists of -invariant elements, if ), ), ) in Theorem 3.5holds for. Especially, we can prove the following; For nested fractals, letbe subgroup ofgenerated by all the reflection maps and suppose that
. Thenthe support of is inin Theorem3.5is constant timechange of Brownian motion on the nested fractal. (This is because, it
is known that non-degenerate fifixed point for-invariant resistors on
nested fractals is unique up to constant multiples; see [16, 13, 15].)3) Note that non-degenerate fifixed points ofis not necessarily uniqueeven for nested fractals. In [1] Example6.13, one parameter family of
non-degenerate fixed points on the Vicsek set are given. The homogenization problem for this particular fractal is studied in [7].
4) By Theorem3.4 (or Proposition4 1), we see that Assumption2.3 in[5] always holds under our Assumption2.3.
$H\wedge Q_{0}$$\hat{H}_{Q_{0}}^{*}$$a$$I_{I}$$I$$\sigma$$I_{I}$$a$$\mathcal{X}_{M}^{G}$

$a$

$b$

$\mathcal{X}_{M}$

$a$

$c$

$\mathcal{X}_{M}^{G}$$G_{0}$$I_{I}$$a$$\mathcal{X}_{M}^{G_{0}}$

$\mu$

$a$

$P_{\mu}$$G_{0}$$a$$\hat{F}$$\cdot$Homogenization on fractals

199

Convergence of Dirichlet forms\S 4.

In this section, we will prove Theorem 3.4 and show a convergenceof the corresponding forms (Proposition 4.4).
The next proposition is arestricted version of the result by Peirone([15]), whose original ideas come from Sabot ([16]). For completeness,
we give the proof in Appendix $A$ .
Proposition 4.1. Under Assumption2.3, for each $M\in Q_{irr}$ , there
exists $Q_{M}\in Int(Q_{M})$ such that
(4.1)

$Q_{M}=\lim_{n\rightarrow\infty}\hat{F}^{n}(M)$For the proof of Theorem 3.4, we use two lemmas in[10]. Let.
be the -th iteration of$n$$H_{Q_{*}}^{n}$$H_{Q_{*}}$

Lemma 4.2. Assume that $Q_{*}\in Int(Q_{M})$ satisfies $\hat{F}(Q*)=Q*\cdot$
Then, there exist $c_{4.1}>0$ and $0<\epsilon<1$ such that$E^{\mu}[||H_{Q_{*}}^{n}(X)(\omega)-H_{Q_{*}}^{n}(E^{\mu}[X])(\omega)||^{2}]\leq c_{4.1}(1-\epsilon)^{n}$,$\forall\omega\in\Omega_{+}$,$n$$\geq 1$.

In particular,$\lim_{r\iota\rightarrow\infty}||H_{Q_{*}}^{n}(X)(\omega)-H_{Q_{*}}^{n}(E^{\mu}[X])(\omega)||=0$, $\mu-a.e$ . $X$ ,$\forall\omega\in\Omega_{+}$.

By the linearity of $HQ_{*}$ , $E^{\mu}[HQ_{*}(X)(\omega)]=HQ_{*}(E^{\mu}[X])(\omega)$ .
Then the proof is basically the same as that of Lemma 4.1 in [10].
$Q$ .E.D.
$Proo/$.

Lemma 4.3. ([10], Lemma 4.2) Let
be random variables$\sup_{n}E[Y_{n}^{2}]<\infty$. Let $Y=\lim\sup_{n\rightarrow\infty}Y_{n}$ and assume thatsuch that$\lim_{n\rightarrow\infty}E[Y_{n}]=E[Y]$ .
Then $\lim_{n\rightarrow\infty}E[|Y-Y_{n}|]=0$ .
$\{Y_{n}\}_{n=1}^{\infty}$

Proof of Theorem3.4, Let $R_{m}=E^{\mu}[F^{m}(X)(\omega)](R_{m}$ is independentof ). By Proposition4.1, for each $m\in N$ , there exists $Q_{m}\in Int(Q_{M})$and $F\wedge(Q_{m})=Q_{m}$ . On the othersuch that
hand, by Proposition3.36) we see$\omega$$\lim_{n\rightarrow\infty}\hat{F}^{n}(R_{m})=Q_{m}$(4.2)$\hat{F}^{n}(R_{m})\geq R_{n+m}$$\forall m$, $n\in N\cup\{0\}$ ,, then $F\wedge(Q+)=$so that $Q_{m}\geq Q_{n+m}$ . Denote the limit of $\{Q_{m}\}$ as. In particular, $Q_{+}\in Int(Q_{M})$ due to Assumption2.3 1). For any, there exists
such that$Q_{+}$$Q_{+}$$\epsilon>0$(4.3)$N_{\epsilon}\in N$$(1+\epsilon)Q_{+}\geq R_{m}$$\forall m\geq N_{\epsilon}$

.

200

T. Kumagai

Indeed, if this does not hold, then because
is compact,
such that
andthere existsa subsequence
exists. On the other hand, by (4.2), we have
, which is a contradiction. By defifor all $j’\geq j$ so that$\{Q_{m}\}$$m$, for eachnition ofand $\epsilon>0$ , there exists
such that$(1 -\epsilon)Q_{m}\leq\hat{F}^{n}(R_{m})$ for all. Combining these facts and, we have(。)noting$Q_{(1+\epsilon)Q_{+},C_{2}Q_{0}}$$R_{l_{j}}\geq(1+\epsilon)Q_{+}$$\{l_{j}\}$$=:\overline{R}$$\lim_{j\rightarrow\infty}R_{l_{j}}$$\hat{F}^{l_{j’}-l_{j}}(R\iota_{j})\geq R_{\iota_{j’}}$$Q_{+}\geq\overline{R}$$L_{m,\epsilon}$$n$$\geq\hat{F}^{n}(R_{m})$$H\wedge Q_{+}n$(4.4)$\geq L_{m,\epsilon}$$(R_{m})\leq(1+\epsilon)Q_{+}$$(1-\epsilon)Q_{+}\leq\hat{H}_{Q}^{n}$$\forall n\geq L_{m,\epsilon}$ヤ,$m\geq N_{\epsilon}$.

On the other hand, by Lemma4.2, we have$\lim_{n\rightarrow\infty}||H_{Q_{+}}^{n}(F^{m}(X))(\omega)-H_{Q_{+}}^{n}(\iota(R_{m}))(\omega)||$$(4.5)$$=$$\lim_{n\rightarrow\infty}||H$蝉 $(F^{m}(X))(\omega)-\hat{H}_{Q_{+}}^{n}(R_{m})||=0$. Since $H_{Q_{+}}^{n}(F^{m}(X))(\omega)\geq F^{n+m}(X)(\omega)$ ,and for all
we see that the following holds for some,$\mu-a.e$.

$X$$\omega\in\Omega_{+}$$N_{\epsilon,\omega}’\in N$(4.6)$(1+\epsilon)Q_{+}\geq F^{m}(X)(\omega)$

We now consider more about$ H\wedge Q+\cdot$$\mu- a.e$

. $X$ ,$\forall\omega\in\Omega_{+}$,$m\geq N_{\epsilon,\omega}’$.

It is easy to see$\sup_{n}|||\hat{H}_{Q_{+}}^{n}|||:=\sup_{7l}\sup_{Q\in Q_{M},||Q||=1}||\hat{H}_{Q_{+}}^{n}(Q)||<\infty$(see Lemma 4.3 in [10]). Using this, we see that the size of each Jordan
cell corresponding to the largest eigenvalue of $H\wedge Q+is1$ . We thus obtainthat there exists an orthogonal projection:
so that foreach $k\in N$ , there exists $nk\in N$ such that
$Q_{M}\rightarrow Q_{M}$$P_{0}$(4.7)$|||\hat{H}_{Q_{+}}^{n_{k}}-P_{0}|||\leq 2^{-k}$.

By(4.4) and(4.7), we have $R_{m}\geq P_{0}R_{m}\geq(1-\epsilon)Q_{+}$ for all
Together with(4.3), we have
(4.8)$\lim_{n\rightarrow\infty}R_{n}=Q_{+}$$m\geq N_{\epsilon}$.

Now, by Fatou’s Lemma and (4.8),(4.9)$E^{\mu}[\lim_{n\rightarrow}\sup_{\infty}\hat{S}_{F^{n}(X)(\omega)}(u, u)]\geq\lim_{n\rightarrow}\sup_{\infty}\hat{S}_{R}$$(u, u)=\hat{S}_{Q_{+}}(u, u)$、,.

Homogenization on fractals,
for all
whose address is$\omega\in\Omega_{+}$, where $V_{\omega}:=\{\pi([\omega, i]) : i\in I_{F}\}$ is a O-cell. By (4.6) and (4.9), we have$u\in l(V_{\omega})$$\omega$$\lim_{n\rightarrow}\sup_{\infty}\hat{S}_{F^{n}(X)(\omega)}$$\mu-a.e$.

$X$$(u, u)=\hat{S}_{Q_{+}}(u, u)$,, $u\in l(V_{\omega})$ . Applying Lemma 4.3 with$(u, u)$ and $Y=\hat{S}_{Q_{+}}(u, u)$ ( $Y$ is non-random), we haveand for all$Y_{n}=\hat{S}_{F^{n}(X)(\omega)}$201$\lim_{n\rightarrow\infty}E^{\mu}[|\hat{S}_{F^{n}(X)(\omega)}$$\omega\in\Omega_{+}$$(u, u)-\hat{S}_{Q_{+}}(u, u)|]=0$$\forall\omega\in\Omega_{+}$,$u\in l(V_{\omega})$.

is finite dimensional, we obtain (3.2)where $Q_{\mu}=Q_{+}$ .
Q.E.D.
Using Theorem 3.4, we can prove the convergence of forms.

Since$l(V_{\omega})$

Proposition 4.4. For all $u\in L^{2}$$(V, d\nu_{0})$,$\lim_{n\rightarrow\infty}E^{\mu}[|S_{F^{n}(X)} (u, u)-S_{\iota(Q_{\mu})}(u, u)|]\rightarrow 0$.

is in one 0-cell whose address is ,then the result is clear by Theorem3.4. 珂 hen is compactly supported,
we can decompose the form into finite number of forms on O-cells, so theresult still holds. It is thena routine work to show the result for all$u\in L^{2}$ $(V, d\nu_{0})$Q.E.D.
.

Proof. When the support of$u$$\omega$$u$

We note here that there are several errors in [10] Section4, where theproof of the theorem corresponding to our Theorem 3.4 is given. They
can be fixed, but since we have given aproof of the improved theorem,
we omit mentioning where the errors are and how to fix them.
Proof of Theorem 3.5\S 5.

Now that we obtain Proposition 4.4, the proof of Theorem 3.5 isbasically the same as the proof of Theorem3.6 in [10]. Here we will juststate key propositions and briefly comments how to prove them. For
detailed arguments, we refer to Section 5 in [10].
As before, define $K_{m}=\alpha^{m}\hat{K}$ . We consider processes killed at. Set $\mathcal{F}_{n,m}=\{u\in l(V_{n}) : u|_{K\backslash K_{m}}=0\}$ . For ,、, $m$$(u, v)=\rho_{Qo}^{n}S_{X}(uo\Psi_{1}^{n}, vo\Psi_{1}^{n})$ .
Then,, we setand(, 、, $m$ ) is a regular Dirichlet form. We denote the corresponding
process $(Y^{X,n,m}, \{P_{X,n,m}^{x}\}_{x\in V_{n}\cap K_{m}})$ and the corresponding generator$L^{(X,n,m)}$. Als05 letdenote the process correspond$ing$ to, wherein Theorem
isaform constructed from$\alpha^{m}\hat{V}_{0}\backslash \{0\}$$X\in \mathcal{X}_{M}$$\mathcal{E}_{X}^{n,m}$$u$$\mathcal{E}_{X}^{n,m}$$\mathcal{F}$$(Y^{\mu,m}, \{P_{\mu,m}^{x}\}_{x\in K_{m}})$$(\mathcal{E}_{Q_{\mu}}, \mathcal{F}_{m})$$\mathcal{E}_{Q_{\mu}}$$Q_{\mu}$$v\in \mathcal{F}$T. Kumagai

202

3.4 and $\mathcal{F}_{m}=\{f\in \mathcal{F} : f|_{K\backslash K_{m}}=0\}$ .
is the corresponding
generator.
The first key proposition is the following convergence of finite dimensional distribution. This can be obtained by using(H-2), Theorem
2.7and Proposition4.4. See Proposition5.7in [10] for the proof.
$L^{(m)}$

Proposition 5.1. Let
be sequence such that$0<t_{1}<\cdots<t_{k}$. Then, for all,$\{x_{n}\}_{n=1}^{\infty}\subset V$$a$$\alpha^{-n}x_{n}$$\rightarrow x_{\infty}\in K_{m}$$E^{P_{X,n,m}^{\alpha x_{n}}}-n[f_{1}(\omega_{t_{1}})\cdots f_{k}(\omega_{t_{k}})]\rightarrow E^{P_{\mu,m}^{x}}\infty[f_{1}(\omega_{t_{1}})\cdots f_{k}(\omega_{t_{k}})]$

in probability under$\mu$, for any$f_{1}$,$\cdots$,, $f_{k}\in C(K_{m}, R)$ .

Fora process $Z$ on $K$ let $T_{0}^{r}(Z)=\inf\{t\geq 0 : Z(t)\in V_{r}\}$ anddefine inductively $T_{i}^{r}(Z)=\inf\{t>T_{i-1}^{r}(Z) : Z(t)\in V_{r}\backslash Z(T_{i-1}^{r}(Z))\}$
for $i\in N$ . Then the following holds.
Lemma 5.2. Let
be as in Proposition 5.1. Then there, $c_{5.2}>0$ such that the following holds for $s\geq 0$ , $\mu-a.e$ . $X$ .
exist ,$\{x_{n}\}_{n}\subset V$$\gamma$$c_{5.1}$$-nx_{n}(T_{i+1}^{r}(Y^{X,n,m})-T_{i}^{r}(Y^{X,n,m})\leq s)\leq c_{5.1}e^{-c_{5.2}(\tau_{Qo}^{r}s)^{-\gamma}}$$\lim\sup\sup_{in\rightarrow\infty\geq 0}P_{X,n,m}^{\alpha}$

To show this, uniform ( 垣 iptic) Harnack inequality for $L^{(X,n,m)}$ isimportant (see [10] Lemma 5.8). In our case, we can obtain it easily
by using Proposition 2.8. See [10] Lemma 5.10 for the detailed proof(thanks to Proposition 2.8, the proof can be shortened).
Using Lemma 5.2, the following tightness is deduced by a standard
argument (see Proposition5.11 in [10]).
$e$Proposition 5.3. Let
be as in Proposition 5.1. Thenis tight(pre-compact)in $D([0, \infty)$ , $K_{m})$ for $\mu- a.e$ . $X$ .
$\{x_{n}\}_{n}\subset V$$\{P_{X,n,m}^{\alpha^{-n}x_{n}} ; n \geq 1\}$

By Proposition 5.1 and Proposition5.3, we have the killed process
version of Theorem 3.5. Using Lemma 5.2 again, it is easy to deduce
the full version of Theorem 3.5.
\S Appendix A.

Proof of Proposition 4.1

To start with, we prepare several results for the proof. First, we(cf. [14]). For $X$ , $Y\in Q_{+}$ , letdefine Hilbert’sprojective metric on$Q_{+}$$h_{+}$$(X, Y)=\inf\{\alpha>0 :Clearly,$h_{-}$

X\leq\alpha Y\}$$(X, Y)\leq h_{+}(X, Y)$ .

, $h_{-}(X, Y)=\sup\{\alpha>0:\alpha Y\leq X\}$ .

Define$h(X, Y)=\log\frac{h_{+}(X,Y)}{h_{-}(X,Y)}$

.

203

Homogenization on fractals

Note that ( $aX$ , bY)=h(X, Y) for all , $b>0$ and $h(X, Y)=0$ if andis not ametric. But
only if $X=aY$ for some $a>0$ , so that
it is ametric on $\{X\in Q_{+} : ||X||=1\}$ . Using Proposition3.3, it iseasy to prove the following(cf. [13] Section3, [15] Remark3.2and [16]Proposition3.3).
$h$

$a$

$h(\cdot, \cdot)$

Proposition A.I.

0

and $h_{-}(\hat{F}(X),\hat{F}(Y))\geq h_{-}(X, Y)$ for
all $X$ , $Y\in Q_{+}$ . In particular, $h(\hat{F}(X),\hat{F}(Y))\leq h(X, Y)$ .
2)If $Q_{*}\in Int(Q_{M})$ satisfifies $F\wedge(Q_{*})=Q_{*}$ , then for each $n\in N\cup\{0\}$$(X, Q_{*})Q_{*}\leq\hat{F}^{n}(X)\leq h_{+}(X, Q_{*})Q_{*}$ .
and each $X\in Q_{irr}$ ,$h_{+}(\hat{F}(X),\hat{F}(Y))\leq h_{+}(X, Y)$$h_{-}$

is non-constant,$(u, u)=h_{\pm}(X, Y)\hat{S}_{Y}(u, u)\}$ . Also, for each $Q\in Q_{irr}$ and,so thatdefine $kl_{n,Q}(u)$ as aunique function on

For $X$ , $Y\in Q_{+}$ , let$A^{\pm}$$(X, Y)=\{u\in l(\hat{V}_{0})$:$u$$u\in l(\hat{V}_{0})$$\hat{S}_{X}$$V\wedge n$$\hat{S}_{\hat{F}}$、(Q)$(u, u)=\hat{\mathcal{E}}_{Q}^{n}(H_{n,Q}(u), \prime\mu_{n,Q}(u))$,

isaQwhere
is defined in (2.2). In other word,. By definition, $A_{j,Q}(u)=H_{1,Q}(u)\circ\Psi j$ .
harmonic extension ofThus the following holds for all $m\geq 0$ and $\geq n\geq 0$ .
$\mathcal{E}\wedge nQ$$\prime\kappa_{n,Q}(u)\in l(\hat{V}_{n})$$(\cdot, \cdot)$$u\in l(\hat{V}_{0})$

$l$

(A.1)

$\mathcal{H}_{m+n,\hat{F}^{l-n}(Q)}(u)\circ\Psi_{i_{1},\cdots,i_{m},j_{1},\cdots,j_{n}}$$=$$A_{j_{n},\hat{F}^{l-n}(Q)}\circ\cdots\circ A_{j_{1},\hat{F}^{1-1}(Q)}(H_{m,\hat{F}^{l}(Q)}(u)\circ\Psi_{i_{1},\cdots,i_{m}})$.

We have the following(cf. [15] Proposition3.3, [13] Lemma5.8).

Lemma A.2. For $X$ , $Y\in Q_{irr}$ , defifine $h_{\pm},n=h\pm(\hat{F}^{n}(X),\hat{F}^{n}(Y))$ .
Then. for each $0\leq m\leq n$ ,(A.2)

$h_{+,n}\leq h_{+,m}\leq h_{+,0}$(A.3)

There exists,$h_{-,n}\geq h_{-,m}\geq h_{-,0}$$\lim_{n\rightarrow\infty}h_{\pm,n}\in(0, \infty)$.

.

Further, if $h_{\pm,n}=h_{\pm,0}$ , then for all $u\in A^{\pm}(\hat{F}^{n}(X),\hat{F}^{n}(Y))$ , we have

(A.4)$\prime\kappa_{n-m,\hat{F}^{m}(X)}(u)\circ\Psi_{i_{1},\ldots,i_{n-m}}$$=$$ H_{n-m,\hat{F}^{m}(Y)}(u)\circ$

重 $i_{1},\ldots,i_{n-m}\in A^{\pm}(\hat{F}^{m}(X),\hat{F}^{m}(Y))$ .

(A.2) is ﬀom Proposition A.I 1). (A.3) is a simple consequence of(A.2) and the fact $h_{-}(X, Y)\leq h_{+}(X, Y)$ . Next, if $h_{\pm},7l=h_{\pm,0}$$Proo/$.

204

T. Kumagai

and $u\in A^{-}(\hat{F}^{n}(X),\hat{F}^{n}(Y))$ , then we have$h_{-,0}\hat{S}_{\hat{F}}$、(Y) (u)$=$$\hat{S}_{\hat{F}^{n}(X)}(u)=\hat{\mathcal{E}}_{\hat{F}^{m}(X)}^{n-m}(H_{n-m,\hat{F}^{m}(X)}(u))$$\geq$$h_{-,m}\hat{\mathcal{E}}_{\hat{F}^{m}(Y)}^{n-m}(H_{n-m,\hat{F}^{m}(X)}(u))$$\geq$$h_{-,m}\hat{\mathcal{E}}_{\hat{F}^{m}(Y)}^{n-m}(H_{n-m,\hat{F}^{m}(Y)}(u))$$=$$h_{-,m}\hat{S}_{\hat{F}^{n}(Y)}(u)\geq h_{-,0}\hat{S}_{\hat{F}^{n}(Y)}(u)$.

Thus all the inequalities above are in fact equalities. By the uniquenessof the harmonic extension, we obtain the $( )$ -version of (A.4). $(+)-$$Q.E.D$ .
version of (A.4) can be proved similarly.
We next mentiona convergence result on positive matrices (cf. [15]Proposition3.5).
,,,
Lemma A.3. Let $B$ be finite set. Suppose$l(B)$
are positive matrices fromto itself and suppose there exists
sequencesuch that$A_{1}$$a$$\cdots$$A_{n}$$\cdots$$a$,$A_{\infty}$sub-$\{\sigma(n)\}_{n}$(A.5)

for all$\lim_{n\rightarrow\infty}(A_{\sigma(n)})_{ij}=(A_{\infty})_{ij}$$i$, $j\in B$ .

Then, for each family of non-negative non-zero vectors $\{v_{n}\}_{n}\subset l(B)$ ,(A.6)$\lim_{n\rightarrow\infty}\frac{A_{1}\circ\cdots\circ A_{n}v_{n}}{||A_{1}\circ\cdots\circ A_{n}v_{n}||}$

exists and it is$a$positive vector.

For , $r\geq 0$ , define $T_{p,r}=A_{p+1}\circ\cdots\circ A_{p+r}$ . By (A.5)
exists for each
and Theorem 3.6 in [17],, $j\in B$ , $p\geq 0$ and it is independent of and (in [17], such aproperty
is called strongly ergodic). Using Lemma 3.3 in [17], we obtain the$Q.E.D$ .
result.

Proof.

$p$$\lim_{r\rightarrow\infty}(T_{p,r})_{ij}/(\sum_{s\in B}(T_{p,r})_{is})$
$i$

$i$

$p$

We now give a key Lemma (cf. [15] Lemma 3.6, [16] Section 5.3).
Lemma A.4. Let $X\in Q_{irr}$ . Then, $F\wedge(X)=X$ if and only if(A.7)$h(\hat{F}^{n}(X),\hat{F}^{n+1}(X))=h(X,\hat{F}(X))$

for all$n$$\in N$.

We will assume(A.7) and prove $F\wedge(X)=X$ (since the otherdirection is clear). First, note that there exist $j\in I_{F}$ anda subsequence$\{s(n)\}_{n}$ so that we can take $u\pm,n\in A^{\pm}(\hat{F}^{s(n)}(X),\hat{F}^{s(n)+1}(X))$ with

Proof.

205

Homogenization on fractals$u\pm,n(a_{j})=\min_{x}u\pm,n(x)=0$($a_{j}$

is afixed point of$\Psi_{j}$).

Then byLemma A.2,
(A.8)
$=$

$u\pm,n,m:=H_{s(n)-m,\hat{F}^{m}(X)}(u\pm,n)o\Psi_{j}^{s(n)-m}$$7\{s(n)-m,\hat{F}^{m+1}(X)(u\pm,n)o\Psi_{j}^{s(n)-m}\in A^{\pm}(\hat{F}^{m}(X),\hat{F}^{m+1}(X))$

for all $m\leq s(n)$ where(A.9)$u\pm,n,m$

isa -th iteration of$n$$\Psi_{j}^{n}$$\Psi_{j}$,. By $(A.1)$ , we have$=$$A_{j,\hat{F}^{m}(X)}\circ\cdots\circ A_{j,\hat{F}^{s(n)-1}}(x)(u\pm,n)$$=$$A_{j,\hat{F}^{m+1}}(x)o\cdots oA_{j,\hat{F}^{\epsilon(n)}}(x)(u\pm,n)$.

large enough so that $F\wedge m(X)\in Int(Q_{M})$ for all $ m\geq$
Now choose(we use Assumption2.3 1)here). Using Lemma 3.1, we see thatare positive matrices ﬀom $l(B)$ to itself where $B:=$. Since all the elements of the matrices are less than 1 (due to, we see that(A.5) holds. We can thus apply
Lemma3.1) andLemma A.3 and obtain that
$N_{0}$$N_{0}$$\{(A_{j,\hat{F}^{m}(X)})|_{B}\}_{m\geq N_{0}}$$\hat{V}_{0}\backslash \{a_{j}\}$$\Downarrow B<\infty$$\lim_{n\rightarrow\infty}\frac{(u\pm,n,m)1B}{||(u\pm,n,m)|_{B}||}$

exists for $m\geq N_{0}$ . By(A.9), this limit is independent of $m\geq N_{0}$ , wethus denote it as. Then, by the choice of
We now regard
as a function on$u\pm,n’ u\pm(a_{j})=0$ so thatis non-constant. By (A.8), we see that$u\pm\in A^{\pm}(\hat{F}^{m}(X),\hat{F}^{m+1}(X))$ for all $m\geq N_{0}$ .
Thus,$ u\pm\cdot$$V\wedge 0$$ u\pm$$ u\pm$$\hat{S}_{\hat{F}^{m}(X)}(u\pm)=$$h_{\pm}(\hat{F}^{m}(X),\hat{F}^{m+1}(X))\hat{S}_{\hat{F}^{m+1}}(X)(u\pm)$.

On the other hand, by (A.7),$h_{+}(\hat{F}^{m}(X),\hat{F}^{m+1}(X))/h_{-}(\hat{F}^{m}(X),\hat{F}^{m+1}(X))\geq 1$$m\geq 0$

which we denote by$\beta$is independent of. Then, we obtain$\frac{\hat{S}_{\hat{F}^{m+1}}((x)u_{+})}{\hat{S}_{\hat{F}^{m+1}}((x)u_{-})}=\beta^{-1}\frac{\hat{S}_{\hat{F}^{m}(X)}(u_{+})}{\hat{S}_{\hat{F}^{m}(X)}(u_{-})}=\cdots=\beta^{-(m+1-N_{0})}\frac{\hat{S}_{\hat{F}^{N_{0}}(X)}(u_{+})}{\hat{S}_{\hat{F}^{N_{0}}(X)}(u_{-})}$,

for all $m\geq N_{0}$ . If $\beta>1$ , it contradicts to Proposition A.l2). So, $\beta=1$
which means $h(X,\hat{F}(X))=0$ (by taking $m=0$ ). Thus, $F\wedge(X)=cX$ for$Q.E.D$ .
some $c>0$ . Using Remark2.41), we have $c=1$ .

Proof of Proposition4 1. First, since$c_{1}Q_{0}\leq\hat{F}^{n}(M)\leq c_{2}Q_{0}$$\cdot$$n\in N$ ,andthere exists asubsequence (which could depend onsuch that$Q_{M}\in Q_{irr}$(A.10)$\lim_{r\iota\rightarrow\infty}\hat{F}^{\sigma(n)}(M)=Q_{M}$

.

$M$ )

for all$\{\sigma(n)\}_{n}$206

T. Kumagai

On the other hand, by Lemma A.2, the following limit exists,$h_{\pm}:=\lim_{n\rightarrow\infty}h_{\pm}(\hat{F}^{n}(M),\hat{F}^{n+1}(M))\in(0, \infty)$

.

Thus,
$h(\hat{F}^{m}(Q_{M}),\hat{F}^{m+1}(Q_{M}))$$=$$\lim_{n\rightarrow\infty}h(\hat{F}^{m+\sigma(n)}(M),\hat{F}^{m+1+\sigma(n)}(M))=\log\frac{h_{+}}{h_{-}}$,

for all $m\in N$ . By Lemma A.4, this implies $F\wedge(Q_{M})=Q_{M}$ . In particular,$Q_{M}\in Int(Q_{M})$ due to Assumption2.3 1). Using Lemma A.2 again,
exists and the limit is 1 due to (A.1O). By
Proposition A.l 2), this implies (4.1).
Q.E.D.
$\lim_{n\rightarrow\infty}h_{\pm}(Q_{M},\hat{F}^{n}(M))$

ACKNOWLEDGEMENTS. The author is gratefirl to Dr. V. Metz and Dr.
R. Peirone for giving him information of the paper [15].

References[1] M.T. Barlow, Diﬀusions on fractals, Lectures in Probability Theory andStatistics (Lect. Notes Math. 1690), Springer, 1998.
[2] M.T. Barlow and R.F. Bass, Stability of parabolic Harnack inequalities,
Preprint 2002.
[3] A. Grigor’yan and A. Teles, Sub-Gaussian estimates of heat kernels on
infinite graphs, Duke Math. J., 109 (2001), 451-510.
[4] A. Grigor’yan and A. Teles, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Annalen, 324 (2002), 521-556.
[5] B.M. Hambly and T. Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries, Proc. of Symposia in Pure Math. (AMS), to appear.
[6] B.M. Hambly and T. Kumagai, Transition density estimates for diﬀusions
on p.c.f. fractals, Proc. London Math. Soc, 78 (1999), 431-458.
[7] B.M. Hambly and V. Metz, The homogenization problem for the Vicsek
set, Stochastic Process. Appl., 76 (1998), 167-190.
[8] V.V. Jikov, S.M. Kozlov and O.A. Ole nik, Homogenization of diﬀerential
operators and integral functionals, Springer, 1994.
[9] J. Kigami, Analysis on Fractals, Cambridge Univ. Press, 2001.
[10] T. Kumagai and S. Kusuoka, Homogenization on nested fractals, Probab.
Theory Relat. Fields, 104 (1996), 375-398.
[11] S. Kusuoka, Diﬀusion processes on nested fractals, Statistical Mechanics
and Fractals (Lect. Notes Math. 1567), Springer, 1993.
[12] T. Lindstrm, Brownian motion on nested fractals, Mem. Amer. Math.
Soc. 420 83, 1990.

Homogenization on fractals

207[13] V. Metz, Renormalization contracts on nested fractals, J. Reine Angew.
Math., 480 (1996), 161-175.
[14] R.D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps,
Mem. Amer. Math. Soc. 391 75, 1988.
[15] R. Peirone, Convergence and uniqueness problems for Dirichlet forms on
fractals, Boll. Unione Mat. Ital. Sez. B Artie. Ric. Mat. (8), 3 (2000),
431-460.
[16] C. Sabot, Existence and uniqueness of diﬀusions on finitely ramified selfsimilar fractals, Ann. Sci. \’Ecole Norm. Sup. (4), 30 (1997), 605-673.
[17] E. Seneta, Non-negative matrices and Markov chains, Springer, 1981.

Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606-8502, Japan
Email: kumagai@kurims.kyoto-u.ac.jp

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 209-232

Representation of Martingales with Jumps
andApplications to Mathematical Finance
Hiroshi Kunita
Dedicated to Professor Kiyosi It\^o on his 88-th birthday
Abstract.

We study representations of martingales with jumps based on thefiltration generated by a L\’evy process. Two types of representation
theorem are obtained. The first formula is valid for any martingale
and written as the sum of the stochastic integral based on the Brownian motion and that based on the compensated Poisson randommeasure. See (0.1). The second formula is valid only for a process
which is a martingale for any equivalent martingale measure. See(0.2). The latter representation formula is then applied to a problem in mathematical finance. The upper hedging strategy and thelower hedging strategy of a contingent claim is obtained through therepresentation kernel.

\S 0.

Introduction

It is a well known fact that any martingale with respect to thefiltration generated by a Brownian motion can be represented as It\^o’s
stochastic integral based on the Brownian motion. On the other hand,martingales with respect to the filtration generated by a L\’evy process
are not always represented by It\^o’s stochastic integrals based on the
L\’evy process, even the latter is a martingale. What is known is thatany square integrable martingale with respect to the filtration generated
by a L\’evy process is represented by stochastic integrals based on the
Brownian motion and the compensated Poisson random measure.
In the first half of this paper, we recall these representation theobe the filrems following Kunita-Watanabe [6] (Section 1). Let
tration generated by a $m$ -dimensional L\’evy process. Then every (lo$cal)martingale$ $M(t)$ with respect to the filtrationis represented$(F_{t})$$(F_{t})$Received April 7, 2003.

H. Kunita

210

by
(0.1)

$M(t)=M(0)+\sum_{i=1}^{m}\int_{0}^{t}\phi_{i}(s)dW^{i}(s)+\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$(dsdz),

where $W(t)=(W^{1}(t), \ldots, W^{m}(t))$ is a standard Brownian motion and(dsdz) is the compensated Poisson random measure, which appear in,
the L\’evy-It\^o decomposition of the L\’evy process. The pair
, $\psi(s, z))$ is a predictable process with parameter satisfying certain integrability conditions (See Theorem 1.1). We are particularly interested in the exponential representation of positive martingales (Theorem 2.1). We apply it to the study of Radon Nikodym density ofequivalent probability measure and extend Girsanov’s theorem to jumpprocesses (Theorem 2.3).
In the second half of the paper, we apply these representation theorems to some problems in mathematical finance. Suppose that we are(e.g., a price process or its
given a stochastic processreturn process in mathematical finance) governed by a L\’evy process. If
the process has jumps, there are infinitely many equivalent probability
measures with respect to which is a localmartingale (called equivalentmartingale measures). Now suppose that $M(t)$ is a localmartingale forany equivalent martingale measure. We will show that under some conditions for , $M(t)$ is represented by a stochastic integral based on ,i.e., it is written as$\overline{N}$$((\phi_{1}(s),$$\phi_{m}(s))$$\ldots$$z$$\xi_{t}=(\xi_{t}^{1}, \ldots, \xi_{t}^{d})$$\xi_{t}$$\xi_{t}$$\xi_{t}$(0.2)$\xi_{t}$$M(t)=M(0)+\sum_{i=1}^{d}\int_{0}^{t}\varphi_{i}(s)d\xi_{s}^{i}$.

The diﬀerence of these two representations (0.1) and (0.2) are big. We
show further that an adapted process $X(t)$ is a supermartingale forany equivalent martingale measure if and only if it admits the unique
Doob-Meyer decomposition (not depending on each equivalent martingale measure) and the localmartingle part $M(t)$ is represented as (0.2).
See Theorem 3.4 in Section 3.
At the end of Section 3, we apply the above representation theorem
to determine the upper hedging price and the lower hedging price of agiven contingent claim (Theorem 3.5).
Finally, we mention that there are several works on determing theupper or the lower hedging prices of contingent claims in the case where
the price processes have jumps. See e.g. Kabanov-Stricker [3] and references therein. In these works, more general price processes are studied
in an abstract manner.

Representation of martingales and mathematical fifinance

211

Representation of localmartingales\S 1.

Let $T$ be a positive number and let $Z(t)$ , $t\in[0,T]$ be an $m$ dimensional
L\’evy process such that $Z(0)=0$ . Then it admits the L\’evy-It\^o decomposition:(1.1)$Z(t)=\sigma W(t)+bt+\int_{(0,t]}\int_{|z|>1}zN(dsdz)$$+\int_{(0,t]}\int_{|z|\leq 1}z$

{ $N(dsdz)-\hat{N}$ (dsdz)},

whereis an $m\times m$ matrix, $W(t)=(W^{1}(t), \ldots, W^{m}(t))$ is an mdimensional standard Brownian motion and $N(dsdz)$ is a Poisson counting measure on $[0, T]\times R^{m}$ with intensity measure (dsdz) $=ds\iota/(dz)$ ,which is independent of $W(t)$ . In the following, we denote$\sigma$$\hat{N}$(1.2)$\tilde{N}$(dsdz) $=N(dsdz)-\hat{N}$ (dsdz).

, $t\in[0, T]$ be the filtration generated by the BrownianLet$W(t)$
motionand the Poisson random measure $N(dtdz)$ . Then both$W(t)$ and(dsdz) are martingales adapted to the filtration. Let$M(t)$ , $t\in[0, T]$ be an-adapted cadlag (right continuous with the leﬃ
hand limits) process. It is called a localmartingale if there exists a nondecreasing sequence of stopping times , $=1,2$ , with values in $[0, T]$isand the stopped process
such that $P(\tau_{n}<T)\rightarrow 0$ as
a martingale for any . In particular if we can choose the sequence suchthat the stopped process
is a square integrable martingale for
any , $M(t)$ is called a locally square integrable martingale. Any continuous localmartingale is a locally square integrable martingale, but it isadaptednot always the case for a localmartingale with jumps. An
cadlag process $X(t)$ is called a semimartingale if it is written as a sumof a localmartingale and a process of bounded variation. In particular
if the corresponding process of bounded variation is locally integrable,$X(t)$ is called a special semimartingale. A special semimartingale is decomposed uniquely to the sum of a localmartingale and a predictableprocess of bounded variation.
We denote by the set of all $m$ dimensional predictable processes$\phi(t)=(\phi_{1}(t), \ldots, \phi_{m}(t))$
such that $\int_{0}^{T}|\phi(s)|^{2}dt<\infty a.s$ . Then thestochastic integral based on the $m$-dimensional Brownian motion $W(t)=$$(W^{1}(t), \ldots, W^{m}(t))$ is well defined for. We use the notation:$(F_{t})$$\int_{|z|\leq 1}z\overline{N}$$(F_{t})$$\tau_{n}$

$n$

$n$

$\ldots$$M(t\wedge\tau_{n})$$\rightarrow\infty$$n$$M(t\wedge\tau_{n})$$n$$(F_{t})$$\Phi$$\phi\in\Phi$(1.3)$\int_{0}^{t}(\phi(s), dW(s))=\sum_{i=1}^{m}\int_{0}^{t}\phi_{i}(s)dW^{i}(s)$212

H. Kunita

It is a continuous locally square integrable martingale.
Let $P$ be the predictable -algebra on $[0, T]$and letbe the$(s,z,\omega)\in[0,T]\timesR^{m}\times\Omega$. A functional,
Borel algebra on$P\timesB$
is called a predictable process if it is-measurable.
We will recall the definition of the stochastic integral of the predictable process $\psi(s, z)$ based on the compensated Poisson random measure (dsdz) following Kunita-Watanabe [6]. Note first that if , ,are disjoint Borel subsets of $[0, T]\times R^{m}$ such that $\hat{N}(E_{1})<\infty$ ,$<\infty$ , then,are independent random variables with,, respectively. Now, let $\psi(t, z)$ be
mean 0 and variancea step process of the form
, where $0=t_{0}<$$<t_{N}=T$ and for each, ,are disjoint subsets ofsatisping $\iota/(F_{ij})<\infty,j=1$ , , and-adapted random
are boundedbyvariables. We define the stochastic integral of based on$B$$\times\Omega$$\sigma$$R^{m}$$\psi(s, z, \omega)$$\tilde{N}$$E_{1}$$\ldots$$E_{n}$$\ldots,\hat{N}(E_{n})$$\tilde{N}(E_{1})$$\ldots,\tilde{N}(E_{n})$$\ldots,\hat{N}(E_{n})$$\hat{N}(E_{1})$$\sum_{i,j}a_{ij}1_{(t_{i},t_{i+1}]}(t)1_{F_{ij}}(z)$$iF_{i1}$$\ldots$$\ldots$$n$$\ldots$$R^{m}$$F_{in}$$F_{t_{i}}$$a_{ij}$$\tilde{N}$$\psi$(1.4)$\int_{0}^{T}\int_{R^{m}}\psi(t, z)\tilde{N}$(dtdz)$=\sum_{i,j}a_{ij}\tilde{N}((t_{i}, t_{i+1}]\times F_{ij})$.

Then we have
(1.5)
$E[(\int_{0}^{T}\int_{R^{m}}\psi(t, z)\overline{N}(dtdz))^{2}]=\sum_{i}E[(\sum_{j}a_{ij}\hat{N}((t_{i}, t_{i+1}]\times F_{ij}))^{2}]$$+\sum_{i<k}E[\{\sum_{j}a_{kj}\hat{N}((t_{k}, t_{k+1}]\times F_{kj})\}\{\sum_{j}a_{ij}\hat{N}((t_{i}, t_{i+1}]\times F_{ij})\}]$

Since $\{a_{ij},j=1,2, \ldots\}$ andare independentand the latters are of mean 0, the first term of the left hand side iscomputed as$\tilde{N}((t_{i}, t_{i+1}]\times F_{ij}),j=1,2_{7}\ldots$(1.6)$\sum_{ij}E[a_{ij}^{2}\tilde{N}((t_{i}, t_{i+1}]\times F_{ij})^{2}]=\sum_{ij}E[a_{ij}^{2}\hat{N}((t_{i}, t_{i+1}]\times F_{ij})]$The last term of (1.5) is 0, since $E[\sum_{j}a_{kj}\tilde{N}((t_{j}, t_{j+1}]\times F_{kj})|F_{t_{i+1}}]=0$ .
Therefore we have
(1.7)
$E[(\int_{0}^{T}\int_{R^{m}}\psi(t, z)\tilde{N}(dtdz))^{2}]=E[\int_{0}^{T}\int_{R^{m}}|\psi(t, z)|^{2}\hat{N}(dtdz)]$.

Representation of martingales and mathematical fifinance213

Now suppose that $\psi(t, z)$ is a predictable process satisfying the condition $ E[\int_{0}^{T}\int_{R^{m}}|\psi(t, z)|^{2}dt\iota/(dz)]<\infty$ . Then we can choose a sequenceof step processes such that(formula (1.4)). Denote the stochastic integral of, as. We denote the limit by $M=$
converges in. Thenby(dtdz). Then it satisfies (1.7) again.
(dsdz) is denoted by
The stochastic integral(dsdz). It is a cadlag process with time and in fact is asquare integrable martingale. This fact can be shown directly in thecase where $\psi(t, x)$ is a step process defined above. Then the martingale
property is extended to any such that (1.7) is finite.
the set of all predictable processes
We denote by$\psi(t, z)$ which are square integrable (resp. integrable) with respect tothe measure (dtdz) $a.s$ . Then we can define the stochastic integral(dsdz) for
as a locally square integrable, we define the stochastic integral bymartingale. For$E[\int_{0}^{T}\int_{R^{m}}|\psi(t, z)-\psi_{n}(t,z)|^{2}dt\iota/(dz)]$$\{\psi_{n}(t,z)\}$$\rightarrow 0$$\psi_{n}$$ n\rightarrow\infty$$L^{2}$$M_{n}$$M_{n}$$\int_{0}^{T}\int_{R^{m}}\psi(t, z)\tilde{N}$$\int_{0}^{T}1_{(0,t]}(s)\psi(s, z)\tilde{N}$$\int_{0}^{t}\psi(s, z)\tilde{N}$$t$$\psi$$\Psi_{2}(\hat{N})(\Psi_{1}(\hat{N}))$$\hat{N}$$\psi\in\Psi_{2}(\hat{N})$$\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$$\psi\in\Psi_{1}(\hat{N})$(1.8)$\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$(dsdz) $:=\int_{0}^{t}\int_{R^{m}}\psi(s, z)N(dsdz)$$-\int_{0}^{t}\int_{R^{m}}\psi(s, z)\hat{N}$(dsdz).

$=$
It is a localmartingale. For a general $\psi(t, z)$ , we set$\psi(t, z)1_{\{|\psi|>1\}}(t, z)$ , $\psi_{2}(t, z)=\psi(t, z)1_{\{\{|\psi|\leq 1\}}(t, z)$ , and we denote by
the set of all predictable process $\psi(t, z)$ such that, the stochastic integral. Then, for any
andand. It is a localis defined as the sum of stochastic integrals ofmartingale.
The following notations will be used
(1.9)$\psi_{1}(t, z)$$\psi_{1}\in\Psi_{1}(\hat{N})$$\Psi_{1,2}(\hat{N})$$\psi\in\Psi_{1,2}(\hat{N})$$\psi_{2}\in\Psi_{2}(\hat{N})$$\psi_{1}$$N_{t}(\psi)=\int_{0}^{t}\int_{R^{m}}\psi(s, z)N(dsdz)$,$\psi_{2}$$\tilde{N}_{t}(\psi)=\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$$\hat{N}_{t}(\psi)=\int_{0}^{t}\int_{R^{m}}\psi(s, z)\hat{N}$(dsdz),(dsdz),

if these are well defined.
Now, we give a representation theorem of localmartingales.
Theorem 1.1. ([6], Example at p.227 and Proposition 5.2) Let$M(t)$ be a localmartingale. Then there exist $\phi(s)\in\Phi$ , $\psi(s, z)\in\Psi_{1,2}(\hat{N})$ ,214

H. Kunita

and $M(t)$ is represented by(1.10)$M(t)=M(0)+\int_{0}^{t}(\phi(s), dW(s))+\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$(dsdz).

The representation kernel $(\phi(s), \psi(s, z))$ is uniquely determined from$M(t),$, if $M(t)$ is represented by (1.10) with another $(\phi’(s), \psi’(s, z))$ ,then we have $\phi(s)=\phi’(s)a.e$ .
and $\psi(s, z)=\psi’(s, z)a.e.\hat{N}\otimes P$ ,where is the Lebesgue measure on $[0, T]$ .
$i.e.$$\lambda\otimes P$$\lambda$

Proof. In the paper [6], the above theorem is proved for square integrable martingale by using the theory of additive functionals of Markov
processes. Here we give a direct and simpler proof by applying It\^o[2]. For simplicity we prove the theorem in the case $m=1$ only. Let$Z=(Z(t))$ be a one dimensional L\’evy process and let (1.1) be the L\’evyIt\^o decomposition. We introduce a random measure $M(E)$ on $[0, T]\times R$
by(1.11)$M(E)=\int_{E(0)}dW(t)+\int_{E-E(0)}\frac{z}{1+|z|}\tilde{N}$where $E(0)=\{(t, 0)).
$\mu(E_{1}\cap E_{2})$ , where(t, O)\in E\}$ .

(dtdz),

Then we have $E[M(E_{1})M(E_{2})]=$$\mu(E)=|E(0)|+\int_{E-E(0)}(\frac{z}{1+|z|})^{2}dt\iota/(dz)$.

For each positive integer , we define the multiple Wiener integral by$p$(1.12)$I_{p}(f)=\int\cdots\int f(\xi_{1}, \ldots, \xi_{p})dM(\xi_{1})\cdots dM(\xi_{p})$.

space overLet
be theand letbe the closed$\{I_{p}(f);f\in
L_{p}^{2}\}$, wherelinear manifold ofspace onis thewith.
the product measure ofThen it is shown in [2] that one has the direct$H_{Z}=\sum_{p\geq0}\oplus
H_{Z}^{(p)}$ . Note that each $I_{p}(f)$ is written as
sum expansion:
the sum of the following terms(1.13)$H_{Z}$$L^{2}$$H_{Z}^{(p)}$$(\Omega, \mathcal{F}_{T}, P)$$L^{2}$$L_{p}^{2}$$R^{p}$$\mu$$\int\cdot\int_{0\leq t_{1}<<t_{p}\leq T,(z_{1},,z_{p})\in R^{p}}\cdots\ldots f((t_{1},z_{1})$,$\ldots$, $(t_{p},z_{p}))dM(t_{1}z_{1})\cdots dM(t_{p}z_{p})$$=\int_{0}^{T}\int_{R}\varphi(t_{p}, z_{p})dM(t_{p}z_{p})$

,

215

Representation of martingales and mathematical fifinance

where$\varphi(t_{p}, z_{p})=\int\cdot\int_{\Lambda(t_{p},z_{p})}f((t_{1}, z_{1}),$$\ldots$, $(t_{p}, z_{p}))dM(t_{1}z_{1})\cdots dM(t_{p-1}z_{p-1})$

and $\Lambda(t_{p}, z_{p})=\{0<t_{1}<\cdots<t_{p-1}<t_{p}, (z_{1}, \ldots, z_{p-1}, z_{p})\in R^{p-1}\}$ .
Setting $\phi(t)=\varphi(t, 0)$ and $\psi(t, z)=\varphi(t, z)\frac{1+|z|}{z}(|z|>0)$ , we find thatthe above is written as(1.14)$\int_{0}^{T}\phi(s)dW(s)+\int_{0}^{T}\int_{R}\psi(s, z)\overline{N}$(dsdz).

with
Therefore any element ofand hence any element $X$ ofmean 0 is written as the above. Now taking the conditional expectation of (1.14), we obtain the representation (1.10) for square integrable
martingale $M(t)=E[X|F_{t}]$ .
The extension to locally square integrable martingales will be obvious. The extension to localmartingales will be discussed after Theorem
2.1 in the next section.
$H_{Z}^{(p)}$$H_{Z}$

Let $M(t)$ and $N(t)$ be two locally square integrable martingales suchthat $M(0)=N(0)=0$ . Then by the Doob-Meyer decomposition ofa supermartingale, there exist adapted continuous increasing processes,
and an adapted continuous process of bounded variations,
andsuch thatand $M(t)N(t)-\langle M, N\rangle_{t}$ are localmartingales. Such bracket
by theprocesses are uniquely determined. Note thatdefinition. If $M(t)$ is represented by (1.10) with $M(0)=0$ and $N(t)$ is, then we have the formula
represented with the kernel$\langle M\rangle_{t}$$\langle N\rangle_{t}$$\langle M\rangle_{0}=\langle N\rangle_{0}=\langle M, N\rangle_{0}=0$$\langle M, N\rangle_{t}$$M(t)^{2}-\langle M\rangle_{t}$$N(t)^{2}-\langle N\rangle_{t}$$\langle M, M\rangle_{t}=\langle M\rangle_{t}$$(\overline{\phi},\tilde{\psi})$(1.15)$\langle M, N\rangle_{t}=\int_{0}^{t}(\phi(s),\tilde{\phi}(s))ds+\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{\psi}(s, z)\hat{N}$We can define the quadratic$X(t)$ and $Y(t)$ by(1.16)$c$(dsdz).

-variation of two semimartingales$[X, Y]_{t}=\exists\lim_{|\triangle|\rightarrow 0}\sum_{k=1}^{n}(X(t_{k})-X(t_{k-1}))(Y(t_{k})-Y(t_{k-1}))$,such that $0=t_{0}<$are partitions of the time interval$t_{1}<\cdots t_{n}=t$ and $|\triangle|=\max_{1\leq k\leq n}|t_{k}-t_{k-1}|$ . We set $[X]_{t}=[X, X]_{t}$ .
If $M(t)$ and $N(t)$ are continuous localmartingales, it is known that theand the quadratic $co$ -variation coincides, i.e.,
bracket process$[M, N]_{t}=\langle M, N\rangle_{t}$ . However if both $M(t)$ , $N(t)$ have jumps, the bracket
where$\triangle$$[0, t]$$\langle M, N\rangle_{t}$216

H. Kunita

process is not equal to the quadratic variation. In the case where the rep, respectively,
resentation kernels of $M(t)$ and $N(t)$ areand
we have$(\overline{\phi},\tilde{\psi})$$(\phi, \psi)$(1.17)$[M, N]_{t}=\int_{0}^{t}(\phi(s),\tilde{\phi}(s))ds+\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{\psi}(s, z)N(dsdz)$.

Two locally square integrable martingales $M(t)$ and $N(t)$ are called
orthogonal if $M(t)N(t)$ is a localmartingale or equivalently, the bracket
process
is identically 0. By the formula (1.15) we see that thecontinuous local martingale
and the discontinuous(dsdz) are orthogonal.
one
Suppose that $M(t)$ and $N(t)$ are not orthogonal. There exists a
predictable process. We definesuch thatnew locally square integrable martingales by $M^{1}(t)=\int_{0}^{t}\varphi(s)dN(s)$ and$M^{2}(t)=M(t)-M^{1}(t)$ . Then $M^{1}(t)$ and $M^{2}(t)$ are orthogonal eachother because of the equality$\langle M, N\rangle_{t}$$\sum_{i=1}^{m}\int_{0}^{t}\phi_{i}(s)dW^{i}(s)$$\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}$$\varphi(t)$$\langle M, N\rangle_{t}=\int_{0}^{t}\varphi(s)d\langle N\rangle_{s}$$\langle M^{1}, M^{2}\rangle_{t}=\langle M^{1}, M\rangle_{t}-\langle M^{1}, M^{1}\rangle_{t}=\int_{0}^{t}\varphi(s)^{2}d\langle N\rangle_{s}-\int_{0}^{t}\varphi(s)^{2}d\langle N\rangle_{s}=0$.

The locally square integrable martingale $M^{1}(t)$ is called the orthogonal
projection of $M(t)$ to $N(t)$ . In the case where $M(t)$ and $N(t)$ are rep, respectively, the kernel of the
resented with kernels
andorthogonal projection is given by$(\phi, \psi)$(1.17)$(\tilde{\phi},\tilde{\psi})$$\varphi$$\varphi(t)=\frac{(\phi(t),\tilde{\phi}(t))+\int_{R^{m}}\psi(t,z)\tilde{\psi}(t,z)_{l/}(dz)}{|\tilde{\phi}(t)|^{2}+\int_{R^{m}}\tilde{\psi}(t,z)^{2}\nu(dz)}$.

(resp.
) the set of all locally square inteWe denote by
grable martingales (resp. localmartingales) $M(t)$ with $M(0)=0$ . It isa vector space. A sequence $\{M_{k}(t), k=1,2, ..\}$ ofis said to con$M(t)$if there exists a nondecreasing sequence of stopping times
verge to, $=1$ , .. such that $P(\tau_{n}<T)\rightarrow 0$ as
and each sequenceof stopped processes $\{M_{k}^{\tau_{n}}(t):=M_{k}(t\wedge\tau_{n}), k=1,2, \ldots\}$ converges to. Thenin
is a complete space by this topology.
, we set
For a given$\mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{1}$$\mathcal{M}_{loc}^{2}$$\tau_{n}$$ n\rightarrow\infty$$n$$M^{\tau_{n}}(t)$$L^{2}$$\mathcal{M}_{loc}^{2}$$M(t)\in \mathcal{M}_{loc}^{2}$(1.19)$\mathcal{L}(M)=\{\int_{0}^{t}\varphi(s)dM(s);\varphi(s)\in\Phi(\langle M\rangle)\}$

,

217

Representation of martingales and mathematical fifinancesuch thatis the set of all predictable processes. Let $N$ be a sub. It is a subset of
if it is a closed vector space
set of. It is called a subspace ofincludingwhenever $N\in N$ .
the smallest closed, we denote by
Given a subset $N$ of$N$
the set of all $ M(t)\in$
subspace containing the set . We denote by$N\inN$. Thenis a closed subspace
which is orthogonal to any, every. Further, if $N$ is a closed subspace ofofis decomposed uniquely to the sum of $M^{1}(t)\in N$ and $M^{2}(t)\in N^{\perp}$ . We
have thus the orthogonal decomposition

where$\Phi(\langle M\rangle)$$\varphi$$\int_{0}^{T}|\varphi(t)|^{2}d\langle M\rangle_{t}<\infty a.s$$\mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{2}$$\mathcal{L}(N)$$\mathcal{L}(N)$$\mathcal{M}_{loc}^{2}$$N^{\perp}$$N^{\perp}$$\mathcal{M}_{loc}^{2}$(1.20)\S 2.

$M(t)\in \mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{2}$$\mathcal{M}_{loc}^{2}=N\oplus N^{\perp}$.

Exponential representation of positive martingales andextension of Girsanov’s theorem

We shall consider the exponential representation of a positive lois called positive if $\alpha_{t}>0$
calmartingale. Here a localmartingale
holds for all $t\in[0, T]a.s$ . For a predictable process $g(t, z)$ , we set$g(s, z)$ is said to belong to$g_{1}=g1_{|g|>1}$ and $g_{2}=g1_{|g|\leq 1}$ as before.
. Then it holds that
if $e^{g_{1}(t,z)}-1\in\Psi_{1}(\hat{N})$ and$\psi\equiv e^{g}-1\in\Psi_{1,2}(\hat{N})$.
if and only if$\alpha_{t}$$g_{2}\in\Psi_{2}(\hat{N})$$\Psi_{e,2}(\hat{N})$$g\in\Psi_{e,2}(\hat{N})$be a positive localTheorem 2.1. ($c.f$. [6], Theorem 6.1) Let
martingale such that $\alpha_{0}=1$ . Then there exists a pair of predictable
and $g(s, z)$ ofsuch that theprocess $f(t)=(f_{1}(t), \ldots, f_{m}(t))$ oflocalmartingale
is represented by$\alpha_{t}$$\Phi$$\Psi_{e,2}$$\alpha_{t}$(2.1)$\alpha_{t}=\exp\{(\int_{0}^{t}(f(s), dW(s))-\frac{1}{2}\int_{0}^{t}|f(s)|^{2}ds)$$+(N_{t}(g_{1})-\hat{N}_{t}(e^{g_{1}}-1))+(\overline{N}_{t}(g_{2})-\hat{N}_{t}(e^{g_{2}}-1-g_{2}))\}$.

Further, the pair $(f, g)$ is uniquely determined from .
Conversely let $(f(t), g(t, z))$ be a pair of predictable processes beby (2.1). Then it is, respectively. Definelonging to andpositive localmartingale.
$\alpha_{t}$$\Psi_{e,2}(\hat{N})$$\Phi$$a$$\alpha_{t}$The aboveis characterized as the solution of the following It\^o’s
stochastic diﬀerential equation starting from 1 at time 0:$\alpha_{t}$(2.2)$d\alpha_{t}=\alpha_{t-}(f(t), dW(t))+\alpha_{t-}\int_{R^{m}}(e^{g(t,z)}-1)\tilde{N}N\{dtdz)$

.

218

H. Kunita

In fact, apply It\^o’s formula ([6], Theorem 5.1) to the function $F(x)=e^{x}$
and $X(t)=\log\alpha_{t}$ , whereis given by (2.1). Note the obvious formula$e^{g}-1=(e^{g_{1}}-1)+(e^{g_{2}}-1)$ . Then we find thatsatisfies the above$g(t,z)$ . We denote the$f(t)$
SDE. It is determined by two integrands
andpositive localmartingale by.
The above theorem is proved in [6] in the case whereis a multiplicative functional of a Markov process. We give here a direct andsimpler proof.
$\alpha_{t}$$\alpha_{t}$$\alpha_{t}=\alpha_{t}(f, g)$$\alpha_{t}$Lemma 2.2. ($cf[6]$, Lemma 6.1.) Let $f(t)=(f_{1}(t), \ldots, f_{m}(t))$ , $g(t,z)$ ,be predictable processes such that $ f\in\Phi$ ,is bounded belonging
to, $gh=0$ and $A(t)$ is a right continuous predictable process ofbounded variation. Set$h$$h(t, z)$$\Psi_{2}(\hat{N})$

(2.3)

.

$\beta_{t}=\exp\{\int_{0}^{t}(f(s), dW(s))+N_{t}(g)+\tilde{N}_{t}(h)-A(t)\}$Then
is a localmartingale if and only if the following two conditions
are satisfied.
$\beta_{t}$

(2.4)

(2.5)

$e^{g}-1\in\Psi_{1}(\hat{N})$,$A(t)=\frac{1}{2}\int_{0}^{t}|f(s)|^{2}ds+\hat{N}_{t}(e^{g}-1)+\hat{N}_{t}(e^{h}-1-h)$.

Proof. By It\^o’s formula, we have$\beta_{t}-1=\int_{0}^{t}\beta_{s-}(f(s), dW(s))+\int_{0}^{t}\int_{R^{m}}\beta_{s-}(e^{g}-1)dN$$+\int_{0}^{t}\int_{R^{m}}\beta_{s-}(e^{h}-1)d\tilde{N}$$+\frac{1}{2}\int_{0}^{t}\beta_{s-}|f(s)|^{2}ds+\int_{0}^{t}\int_{R^{m}}\beta_{s-}(e^{h}-1-h)d\hat{N}-\int_{0}^{t}\beta_{s-}dA(s)$.

Therefore if (2.4) and (2.5) are satisfied, then$\beta_{t}-1=\int_{0}^{t}\beta_{s-}(f(s), dW(s))+\int_{0}^{t}\int_{R^{m}}\beta_{s-}(e^{g}-1)d\tilde{N}$$+\int_{0}^{t}\int_{R^{m}}\beta_{s-}(e^{h}-1)d\tilde{N}$Therefore$\beta_{t}$is a localmartingale.

.

219

Representation of martingales and mathematical fifinanceis a localmartingale. We want to prove
Conversely suppose that
(2.4). Set $g^{+}=\max\{g, 0\}$ and $g^{-}=\max\{-g, 0\}$ . Then $g=g^{+}-g^{-}$We shall prove first $e^{-g^{-}}-1\in\Psi_{1}(\hat{N})$ . It holds by It\^o’s formula$\beta_{t}$$e^{-N_{t}(g^{-})}-1=-\int_{0}^{t}(1-e^{-g^{-}})e^{-N_{s-}(g^{-})}dN$ .

Since $-N_{t}(g^{-})\leq 0$ , the expectation of the above is finite and is equal to$-E$ [(dsdz)]. Therefore, $(1-e^{-g^{-}})e^{-N_{s-}(g^{-})}\in$and this implies $(1-e^{-g^{-}})\in\Psi_{1}(\hat{N})$ . Next, we have by It\^o’s
formula,$\int_{0}^{t}(1-e^{-g^{-}})e^{-N_{s-}(g^{-})}\hat{N}$$\Phi_{1}(\hat{N})$$\int_{0}^{t}\frac{d\beta_{s}}{\beta_{s-}}$$=$$\int_{0}^{t}(f(s), dW(s))+N_{t}(e^{g}-1)+\tilde{N}_{t}(e^{h}-1)$$+\frac{1}{2}\int_{0}^{t}|f(s)|^{2}ds+\hat{N}_{t}(e^{h}-1-h)-A(t)$.

The left hand side is a localmartingale. All terms except $N_{t}(e^{g}-1)$ of
the right hand side are locally integrable. Further we have $N_{t}(e^{g}-1)=$$N_{t}(e^{g^{+}}-1)+N_{t}(e^{-g^{-}}-1)$ and the last term is locally integrable. Then
$N_{t}(e^{g+}-1)$ should be locally integrable, which shows thatis also locally integrable, proving that $e^{g^{+}}-1\in\Psi_{1}(\hat{N})$ . We have thus
proved (2.4).
can beNow since (2.4) holds, the bounded variation part ofwritten as$\int_{0}^{t}(e^{g^{+}}-1)d\hat{N}$$\beta_{t}-1$$\frac{1}{2}\int_{0}^{t}\beta_{s-}|f(s)|^{2}ds+\int_{0}^{t}\beta_{s-}(e^{g}-1)d\hat{N}+\int_{0}^{t}\beta_{s-}(e^{h}-1-h))d\hat{N}-\int_{0}^{t}\beta_{s-}$It should be 0 since$\beta_{t}$dA(s).

is a localmartingale. This implies (2.5).

is a positive localmartinProof of Theorem 2.1. Suppose that
gale. Set $X(t)=\log\alpha_{t}$ . It is a semimartingale. Consider$\alpha_{t}$$P^{n}(t)=\sum_{s\leq t,1\leq|\triangle X(s)|\leq n}\triangle X(s)$,(finite sum).

It is a locally integrable process of bounded variation. There exists a continuous process of boundecd variation $C^{n}(t)$ such that $M^{n}(t)=P^{n}(t)-$$C^{n}(t)$ is a locally square integrable martingale by Doob-Meyer decomposition. Then there exists
such that $M^{n}(t)=\tilde{N}(\psi_{n})$ by
Theorem 1.1 (for locally square integrable martingales). Jump parts of$P^{n}(t)$ andcoincide. Therefore we have $P^{n}(t)=N_{t}(\psi_{n})$ . It holds$\psi_{n}\in\Psi_{2}(\hat{N})$$N_{t}(\psi_{n})$220

H. Kunita

for any $m<n$ . Then there exists
and we have$\psi_{m}=\psi_{n}1_{|\psi_{n}|\leq m}a.e.\hat{N}\times P$

that$\psi_{n}=\psi 1_{|\psi|\leq n}$$N_{t}(\psi)=\sum_{s\leq t,1\leq|\triangle X(s)|<\infty}\triangle X(s)$,$\psi$

such(finite sum).

Now set $Y(t)=X(t)-N_{t}(\psi)$ . It is a semimartingale such that
. Therefore it is a special semimartingale. Then it is decomposed
uniquely to the sum of a martingale $M(t)$ and a predictable process
of bounded variation, denoted by $B(t)$ . Further $M(t)$ is locally square
integrable so that it is written as $M(t)=\int_{0}^{t}(f(s), dW(s))+\tilde{N}_{t}(\eta)$ , where$ f\in\Phi$ andand $Y(t)$ do not. It holds $\psi\eta=0$ sincehave common jumps. Then we get the decomposition:$|\triangle Y(s)|\leq$$1$$\eta\in\Psi_{2}(\hat{N})$

(2.6)

Since

$N_{t}(\psi)$$\alpha_{t}=\exp\{\int_{0}^{t}(f(s), dW(s))+N_{t}(\psi)+\tilde{N}_{t}(\eta)+B(t)\}$$\alpha_{t}$.

is a localmartingale, we have $e^{\psi}-1\in\Psi(\hat{N})$ and$-B(t)=\frac{1}{2}\int_{0}^{t}|f(s)|^{2}ds+\hat{N}_{t}(e^{\psi}-1)+\hat{N}_{t}(e^{\eta}-1-\eta)$by the previous lemma. Now set $ g=\psi+\eta$ . Then we have $ g_{1}=\psi$ and. Therefore we get the formula (2.1). The proof is complete.

$ g_{2}=\eta$

Let $M(t)$ be a martingale. We

Proof of Theorem 1.1 (continued).

set $M^{+}=M(T)\vee 0$ and $M^{-}=(-M)\vee 0$ and define $M_{1}(t)=E[M^{+}|F(t)]$and $M_{2}(t)=E[M^{-}|\mathcal{F}(t)]$ . Then both are nonnegative martingales and$M(t)=M_{1}(t)-M_{2}(t)$ . We consider positive martingales$M_{i}(t)+\epsilon(\epsilon>0)$ . These are represented by, whereare exponential martingales. These satisfy SDE (2.2).
Now setand $\psi_{i}(t, z)=\alpha_{t-}^{i}(e^{g_{i}’(t,z)}-1)$ . Then, since$\sup_{t}\alpha_{t}<\infty a.s.$ ,. Further, we get the repand, $i=1,2$ . Thus we get the representation
resentation (1.10) for.
(1.10) whereand$M_{i,\epsilon}(t)=$$M_{i,\epsilon}(t)=M_{i,\epsilon}(0)\alpha_{t}^{i}$$\alpha_{t}^{i}=\alpha_{t}(f_{i}’, g_{i}’)$$\phi_{i}(t)=\alpha_{t-}^{i}f_{i}’(t)$$\phi_{i}\in\Phi$$\psi_{i}\in\Psi_{1,2}(\hat{N})$$M_{i,\epsilon}(t)$$\psi\in\Psi_{1,2}(\hat{N})$$\phi\in\Phi$be a positive martingale with mean 1.
Let
probability measure $Q$ by the formula$\alpha_{t}$(2.7)$Q(A)=\int_{A}\alpha_{T}dP$,

We can define a$A\in F$ .

Then $((F_{t}), Q)$ and $((F_{t}), P)$ are equivalent (mutually absolutely continuous). Conversely let $((F_{t}), Q)$ be a probability measure equivalentwith
to $((F_{t}), P)$ . Let
be the Radon-Nikodym density of$\alpha_{t}$$(\mathcal{F}_{t}, Q)$

221

Representation of martingales and mathematical fifinancerespect to $(F_{t}, P)$ . Then the stochastic process $\{\alpha_{t}, t\in[0, T]\}$ is a positive martingale with respect to $P$ . Therefore it can be represented as.
A localmartingale with respect to $((F_{t}), Q)$ is called a Qlocalmartingale. The following is an extension of Girsanov’s theorem.
$\alpha_{t}=\alpha_{t}(f, g)$

Theorem 2.3. ($c.f$. [6], Theorem 6.2) With respect to $((F_{t}), Q)$ , we
have1) $W^{f}(t):=W(t)-\int_{0}^{t}f(s)ds$ is a standard Brownian motion.
2) The compensator of $N$ is $\hat{N}^{g}(dsdz)=e^{g(s,z)}ds\nu(dz)$ , that is$:=N(dsdz)-\hat{N}^{g}(dsdz)$ is a martingale measure. Further
if belongs, the stochastic integralto$\tilde{N}^{g}(dsdz)$$\psi$$\Psi_{1,2}(\hat{N}^{g})$(2.8)$\tilde{N}_{t}^{g}(\psi):=\int_{0}^{t}\int_{R^{m}}\psi(s, z)\tilde{N}^{g}(dsdz)$

is well defined as a $Q$ localmartingale.
3) Let $X(t)$ be a $Q$ -localmartingde. Then there exists a pair of, respecpredictable processes $(\phi(t), \psi(t, z))$ belonging to andtively and $X(t)$ is represented by$\Psi_{1,2}(\hat{N}^{g})$$\Phi$(2.8)$X(t)=X(0)+\int_{0}^{t}(\phi(s), dW^{f}(s))+\tilde{N}_{t}^{g}(\psi)$.

Remark. 1) $N(dtdz)$ is no longer a Poisson random measure withrespect to $Q$ unless is a deterministic function.
.
2) Set $F_{t}(f, g)=\sigma(W_{s}^{f},\tilde{N}^{g}(dsdz);s\leq t)$ . Then it holds(2.9)
representation
in
The
is
valid
general.
The equality does not holdbut it is not clear
for localmartingale with respect to the filtration
if we have the similar representations for localmartingales with respect
to the filtration $(F_{t}(f, g))$ .
$g$$\mathcal{F}_{t}(f, g)\subset \mathcal{F}_{t}$$(\mathcal{F}_{t})$Proof. The assertions (1) and (2) are shown in [6] in the case whereis a multiplicative functional of a Markov process. We give here analternative proof. We first show that $W^{f}(t)$ is a $Q$ localmartingale. Set$X(t)=W^{f}(t)$ . Then $X(t)$ is a $Q$ -localmartingale if and only if the
product
is a $P$ -localmartingale. Note the equality$\alpha_{t}$$X(t)\alpha_{t}$(2.10)$X(t)\alpha_{t}=\int_{0}^{t}X(s-)d\alpha_{s}+\int_{0}^{t}\alpha_{s-}dX(s)+[X, \alpha]_{t}$.

The first term of the right hand side is a $P$ localmartingale. Since, we have $\int_{0}^{t}\alpha_{s-}dX(s)+[X, \alpha_{t}]=$is a, which is also a $P$ -localmartingale. Therefore$[X, \alpha]_{t}=\langle X, \alpha\rangle_{t}=\int_{0}^{t}\alpha_{s-}f(s)ds$$\int_{0}^{t}\alpha_{s-}dW(s)$$X(t)\alpha_{t}$222

H. Kunita-localmartingale or equivalently $X(t)$ is a continuous $Q$ -localmartingale.
is 0.
It holds $[X]_{t}=[W]_{t}=t$ , since the quadratic variation ofHence $X(t)=W^{f}(t)$ is a Brownian motion with respect to $Q$ .
We will next prove (2). Suppose first that $\psi(t, z)$ is bounded and, sinceis satisfied. Then it holds valid
is decomposed as $X(t)=\overline{N}_{t}(\psi)-$. Then. It holds (2.10) again. We have$P$$\int_{0}^{t}f(s)ds$$\int_{0}^{T}|\psi|e^{g}d\hat{N}<\infty$$\int_{0}^{T}|\psi|d\hat{N}<\infty$$X(t):=\tilde{N}_{t}^{g}(\psi)$$g\in\Psi_{e,2}(\hat{N})$$\int_{0}^{t}\int\psi(e^{g}-1)d\hat{N}$$\int_{0}^{t}\alpha_{s-}dX(s)+[X, \alpha]_{t}=\int_{0}^{t}\alpha_{s-}\psi d\tilde{N}$$-\int_{0}^{t}\int_{R^{m}}\alpha_{s-}\psi(e^{g}-1)d\hat{N}+\int_{0}^{t}\int_{R^{m}}\alpha_{s-}\psi(e^{g}-1)dN$,

is again a Pwhich is a $P$ -localmartingale. Consequently$Q$
localmartingale, proving thatis a -localmartingale. It.
can be extended to any
We will prove (3). Suppose first that $X(t)$ is a $Q$ -localmartingale
such that its jumps are bounded. Then $M(t):=X(t)\alpha_{t}$ is a Plocalmartingale. Since
is a $P$ -semimartingale, the product $X(t)=$is a $P$ -semimartingale. Note that jumps of $X(t)$ are bounded.
Then $X(t)$ is a $P$ special semimartingale. Then it is decomposed uniquely
as $X(t)-X(0)=N(t)+A(t)$ , where $N(t)$ is a $P$ locally square integrable martingale and $A(t)$ is a right continuous predictable process of, wherebounded variation. Now, $N(t)$ is represented by$X(t)$
asis a bounded predictable process. Then we can rewrite$X(t)\alpha_{t}$$X(t)=\tilde{N}_{t}^{g}(\psi)$$\psi\in\Psi_{1,2}(\hat{N}^{g})$$\alpha_{t}^{-1}$$M(t)\alpha_{t}^{-1}$$\int\phi dW+\int\psi d\tilde{N}$$\psi$(2.11)$X(t)=X(0)+\int_{0}^{t}(\phi(s), dW^{f}(s))+N_{t}^{g}(\psi)$$+\{\int_{0}^{t}(\phi(s), f(s))ds+\int_{0}^{t}\int_{R^{m}}\psi(e^{g}-1)d\hat{N}+A(t)\}$.

The first and the second integrals of the right hand side are both Qlocalmartingales. The last term
is a right continuous predictableprocess of bounded variation, which should be 0, since $X(t)$ is a Qmartingale. Therefore we get the representation of $X(t)$ .
The representation can be extended to any $Q$ locally square integrable martingale. Finally, the representation is valid for any Qlocalmartingale. This can be verified through getting the exponential
representation of positive $Q$ -localmartingale similarly as in Theorem 2.1.
$\{\cdots\}$

223

Representation of martingales and mathematical fifinance

It is an interesting problem to find a condition for $(f, g)$ which enis a martingale. We give here asures that the localmartingale
suﬃcient condition.
$\alpha_{t}(f, g)$

Theorem 2.4. Suppose that $ f\in\Phi$ and(2.12) $E[\exp\{$$g\in\Psi_{e.2}(\hat{N})$satisfy$\int_{0}^{T}((1+\epsilon)|f|^{2}$$+\int_{g^{+}>\delta}e^{2(1+\epsilon)g^{+}}d\nu+2(1+\epsilon)e^{2(1+\epsilon)\delta}\int_{|g|\leq\delta}g^{2}d\nu)ds\}]$$<\infty$

for some,$\epsilon>0$

and $\delta>0$ , where $g^{+}=\max(g, 0)$ . Thenmartingale.
In particular,$a.s$$\alpha_{t}(f, g)$

is a martingale if 1). and 2) $g^{+},\hat{N}(g^{+}>1)$ and$\int_{0}^{T}\int_{|g|\leq 1}|g|^{2}\hat{N}$$\alpha_{t}(f, g)$$\int_{0}^{T}|f(s)|^{2}ds$

is

$a$

is bounded(dsdz) are bounded $a.s$ .

, $=1$ , 2,
be an increasing sequence of stopping
Proof. Let$P(\tau_{n}<T)\rightarrow0$
and each stopped process
times such thatasis a martingale with mean 1. We want to prove that the above
sequence of random variables ( is fixed) is uniformly integrable. If thisis also a martingale. For thisproperty is verified, the limit process
purpose it is suﬃcient to prove that
holds for some$p>1$ . By a direct computation we can show that,
under the condition (2.12). Details are omitted.
$\tau_{n}$$n$$\ldots$$n$$\rightarrow\infty$$\alpha_{t\wedge\tau_{n}}$$t$$\alpha_{t}$$\sup_{n}E[\alpha_{t\wedge\tau_{n}}^{p}]<\infty$$\sup_{n}E[\alpha_{t\wedge\tau_{n}}^{1+\epsilon}]<\infty$\S 3. Processes with jumps and equivalent martingale measures
be a $d\times m$ matrix valued predictable process,$b(t)=(b^{i}(t))$ be a -vector predictable process and $v(t, z)=(v^{i}(t, z))$ be
a -vector predictable process continuous in $z\in R^{m}$ , which satisfy the
integrability conditionLet$\sigma(t)=(\sigma_{j}^{i}(t))$
$d$

$d$

(3.1)

$\int_{0}^{T}|\sigma(t)|^{2}+|b(t)|dt<\infty$,$\int_{0}^{T}\int_{|z|\leq 1}|v(s, z)|^{2}ds\nu(dz)<\infty$.

We shall consider a -dimensional stochastic process with jumps defined
by
$d$

(3.2)

$\xi_{t}=\int_{0}^{t}\sigma(t)dW(t)+\int_{0}^{t}b(t)dt$$+\int_{0}^{t}\int_{|z|\leq 1}v(t, z)\tilde{N}(dtdz)+\int_{0}^{t}\int_{|z|>1}v(t, z)N(dtdz)$

,

224

H. Kunita

where $W(t)$ is a $m$-dimensional standard Brownian motion and $N(dtdz)$. To make the problem
is a Poisson counting measure on $[0, T]$
simple, we assume $d=m$ in this paper.
An equivalent probability measure $Q$ such that is a -vector localis called an equivalent martingale
martingale with respect tomeasure. We denote by the set of all equivalent martingale measuresand by the set of all $(f, g)$ such that $\alpha_{t}(f, g)dP\in\Gamma$ . We shall characterize all equivalent martingale measures of a given process by meansof the pair $(f, g)$ .
$\times R^{m}$$\xi_{t}$$d$$((\mathcal{F}_{t}), Q)$$\Gamma$$\overline{\Gamma}$$\xi_{t}$Theorem 3.1. Let $((F_{t}), Q)$ be an equivalent probability measure
be the density such that $dQ=\alpha_{t}(f, g)dP$ , where $ f\in\Phi$and letand $g\in\Psi_{e,2}(\hat{N})$ . Then the stochastic process
defined by (3.2) is$Q$ -localmartingale
if and only if $v(s, z)(e^{g(s,z)}-1_{\{|z|\leq 1\}})\in\Psi_{1}(\hat{N})$ and$\alpha_{t}(f, g)$$a$$\xi_{t}$

(3.3)
$a.e$

.

$b(s)+\sigma(s)f(s)+\int_{R^{m}}v(s, z)(e^{g(s,z)}-1_{\{|z|\leq 1\}})\nu(dz)=0$$\lambda\otimes P$, where$\lambda$,

is the Lebesgue measure.

Proof. In vector notation, we have by (2.2) and (3.2),$\xi_{t}\alpha_{t}$$=$$\int_{0}^{t}\xi_{s-}d\alpha_{s}+\int_{0}^{t}\alpha_{s-}d\xi_{s}+[\xi, \alpha]_{t}$$=$a $1oca1martinga1e+\int_{0}^{t}\alpha_{s-}b(s)ds+\int_{0}^{t}\alpha_{s-}\sigma(s)f(s)ds$$+\int_{0}^{t}\int_{R^{m}}\alpha_{s-}v(s, z)(e^{g(s,z)}-1_{\{|z|\leq 1\}})N(dsdz)$.

If it is a -vector localmartingale, the integrand with respect to $N(dsdz)$
should be integrable with respect to (dsdz) and the equality
(3.4)
$d$

$\hat{N}$$\alpha_{s-}b(s)+\alpha_{s-}\sigma(s)f(s)+\alpha_{s-}\int_{R^{m}}v(s, z)(e^{g(s,z)}-1_{\{|z|\leq 1\}})\nu(dz)=0$,

holds $a.e$ . (Theorem 1.1). Then we have (3.3), since $\inf_{s}\alpha_{s-}>0a.s$ .
The converse will be shown similarly. The proof is complete.

An equivalent martingale measure $Q^{0}=\alpha_{t}(f^{0}, g^{0})dP$ is said to be.
standard if is a locally square integrable martingale with respect to
We will show the existence of such an equivalent martingale measure.
$Q^{0}$$\xi_{t}$is invertible andand $b(t)$ are
Lemma 3.2. Assume that
. Then there exists a standard equivalent martingale
bounded $a.e$ .
, thereandmeasure. Further, for any given pair of$\sigma(t)^{-1}$$\sigma(t)$$\lambda\otimes P$$\phi\in\Phi$$\psi\in\Psi_{1,2}(\hat{N})$Representation of martingales and mathematical fifinance

225

exists a standard equivalent martingale measure $Q^{0}=\alpha_{t}(f^{0}, g^{0})dP$ such
that
(3.5)

$M^{Q^{0}}(t):=\int_{0}^{t}(\phi(s), dW^{f^{o}}(s))+\tilde{N}_{t}^{g^{0}}(\psi)$

is well defined as a locally square integrable martingale with respect to$Q^{0}$.

Proof. We will show that there exists a predictable pair $(f^{0}(s), g^{0}(s,z))$of satisfying$\tilde{\Gamma}$(3.6)$\int_{0}^{T}\int_{R^{m}}\{|v(s, z)|^{2}+|\psi(s, z)|^{2}\}e^{g^{0}(s,z)}ds\nu(dz)<\infty$.

For each $s\in[0, T]$ , set $E(s)=E(s, \omega)=\{z:|\psi(s, z)|>1\}\cup\{|z|>1\}$ .
Then $\int_{0}^{T}i/(E(s))ds$ $<\infty$ . Take first a nonpositive predictable process$g’(s, z)$ supported by $E(s)$ , $|g’(s, z)|>1$ on $E(s)$ and$\int_{E(s)}\{1+|v(s, z)|+|v(s, z)|^{2}+|\psi(s, z)|^{2}\}e^{g’(s,z)}\nu(dz)$is bounded in $(s, \omega)a.e$ . Take next a bounded predictable process $g’’(s, z)$
supported by $E(s)^{c}$ , $|g’’(s, z)|<1$ and $\int_{E(s)^{c}}(1+|v(s, z)|)|g’’(s, z)|\nu(dz)$is bounded in $(s, \omega)a.e$ . Define $g^{0}=g’1_{E}+g’’1_{E^{c}}$ . Then
and$\int_{0}^{T}\int_{E(s)}(|v|^{2}+|\psi|^{2})e^{g^{O}}ds\nu(dz)<\infty$

Since$\int_{0}^{T}\int_{|z|\leq 1}|v|^{2}ds\nu(dz)<\infty$

and

,

$a.s$

$g^{0}\in\Psi_{e,2}(\hat{N})$.

$\int_{0}^{T}\int_{R^{m}}|\psi_{2}|^{2}ds\nu(dz)<\infty$for$\psi_{2}=$, we have $\int_{0}^{T}\int_{E(s)^{c}}(|v|^{2}+|\psi|^{2})e^{g^{0}}ds\nu(dz)<\infty$ , $a.s$ . Therefore(3.6) is satisfied.
The process $a(s)=\int_{R^{m}}v(s, z)(e^{g^{0}(s,z)}-1_{\{|z|\leq 1\}})\nu(dz)$ is well defined since$\psi 1_{|\psi|\leq 1}$(3.7)$|a(s)|\leq\int_{E(s)}|v(s, z)(e^{g’(s,z)}-1_{|z|\leq 1})|\nu(dz)$$+\int_{E(s)^{c}}|v(s, z)(e^{g^{ll}}-1_{|z|\leq 1})|\nu(dz)$$\leq\int_{E(s)}|v|e^{g’}\nu(dz)+\int_{E(s)^{c}}|v||g’’|\nu(dz)$bounded in$(s, \omega)a.e$

.

226

H. Kunita

by $b(s)+\sigma(s)f^{0}(s)+a(s)=0$ . The pair
Then we can definesatisfies (3.3). Further, it satisfies conditions $(1),(2)$ of Theorem
2.4. Indeed, we took
so that it satisfies (2). By the estimation (3.7),$|a(s)|$ is bounded $a.s$ . Since $|b(s)|$ is bounded andis uniformly
positive definite, $|f^{0}(s)|$ is also bounded $a.s$ . Thus $f^{0}(s)$ satisfies (1) of
the theorem. Then
is a martingale.
Let $Q^{0}=\alpha_{T}(f^{0}, g^{0})dP$ . We will show thatis a locally square
integrable martingale with respect to. Observe (3.2) and (3.3). Thenis written as$f^{0}(s)$$(f^{0}, g^{0})$$g^{0}$$\sigma(s)\sigma(s)^{T}$$\alpha_{t}(f^{0}, g^{0})$$\xi_{t}$$Q^{0}$$\xi_{t}=(\xi_{t}^{1}, \ldots, \xi_{t}^{d})$$\xi_{t}=\int_{0}^{t}\sigma(s)dW^{f^{o}}(s)+\int_{0}^{t}\int_{R^{m}}v(s, z)\tilde{N}^{9^{0}}$The bracket process with respect to$Q^{0}$

is given by a(dsdz).
$d\times d$matrix$(\langle\xi^{i}, \xi^{j}\rangle_{t}^{Q^{0}})=\int_{0}^{t}\sigma(s)\sigma(s)^{T}ds+\int_{0}^{t}\int_{R^{m}}v(s, z)v(s, z)^{T^{0}}\hat{N}^{g}$(dsdz).

It is finite $a.s$ . This proves that is a locally square integrable martingale.
Finally, $M^{Q^{O}}(t)$ of (3.5) is well defined as a locally square integrable
martingale, becauseby (3.6). The proof is complete.
$\xi_{t}$$\psi\in\Psi_{2}(\hat{N}^{g^{0}})$We will fix the equivalent martingale measureof Lemma 3.2. Set$\hat{f}=f-f^{0},\hat{g}=g-g^{0}$ . Thenof Theorem 3.1 is decomposed tothe product of two exponential semimartingales;$Q^{0}$$\alpha_{t}(f, g)$(3.8)$\alpha_{t}(f, g)=\alpha_{t}(f^{0}, g^{0})\alpha_{t}^{0}(\hat{f},\hat{g})$

,

(3.9)
$\alpha_{t}^{0}(\hat{f},\hat{g})=\exp\{\int_{0}^{t}(\hat{f}(s), dW^{f^{o}}(s))-\frac{1}{2}\int_{0}^{t}|\hat{f}(s)|^{2}ds$$+N_{t}(\hat{g}_{1})-\hat{N}_{t}^{g^{0}}(e^{\hat{9}1}-1)+\overline{N}_{t}^{g^{O}}(\hat{g}_{2})-\hat{N}_{t}^{g^{O}}(e^{\hat{g}_{2}}-1-\hat{g}_{2})\}$

Since $dQ=\alpha_{T}(f, g)dP$ and $dQ^{0}=\alpha_{T}(f^{0}, g^{0})dP$ , we have $dQ=$. Hence $Q$ is an equivalent martingale measure with respect toandis its density process. Then ,andare all localmartingales with respect to.
Conversely if $Q$ is an equivalent martingale measure with respect. The density process
of $Q$ with respect tois represented by
the set of all such density processes(3.9). We denote by
and by$\alpha_{T}^{0}(\hat{f},\hat{g})dQ^{0}$$Q^{0}$$\alpha_{t}^{0}(\hat{f},\hat{g})$$\xi_{t}$$\xi_{t}\alpha_{t}^{0}(\hat{f},\hat{g})$$\alpha_{t}^{0}(\hat{f},\hat{g})$$Q^{0}$$Q^{0}$$\alpha_{t}^{0}$$Q^{0}$$\hat{\Gamma}^{0}$$\alpha_{t}^{0}$

.

227

Representation of martingales and mathematical fifinanceis the set ofsuch that these are all locally square integrable.
martingales with respect tobe the set of all locally square integrable martingalesLet$M(t)$ with $M(0)=0$ with respect to. Then the -vector process. Further,are orthogonal with respectandbelongs to. We claim;, iftois locally square integrable with respect to$\alpha_{t}^{0}\in\hat{\Gamma}^{0}$$\hat{\Gamma}_{2}^{0}$$Q^{0}$$\mathcal{M}_{loc}^{2}(Q^{0})$$d$$Q^{0}$$\mathcal{M}_{loc}^{2}(Q^{0})$$Q^{0}$$\xi_{t}$$\xi_{t}$$\alpha_{t}^{0}-1$$Q^{0}$$\alpha_{t}^{0}$is invertible andand $b(t)$ are
Lemma 3.3. Assume that. Let
be a standard equivalent martingale meabounded $a.e$ .
, we have the orthogonal decomposition ofsure. Then, with respect to$\sigma(t)^{-1}$$\sigma(t)$$\lambda\otimes P$$Q^{0}$$Q^{0}$$\mathcal{M}_{loc}^{2}(Q^{0})$.

(3.10)$\mathcal{M}_{loc}^{2}(Q^{0})=\mathcal{L}(\xi_{t}^{1}, \ldots, \xi_{t}^{d})\oplus \mathcal{L}(\alpha_{t}^{0}-1;\alpha_{t}^{0}\in\hat{\Gamma}_{2}^{0})$.

and let $M$ be any element of
Proof. Let
sented by $M=\int(\phi, dW^{f^{o}})+\tilde{N}^{g^{O}}(\psi)$ . Since it is orthogonal to,with respect to$\mathcal{K}=\mathcal{L}(\xi_{t}^{1}, \ldots, \xi_{t}^{d})^{\perp}$$\mathcal{K}$$\xi_{t}^{1}$

repre,$\ldots$,$\xi_{t}^{d}$$Q^{0}$$\langle\xi^{i}, M\rangle_{t}^{Q_{0}}=\int_{0}^{t}(\sigma^{i}(s), \phi(s))ds+\int_{0}^{t}(\int v^{i}(s, z)\psi(s, z)e^{g^{O}}\nu(dz))ds=0$$i=1$ ,

Therefore, setting $\phi(t)=(\phi^{1}(t), \ldots, \phi^{d}(t))$ and $v(t,z)=(v^{1}(t,z),$we get$\sigma(t)\phi(t)+\int_{R^{d}}\psi(t, z)v(t, z)e^{g^{O}(t,z)}\nu(dz)=0$,$\forall t$$\ldots$, .
$d$$\ldots$, $v^{d}(t,z))$ ,.

We will show that$\mathcal{H}:=\{\int(\phi, dW^{f^{o}})+\tilde{N}^{g^{0}}(\psi)\in \mathcal{K};\psi$

are $bounded\}$is dense in . Let $M=\int(\phi, dW^{f^{0}})+\overline{N}^{g^{O}}(\psi)$ be any element of .
We define trancated functions by $\psi_{n}=(\psi\wedge n)\vee(-n)$ . Next define dvector functions by $\phi_{n}(t)=-\sigma(t)^{-1}\int\psi^{n}(t, z)v(t, z)e^{g^{0}(t,z)}\nu(dz)$ . Thenit holds$\mathcal{K}$$\mathcal{K}$$\sigma(t)\phi^{n}(t)+\int\psi^{n}(t, z)v(t, z)e^{g^{O}(t,z)}\nu(dz)=0$,$\forall t$.

belongs to . Further, since,holds valid as$\{M^{n}\}$converges to $M$ with respect. Therefore the sequenceasto the topology of. We have thus shown that is dense in .

Therefore$M^{n}=\int(\phi^{n}, dW^{f^{o}})+\overline{N}^{g^{0}}(\psi^{n})$$\int_{0}^{T}\int|\psi_{n}-\psi|^{2}e^{g^{0}}\nu(dz)ds\rightarrow 0$$n$$\mathcal{K}$$ n\rightarrow\infty$$\int_{0}^{T}|\phi^{n}-\phi|^{2}ds\rightarrow 0$$\rightarrow\infty$$\mathcal{M}_{loc}^{2}(Q^{0})$$\mathcal{H}$,$\mathcal{K}$H. Kunita

228

which is bounded from the below.
Let $J$ be the set of all
Then we have
} . Further it holds$\{M^{\tau}(t);M\inH\}\subset\mathcal{L}(J)$, where are stopping. Indeed, we have$M^{\tau}(t):=M(t\wedge\tau)$
are bounded localmartingales. We
times such thathave thus proved$M\in \mathcal{K}$$\mathcal{L}(J)=\mathcal{L}(\alpha_{t}^{0}-1;\alpha_{t}^{0}\in\hat{\Gamma}_{2}^{0}$$\mathcal{L}(J)\supset$$\cdot$$\mathcal{L}(7\{)$$\tau$$\mathcal{L}(\xi_{t}^{1}, \ldots, \xi_{t}^{d})^{\perp}=\mathcal{K}=\mathcal{L}(H)\subset \mathcal{L}(J)=\mathcal{L}(\alpha_{t}^{0}-1;\alpha_{t}^{0}\in\hat{\Gamma}_{2}^{0})$.

The proof is complete.

We are now in a position of stating a main result of the paper.
Theorem 3.4. Assume thatis invertible andand $b(t)$. If $X(t)$ is a supermartingale for any equivalent
are bounded $a.e$ .
martingale measure $Q$ , then it is represented by$\sigma(t)^{-1}$$\sigma(t)$$\lambda\otimes P$(3.11)$X(t)=X(0)+\int_{0}^{t}(\varphi(s), d\xi_{s})-A(t)$ .

Here, $A(t)$ is a predictable increasing process andis a predictableprocess such that $\sigma(s)\varphi(s)\in\Phi$ and $(\varphi(s), v(s, z))\in\Psi_{1,2}(\hat{N}^{g})$ for any$\varphi(s)$$(f, g)\in\tilde{\Gamma}$.

If $X(t)$ is a localmartingale for any equivalent martingale measure,then it is represented by(3.12)$X(t)=X(0)+\int_{0}^{t}(\varphi(s), d\xi_{s})$.

Proof. For each $ Q\in\Gamma$ , the supermartingale $X(t)$ is decomposed
as $X(0)+M^{Q}(t)-A^{Q}(t)$ , where $M^{Q}(t)$ is a $Q$ localmartingale with$M^{Q}(0)=0$ and $A^{Q}(t)$ is a natural $(=predictable)$ increasing process,by Doob-Meyer decomposition. The $Q$ localmartingale $M^{Q}(t)$ is represented by $M^{Q}(t)=\int_{0}^{t}(\phi(s), dW^{f}(s))+\tilde{N}_{t}^{g}(\psi)$ . We will show that thekerneldoes not depend on the choice of $Q$ . Let
be anotherequivalent martingale measure. Thenis represented by $M^{Q^{*}}=$. Since $M^{Q}(t)-A^{Q}(t)=M^{Q^{*}}(t)-A^{Q^{*}}(t)$ ,we have$(\phi, \psi)$$Q^{*}$$M^{Q^{*}}$$\int(\phi^{*}(s), dW^{f^{*}}(s))+\tilde{N}^{g^{*}}(\psi^{*})$$\tilde{N}_{t}^{g^{*}}(\psi^{*})-\tilde{N}_{t}^{g}(\psi)=(\int_{0}^{t}(\phi(s), dW^{f}(s))-\int_{0}^{t}(\phi^{*}(s), dW^{f^{*}}(s)))$$-(A^{Q}(t)-A^{Q^{*}}(t))$ .

The right hand side is a predictable process, so that it has no common
jumps with the Poisson random measure $N(dtdz)$ . So both sides of the229

Representation of martingales and mathematical fifinanceabove can not have jumps. This shows $\psi=\psi^{*}a.e.\hat{N}\otimes P$ . Hence theright hand side should be a predictable process of bounded variation.
Therefore $\int_{0}^{t}(\phi(s), dW^{f}(s))-\int_{0}^{t}(\phi^{*}(s), dW^{f^{*}}(s))$ is also a predictable. We haveprocess of bounded variation, which shows $\phi=\phi^{*}a.e$ .
.
for any
and$A^{Q}=A^{Q^{0}}$in the case where both $Q$ andWe want to prove
are standard equivalent martingale measures such that. Comparing two equations for $M^{Q}(t)$ and $M^{Q^{0}}(t)$ , $M^{Q^{O}}(t)$ canbe written as$\lambda\otimes P$$\psi\in\Psi_{1,2}(\hat{N}^{g})$$\phi\in\Phi$$(f, g)\in\tilde{\Gamma}$$Q^{0}$$\psi\in\Psi_{2}(\hat{N}^{g})\cap$$\Psi_{2}(\hat{N}^{g^{0}})$$M^{Q^{O}}(t)=M^{Q}(t)+\int_{0}^{t}(\phi(s),\hat{f}(s))ds$$+\int_{0}^{t}\int_{R^{m}}\psi(s, z)(e^{\hat{g}(s,z)}-1)e^{g^{O}(s,z)}ds\nu(dz)$where $\hat{f}=f-f^{0}$ and $\hat{g}=g-g^{0}$ , because
we have$\psi$,$e^{\hat{g}}-1\in\Psi_{2}(\hat{N}^{g^{0}})$,. Therefore$A^{Q}(t)=\int_{0}^{t}\{(\phi,\hat{f})+\int_{R^{m}}\psi(e^{\hat{g}}-1)e^{g^{0}}d\nu\}ds+A^{Q^{0}}(t)$.

We claim(3.13)$(\phi,\hat{f})+\int_{R^{\mathfrak{m}}}\psi(e^{\hat{g}}-1)e^{g^{0}}d\nu=0$,, in the case where $g\leq g^{0}$ or equivalently
the case, then either the set

$a.e$

.

$\lambda\otimes P$$\hat{g}\leq 0$$F=\{(s, \omega);(\phi,\hat{f})+\int_{R^{m}}\psi(e^{\hat{g}}-1)e^{g^{O}}d\nu>0\}$. If it is notor$F’=\{(s, \omega);(\phi,\hat{f})+\int_{R^{m}}\psi(e^{\hat{g}}-1)e^{g^{0}}d\nu<0\}$. Suppose that $F$ is of positive
is of positive measure with respect tomeasure. We define $(f’, g’)$ by $f’=f^{0}-n\hat{f}1_{F}$ and $g’=g^{0}+\log\{1-$$n(e^{\hat{g}}-1)1_{F}\}$ . Then it holds $e^{\hat{g}^{l}}-1=-n(e^{\hat{g}}-1)1_{F}$ , where $\hat{g}’=g’-g^{0}$ .
is a positive localmartingale
Set $\hat{f}’=f’-f^{0}$ . Then
is a localmartingale with respect towith respect to. Further. Indeed, equalities$\lambda\otimes P$$\alpha_{t}’:=\alpha_{t}^{0}(\hat{f}’,\hat{g}’)$$Q^{0}$$\xi_{t}\alpha_{t}’$$Q^{0}$$\langle\xi_{t}^{i}, \alpha_{t}’\rangle_{t}^{Q^{0}}=-n\int_{0}^{t}\alpha_{s-}’1_{F}\{(\sigma^{i},\hat{f})+\int_{R^{m}}v^{i}(e^{\hat{g}}-1)e^{g^{0}}\nu(dz)\}ds=0$$i=1$ ,$\ldots$

,

$d$

,

230

H. Kunita

hold valid sinceand.
are orthogonal with respect to
Let $\{\tau_{k}, k=1,2, \ldots\}$ be an increasing sequence of stopping times suchthat $P(\tau_{k}<T)\rightarrow 0$ as
isand each stopped process-martingale. Define a sequence of probability measures
bya. Then eachis an equivalent martingale measureprocess.
stoppedfor the
Then the stopped process
is asupermartingale with respect tofor each . Its Doob-Meyer decomposition is represented by$\xi_{t}^{i}$$\alpha_{t}^{0}(\hat{f},\hat{g})$$Q^{0}$$ k\rightarrow\infty$$\alpha_{t\wedge\tau_{k}}’$$Q^{0}$$Q_{k}’$$dQ_{k}’=\alpha_{\tau_{k}}’dQ^{0}$$Q_{k}’$$X(t\wedge\tau_{k})$$\xi_{t\wedge\tau_{k}}$$k$$Q_{k}’$$X(t\wedge\tau_{k})=\int_{0}^{t\wedge\tau_{k}}(\phi(s), dW^{f’})+\tilde{N}_{t\wedge\tau_{k}}^{g’}(\psi)-A^{\alpha’}(t\wedge\tau_{k})$where$A^{\alpha^{l}}(t)$$k=1,2$ ,,$\ldots$

is a suitable predictable increasing process. It satisfies$A^{\alpha’}(t)=-n\int_{0}^{t}1_{F}\{(\phi,\hat{f})+\int_{R^{m}}\psi(e^{\hat{g}}-1)e^{g^{0}}d\nu\}ds+A^{Q^{O}}(t)$.

This makes a contradiction since the right hand side is negative for
suﬃciently large . Therefore we get $A^{Q}(t)=A^{Q^{O}}(t)$ .
Now if $F’$ is of positive measure instead of the set $F$ , interchange
the role ofand $Q$ in the above discussion. Then we get the sameconclusion. Further in the case where $g\geq g^{0}$ , we get the same equality(3.13) by interchanging the role ofand $Q$ .
We have thus seen that $M^{Q}(t)=M^{Q^{0}}(t)$ holds for any standard
equivalent martingale measure $Q$ such that its density processwithrespect tois a locally square integrable martingale and $g\leq g^{0}$ or$g\geq g^{0}$be the set of allis satisfied. Let
such that $Q$
is a standard equivalent martingale measure andwith$M^{Q^{0}}(t)$. Thenis orthogonal to any element of$n$$Q^{0}$$Q^{0}$$\alpha_{t}^{0}$$Q^{0}$$\alpha_{t}^{0}(\hat{f},\hat{g})\in\Gamma_{2}^{0}$$\overline{\Gamma}_{2}^{0}$$dQ=\alpha^{0}(\hat{f},\hat{g})dP^{0}$$\psi\in\Psi_{2}(\hat{N}^{g})$$N$ $=\{\alpha_{t}^{0}(\hat{f},\hat{g})-1;\alpha^{0}(\hat{f},\hat{g})\in\tilde{\Gamma}_{2}^{0},\hat{g}\leq 0$or$\hat{g}\geq 0\}$,. Observe thatwith respect to. Thenby the decomposition forwe see that $M^{Q^{0}}(t)$ belongs tomula (3.10). Then it is represented bywith respect to. Setting $A(t)=A^{Q^{O}}(t)$ , we get the decomposition formula (3.11).
Further, representation (3.11) should hold valid for any $Q$ of . Then,comparing this with the representation of Theorem 2.3,.
and $(v(s, z)$ ,The second assertion of the theorem follows immediately from theabove discussion by setting $A^{Q^{0}}(t)=A^{Q}(t)\equiv 0$ .
$Q^{0}$$\mathcal{L}(N)=\mathcal{L}(\alpha_{t}^{0}-1;\alpha_{t}^{0}\in\Gamma_{2}^{0})$$\mathcal{L}(\xi_{t}^{1}, \ldots, \xi_{t}^{d})$$\int_{0}^{t}(\varphi(s), d\xi_{s})$$Q^{0}$$\Gamma$$\sigma(s)\varphi(s)\in\Phi$$\varphi(s))\in\Psi_{1,2}(\hat{N}^{g})$Representation of martingales and mathematical fifinance231

Applications to mathematical financeWe consider a simple market model, where the return process isof (3.2) and the interest rate $r(t)$ isgiven by a stochastic process
identically 0. Let
be a predictable process called a strategy or portfolio and be a right continuous predictable increasing process called acumulative consumption process. A stochastic process $X(t)=X^{x,\pi,C}(t)$defined by$\xi_{t}$$\pi(t)$$C_{t}$$X(t)=x+\int_{0}^{t}(\pi_{s}, d\xi_{s})-C(t)$is called a wealth process. We introduce admissible classes for pairs of
portofolios and consumptions. Let $x>0$ . We denote by $A^{+}(x)$ the set ofthe pair $(\pi, C)$ such that $X^{x,\pi,C}(t)\geq 0$ holds $a.s$ . for any $0\leq t\leq T$ . We
denote by $A^{-}(-x)$ the set of the pair $(\pi, C)$ such that $X^{-x,\pi,C}(t)\leq 0$
for any $0\leq t\leq T$ .
-measurable ranA Europian contingent claim $Y$ is a nonnegative
alwaysattainable,
isnotsince thedom variable. The contingent claim
model is not complete due to jumps of the return process. We shallstudy the upper and lower hedging price. The upper hedging price andlower hedging price of the contingent claim $Y$ are defined respectivelyby$\mathcal{F}_{T}$$h_{up}=\inf\{x\geq 0$;

there exists $(\pi, C)\in A^{+}(x)$ such that $X^{x,\pi,C}(T)\geq Ya.s.\}$$h_{low}=\sup\{x\geq 0$;

there exists $(\pi, C)\in A^{-}(-x)$ such that $X^{-x,\pi,C}(T)\geq-Ya.s.\}$is uniformly positive defifinite,
Theorem 3.5. Assume that$v(t, z)$ is greater than -1 and $v(t, z)\neq 0a.e$ .
. Let $Y$ be an
Europian contingent claim. We have$\sigma(t)\sigma(t)^{T}$$\lambda\times\nu\times P$(3.14)$h_{up}$(3.15)$h_{low}$

$=$

$=$

$\sup_{Q\in\Gamma}E_{Q}[Y]=:h$$\inf_{Q\in\Gamma}E[Y]=:f$,.

is finite (resp.
is positive), there exists a pair $(\pi, C)\in A^{+}(h)$(resp. ( , $C’)\in A^{-}(-f)$ ) such that

If

$h$

$f$

$\pi’$(3.16)$X^{h,\pi,C}(t)$(3.17)$-X^{-f^{\pi’,C’}}’(t)$$ess\sup_{Q\in\Gamma}E_{Q}[Y|\mathcal{F}_{t}]$,$ess\inf_{Q\in\Gamma}E_{Q}[Y|F_{t}]$

,

$=$

$=$

232

H. Kunita

holds for any . In particular, $X^{h,\pi,C}(T)=Y$ and $X^{-f,\pi’,C’}(T)=-Y$,
$t$

$a.s$

.

We consider the upper hedging price only. Set $h=$
. We want to prove $h=h_{up}$ . We first show $h\leq h_{up}$ .
The inequality is obvious if $ h_{up}=\infty$ . If $ h_{up}<\infty$ , let be an arbitrary
appearing in the defintion ofelement in the set. Then there exists a pair $(\pi, C)$ of $A^{+}(x)$ such that $X^{x,\pi,C}(T)\geq Y$ . Then $X^{x,\pi,C}(t)$ is a
. Therefore, $E_{Q}[Y]\leq E_{Q}[X^{x,\pi,C}(T)]\leq x$
supermartingale for any. Then we have $h\leq x$ , so that we have $h\leq h_{up}$ .
holds for any
In order to prove the reverse inequality $h\geq h_{up}$ , it is suﬃcient
to construct $(\pi, C)\in A^{+}(h)$ such that $X^{h,\pi,C}(t)=X(t)$ , where $ X(t)\equiv$$ess\sup_{Q\in\Gamma}E_{Q}[Y|F_{t}]$ . It is known that the process $X(t)$ is a supermartingale for any $Q$ . We shall apply Theorem 3.4 to the return process .
Then $X(t)$ admits the decomposition (3.11) by Theorem 3.4. This implies $X(t)=X^{h,\pi,C}(t)$ , by setting
and $A(t)=C(t)$ . It is clearly$a.s$$t\in(0,T]$. for any
nonnegative. Therefore $(\pi, C)$ belongs to $A^{+}(h)$ .
Proof.

$\sup_{Q\in\Gamma}E_{Q}[Y]$$x$$\{\cdots\}$$h_{up}$$ Q\in\Gamma$$ Q\in\Gamma$$\xi_{t}$$\varphi=\pi$

References[1] H. H\"ollmer, Y. Kabanov, Optional decomposition and Lagrange multpliers, Finance and Stochastics, 2(1998), 69-81.
[2] K. It\^o, Spectral type of the shift transformation of diﬀerential processes
with stationary increments, Trans. Amer. Math. Soc. 81(1956), 253263.
[3] Yu.M.Kabanov, Ch.Strieker, Hedging of contingent claims under transaction costs, 2002 (preprint).
Karatzas,[4] I.
Lectures on the mathematics of finance, CRM 8, AMS, 1997.
[5] I. Karatzas, S.E. Shreve, Methods of Mathematical Finance, Springer,
1998.
[6] H. Kunita, S. Watanabe, On square integrable martingales, Nagoya Math.
J. 30(1967), 209-245.
[7] P.A. Meyer, Un cours sur integrates stochastiques, Seminaire Proba. X,
Lecture Notes in Math. 511, 246-400, Springer, Berlin Heidelberg New
York, 1976.
Department of Mathematical Science,
Nanzan University,
Seto, A ichi 489-0863,
JapanAdvanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 233-246

Stochastic Newton Equation
with Reflecting Boundary Condition
Shigeo Kusuoka

Introduction\S 1.

with a smooth boundary and
Let $D$ be a bounded domain in$n(x)$ , $x\in\partial D$ , be an outer normal vector. Let:, $i,j=$, be smooth functions such that $a^{ij}(x)=a^{ji}(x)$ , $x\in R^{d}$ . Also,,:, $i=1$ ,, be bounded measurable functions. We
let,such thatassume that there are positive constants$R^{d}$$R^{d}\rightarrow R$$a^{ij}$$1$$\ldots d$$b^{i}$$R^{2d}\rightarrow R$$\ldots d$$C_{1}$$C_{0}$$C_{0}|\xi|^{2}\leq\sum_{i,j=1}^{d}a^{ij}(x)\xi_{i}\xi_{j}\leq C_{1}|\xi|^{2}$

Let$L_{0}$,$x$, $\xi\in R^{d}$ .

be a second order linear diﬀerential operator in$R^{2d}$given by

$L_{0}=\sum_{i=1}^{d}v^{i}\frac{\partial}{\partial x^{i}}+\frac{1}{2}\sum_{i,j=1}^{d}a^{ij}(x)\frac{\partial^{2}}{\partial v^{i}\partial v^{j}}+\sum_{i=1}^{d}b^{i}(x, v)\frac{\partial}{\partial v^{i}}$Let $\tilde{W}^{d}=C([0, \infty);R^{d})\times D([0, \infty);R^{d})$ . Now let :be a smooth map satisfying the following.
(i):.
is linear for all
and $v\in T_{x}(\partial D)$ , i.e., $\Phi(v, x)=v$ if(ii) $\Phi(v, x)=v$ for any, $v\in R^{d}$ and . $n(x)=0$ .
.
(iii) $\Phi(\Phi(v, x),$ $x)=v$ for all $v\in R^{d}$ and.
(iv) $\Phi(n(x), x)\neq n(x)$ for any
The main theorem in the present paper is the following.
$\Phi$$\Phi(\cdot, x)$$R^{d}\rightarrow R^{d}$$R^{d}\times\partial D\rightarrow R^{d}$$x\in\partial D$$x\in\partial D$$x\in\partial M$$v$$x\in\partial D$$x\in\partial D$. Then there exists a unique
Theorem 1. Let
satisfying the following conditions.
probability measure over
(1) $\mu(w(0)=(x_{0}, v_{0}))=1$ .
(2) $\mu(w(t)\in D^{c}\times R^{d}, t \in[0, \infty))=1$ .
$(x_{0}, v_{0})\in(\overline{D})^{c}\times R^{d}$$\tilde{W}^{d}$$\mu$

Received April 4, 2003.

234

S. Kusuoka(3) For any, $\{f(w(t))-\int_{0}^{t}L_{0}f(w(s))ds;t\geq 0\}$ isa martingale under $\mu(dw)$ .
(4) $\mu(1_{\partial D}(x(t))(v(t)-\Phi(v(t-), x(t)))=0$ for all $t\in[0, \infty))=1$ .
Here $w(\cdot)=(x(\cdot), v(\cdot))\in\tilde{W}^{d}$ .
$f\in C_{0}^{\infty}((\overline{D})^{c}\times R^{d})$Now let us think of the following Stochastic Newton equation$dX_{t}^{\lambda}$$=$$V_{t}^{\lambda}dt$$dV_{t}^{\lambda}$$=$$\sigma(X_{t}^{\lambda})dB(t)+(b(X_{t}^{\lambda}, V_{t}^{\lambda})-\lambda\nabla U(X_{t}^{\lambda}))dt$$X_{0}^{\lambda}=x_{0}$,$V_{0}^{\lambda}=v_{0}$.

Here $B(t)$ is a -dimensional Brownian motion,,:
is a bounded Lipschitz continuous function, and $ U\in$
$d$

$b$

$\sigma\in C^{\infty}(R^{d};R^{d})$$R^{2d}\rightarrow R^{d}$$C_{0}^{\infty}(R^{d})$.

We assume the following also.
(A-1) There are positive constants$C_{0}$,$C_{1}$$C_{0}|\xi|^{2}\leq|\sigma(x)\xi|^{2}\leq C_{1}|\xi|^{2}$such that,$x$, $\xi\in R^{d}$ .

(A-2) Let $D=\{x\in R^{d};U(x)>0\}$ . Then there $are\in 0>0$ ,and a non-increasing
function :satisfying thefollowing.
, if and only if $U_{0}(x)=0$ and $dis(x, \partial D)<\in 0$ .
(1).
(2) $\nabla U_{0}(x)\neq 0$ ,(3) $\rho(t)=0$ , $t\geq 0$ , $\rho(t)>0$ , $t<0$ , and $U(x)=\rho(U_{0}(x))$ for $x\in R^{d}$with $dis(x, \partial D)<\in 0$ .
$ U_{0}\in$$C^{1}-$$C^{\infty}(R^{d};R)$$\rho$$R\rightarrow R$$x\in\partial D$$x\in\partial D$(4)$\lim_{t\uparrow 0}\frac{\rho^{/}(t)}{\rho(t)}=-\infty$

Now let$d\overline{i}s$.

be a metric function on$\tilde{W}^{d}$given by$d\overline{i}s(w_{0}, w_{1})$$=\sum_{n=1}^{\infty}2^{-7l}(1\wedge((\max_{t\in[0,n]}|x_{0}(t)-x_{1}(t)|)+(\int_{0}^{n}|v_{0}(t)-v_{1}(t)|^{n})^{1/7l}))$for,, $i=0,1$ .
Then we will show the following.

$w_{i}(\cdot)=(x_{i}(\cdot), v_{i}(\cdot))\in\tilde{W}^{d}$

Theorem 2. Let ,, be the probability law of,$t\in[0, \infty)$ , on
, and be the probability measure given in Theorem 1
in the case when $\Phi(v, x)=v-2(v\cdot n(x))n(x)$ , $v\in R^{d}$ ,. Thenconveges to weakly as.
as probability measures on$\nu^{\lambda}$$\lambda\in[1, \infty)$$(X_{t}^{\lambda}, V_{t}^{\lambda})$$\overline{W}_{0}$$\mu$$x\in\partial D$$\nu^{\lambda}$$\lambda\rightarrow\infty$$\mu$$(\tilde{W}_{0},\tilde{d}is)$235

Stochastic Newton equation with reflecting boundary condition\S 2.

Basic lemmas

be a filtered probability space, and $B(t)=$$(B^{1}(t), \ldots, B^{d}(t))$ be a-dimensional Brownian motion. Let $B^{0}(t)=t$ ,$t\in[0, \infty)$ . Let, , be Lipschitz continuous, $i=0,1$ ,:$X$
be the solution to thefunctions, and let:
following SDE

Let$(\Omega, F, \{\mathcal{F}_{t}\}_{t\in[0,\infty)}, P)$$d$$R^{N}\rightarrow R^{N}$$d$$\sigma_{i}$$\ldots$$\times R^{N}\times\Omega\rightarrow R^{N}$$[0, \infty)$$X(t, x)=x+\sum_{i=0}^{d}\int_{0}^{t}\sigma_{i}(X(s, x))dB^{i}(s)$,$t\geq 0$, $x\in R^{N}$ .

We may assume that $X(t, x)$ is continuous in $(t, x)$ (cf. Kunita [2]).
Then we have the following.
, $\ldots,p_{m}\in(1, \infty)$ , $m\geq 1$ , with
Lemma 3. For any $T>0$ and, there is a constant $C>0$ such that
$p_{0},p_{1}$$\sum_{k=0}^{m}p_{k}^{-1}=1$$E[\int_{R^{N}}\prod_{k=0}^{m}|f_{k}(X(t_{k}, x))|dx]\leq C\prod_{k=0}^{m}||f_{k}||_{L^{p_{k}}(R^{N},dx)}$, $k=0,1$ ,for all $0=t_{0}<t_{1}<\ldots<t_{m}\leq T$ , andProof. From the assumption, there is a $C_{0}>0$ such that$f_{k}\in C_{0}^{\infty}(R^{N})$$|\sigma_{i}(x)-\sigma_{i}(y)|\leq C_{0}|x-y|$,$x$$\ldots$, $m$ .

, $y\in R^{N}$ .

such that $\int_{R^{N}}\varphi(x)dx=1$ . Let $\varphi_{n}(x)=n^{N}\varphi(nx)$ ,$x\in R^{N}$ , for $n\geq 1$ , and let, $i=0$ ,, . Then$C^{\infty}(R^{N}; R^{N})$ . LetLet$\varphi\in C_{0}^{\infty}(R^{N})$$\sigma_{i}^{(n)}=\varphi_{n}*\sigma_{i}$$\sigma_{i}^{(n)}\in$$d$$\ldots$$W_{i,k}^{(n),j}(x)=\frac{\partial}{\partial x^{k}}\sigma_{i}^{(n),j}(x)$,$x\in R^{N}$,$j$, $ k=1\ldots$ , $N$ , $i=0,1$ ,Then we see that $|W_{i,k}^{(n),j}(x)|\leq C_{0}$ , $x\in R^{N}$ . Let
be the solution to the following SDE$X^{(n)}$, , $n\geq 1$ .
$d$$\ldots$:$[0, \infty)$$\times R^{N}\times$$\Omega\rightarrow R^{N}$$X^{(n)}(t, x)=x+\sum_{i=0}^{d}\int_{0}^{t}\sigma_{i}^{(n)}(X^{(n)}(s, x))dB^{i}(s)$Then we may think thatwith probability one. Let$X^{(n)}(t, \cdot)$,$t\geq 0$, $x\in R^{N}$ .

: $R^{N}\rightarrow R^{N}$ is a difTeomorphism. Let $W_{i}^{n}(x)=$$J_{k}^{(n),j}(t, x)=\frac{\partial}{\partial x^{k}}X^{(n),j}(t, x)$236

S. Kusuoka

and $ J^{(n)}(t, x)=(J_{k}^{(n),j}(t, x))_{k,j=1,N}\ldots,\cdot$ Then the$N\times N$ -matrix valued process $J^{(n)}(t, x)$ satisfies the following SDE$(W_{i,k}^{(n),j}(x))_{k,j=1,N}\ldots$,$J^{(n)}(t, x)=I_{N}+\sum_{i=0}^{d}\int_{0}^{t}W_{i}^{(n)}(X^{(n)}(s, x))J^{(n)}(s, x)dB_{i}(s)$Also, we see th.

$e$$J^{(n)}(t, x)^{-1}$$=I_{N}-\sum_{i=0}^{d}\int_{0}^{t}J^{(n)}(s, x)^{-1}W_{i}^{(n)}(X^{(n)}(s, x))dB_{i}(s)$$+\frac{1}{2}\sum_{i=1}^{d}\int_{0}^{t}J^{(n)}(s, x)^{-1}W_{i}^{(n)}(X^{(n)}(s, x))^{2}ds$

Then we see th.

$e$$ C_{T}=\sup\{E[\det J^{(n)}(t, x)^{-po+1}];t\in[0, T], x\in R^{N}, n\geq 1\}<\infty$ .

So we have
$E[\int_{R^{N}}\prod_{k=0}^{n}|f_{k}(X^{(n)}(t_{k}, x))|dx]$$\leq E[\int_{R^{N}}|f_{0}(x)|_{0}^{p}(\prod_{k=1}^{m}\det J^{(n)}(t_{k}, x)^{-p_{0}/pk})dx]^{1/p0}$$\times\prod_{k=1}^{m}E[\int_{R^{N}}|f_{k}(X^{(n)}(t_{k}, x))|^{pk}\det J^{(n)}(t_{k}, x)dx]^{1/Pk}$$\leq C_{T}(\int_{R^{N}}|f(x)|_{0}^{p}dx)^{1/po}\prod_{k=1}^{m}(\int_{R^{N}}|f_{k}(x))|^{pk}dx)^{1/pk}$

Letting, we have our assertion.
$D$:, $j=1,2$ ,Now let be a bounded domain in $R^{N}$ andbefunctions satisfying the following assumptions (F1) , (F2), furthermore.
, ,(F1) For $x\in D$ and $i=1$ ,$ n\rightarrow\infty$$I$$F^{j}$$C^{2}$$d$$\ldots$$\sum_{j=1}^{N}\sigma_{i}^{j}(x)\frac{\partial}{\partial x^{j}}F^{1}(x)=0$.

(F2) $\inf\{\det(\nabla F^{i}(x)\cdot\nabla F^{j}(x))_{i,j=1,2;}x\in D\}>0$ .
Then we have the following$R^{N}\rightarrow R$

237

Stochastic Newton equation with reflecting boundary condition

Lemma 4. For a.e.x,$P(X(t, x)\in D$ , $F(X(t, x))=0$

Here $F=(F^{1}, F^{2})$ :$R^{N}\rightarrow R^{2}$for some $t>0$ ) $=0$ .

.

Proof. Let$\tau(s, x)=\inf\{t\geq s;X(t, x)\in D^{c}\}\wedge(s+1)$,$x\in R^{N}$, $s>0$ .

Also, let$p(x, s)=P(F(X(t, x))=0$ for some $t\in[s,$ $\tau(s, x)))$ ,$x\in R^{N}$, $s>0$ .

Then we see that
$P(X(t, x)\in D$ , $F(X(t, x))=0$ for some $t>0$ ) $\leq\sum_{r\in Q+}p(x, r)$ ,is the set of positive rational numbers. Let $ V(m)=\{x\in$}, $m\geq 1$ . Let us define random variables $Z_{T,m}$ , $T>0$ ,$m\geq 1$ , and constant
by

where
$R^{N}$;$Q_{+}$$|x|\leq m$$C_{1}$$Z_{T,m}=\sup\{|t-s|^{-1/3}|X(t, x)-X(s, x)|;0\leq s<t\leq T, x\in V(m)\}$ ,and$C_{1}=\sup\{|\sigma_{0}(x)||\nabla F^{1}(x)|+\frac{1}{2}\sum_{i=1}^{d}|\nabla^{2}F^{1}(x)||\sigma_{i}(x)|^{2}+|\nabla F^{2}(x)|;x\in\overline{D}\}$.

Then we see that $P(Z_{T,m}<\infty)=1$ (cf. Kunita[2]). By the assumtion(F1), we see that$F^{1}(X(t, x))=F^{1}(x)+\int_{0}^{t}(\sigma_{0}(X(s, x))\nabla F^{1}(X(s, x))$$+\sum_{i=1}^{d}\frac{1}{2}\nabla^{2}F^{1}(X(s, x))(\sigma_{i}(X(s, x)),$$\sigma_{i}(X(s, x)))ds$.

So we see that$|F^{1}(X(t, x))-F^{1}(X(s, x))|\leq C_{1}|t-s|$ ,$t\in[s,$ $\tau(s, x))$

, $s\geq 0$ , $x\in R^{N}$ ,and$|F^{2}(X(t, x))-F^{2}(X(s, x))|\leq C_{1}Z_{T,m}|t-s|^{1/3}$$t$, $s\in[0, T]$ , $x\in V(m)$ .

238

S. Kusuoka

Also, by the assumption (F2), we see that there is a constant $C_{2}>0$such that$\int_{D}1_{A}(F(x))dx\leq C_{2}|A|$, where $|A|$ denotes the area of $A$ .
for any Borel set $A$ in
Let $\Delta_{\ell,n,k}=[-C_{1}n^{-1}, C_{1}n^{-1}]\times[-\ell C_{1}n^{-1/3}, \ell C_{1}n^{-1/3}]$ , , $n\geq 1$ ,$k=1$ ,,
, . Then we have for any$R^{2}$$\ell$$\ldots$$\ell\geq 1$$n$$\int_{V(m)}dxP(F(X(t, x))=0$ for some $t\in[s,$ $\tau(s, x))$ , $Z_{s+1,m}\leq\ell)$$\leq\sum_{k=1}^{n}\int_{V(m)}dxP(X(s, x)\in D,$$X(s+(k-1)/n, x)\in D$ ,$F(X(s+(k-1)/n, x))\in\triangle_{\ell,n,k})$$=\sum_{k=1}^{n}E[\int_{R^{N}}dx1_{V(m)}(x)1_{D}(X(s, x))$$1_{D}(X(s+(k-1)/n, x))1_{\triangle\ell},n,k(F(X(s+(k-1)/n, x)))]$$\leq C\sum_{k=1}^{n}|V(m)|^{1/10}|D|^{1/10}(\int_{D}1_{\Delta_{\ell,n,k}}(F(x))dx)^{4/5}$$\leq CC_{2}n|V(m)|^{1/10}|D|^{1/10}(4C_{1}^{2}\ell n^{-4/3})^{4/5}$.

Here $C$ is the constant in Lemma 3 for $T=s+1$ , $p_{0}=p_{1}=10$ and$p_{3}=5/4$ . Since $n\geq 1$ is arbitrary, we see that$\int_{V(m)}dxP(F(X(t, x))=0$ for some $t\in[s,$ ’ $\tau(s, x))$ , $Z_{s+1,m}\leq\ell)=0$ ,$\ell\geq 1$.

This implies that $\int_{R^{N}}p(x, s)=0$ , $s>0$ .
Therefore we have our assertion.
, , $i=0$ ,Corollary 5. Suppose moreover that
, , are$dimLie[\frac{\partial}{\partialt}-V_{0},V_{1},\ldots,V_{d}](0,x_{0})=N+1$.
smooth aroundand thatHere$x_{0}\in(\overline{D})^{c}$$d$$\sigma_{i}$$\ldots$$x_{0}$$V_{i}(x)=\sum_{j=1}^{d}\sigma_{i}^{j}(x)\frac{\partial}{\partial x^{j}}$,$i=1$ ,

, ,$d$$\ldots$and$V_{0}(x)=\sum_{j=1}^{d}(\sigma_{0}^{J}(x)-\frac{1}{2}\sum_{i=1}^{d}\sum_{k=1}^{N}\sigma_{i}^{k}(x)\frac{\partial\sigma_{i}^{j}}{\partial x^{k}}(x))\frac{\partial}{\partial x^{j}}$.

Stochastic Newton equation with reflecting boundary condition

239

Then
$P(X(t, x_{0})\in D$ , $F(X(t, x_{0}))=0$

for some $t>0$ ) $=0$ .

Let $U$ be an open neighborhood of,, , are smooth aroundand that$0;X(t, x_{0})\in U^{c}\}$ . Then we see that

Proof.

$\overline{U}\cap\overline{D}=\emptyset$$\overline{U}$

$d$

$0$

$\ldots$$P(X(t, x_{0})\in D,$ $F(X(t, x_{0}))=0$$\leq\sum_{n=1}^{\infty}P(X(t, x_{0})\in D,$

such that , $=$. Let $\tau=\inf\{t>$$i,$$x_{0}$

for some $t>0$ )$F(X(t, x_{0}))=0$ for some $t>\frac{1}{n}$ ,$\leq\sum_{n=1}^{\infty}\int_{U}P(X(\frac{1}{n}, x_{0})\in dx$$\sigma_{i}$$\tau>\frac{1}{n}$), $\tau>\frac{1}{n})P(X(t, x)\in D,$ $F(X(t, x))=0$

for some $t>0$ ).
However, by [3], we see that $P(X(\frac{1}{n}, x_{0})\in dx$ ,continuous. So by Lemma 4, we have our assertion.

$\tau>\frac{1}{n})$is absolutely
1

Proof of Theorem 1\S 3.

Since the proof is similar, we prove Theorem 1 in the case that $D=$$\{x=(x^{1}, \ldots, x^{d})\in R^{d};x^{1}<0\}\subset R^{d}$ , and $\Phi(v, x)=(-v^{1}, v^{2}, \ldots, v^{d})$for $v=(v^{1}, v^{2}, \ldots, v^{d})$ and $x\in\partial D$ . In general, if we take a double
and change the coordinate functions, we can apply a similar
cover of:, be bounded Lipschitz continuous, $i,j=1$ ,proof. Let$x\inR^{d}$and that there are positive
function such that $a^{ij}(x)=a^{ji}(x)$ ,,constants
such that$D^{c}$$a^{ij}$$C_{0}$$R^{d}\rightarrow R$$\ldots d$$C_{1}$$C_{0}|\xi|^{2}\leq\sum_{i,j}^{d}a^{ij}(x)\xi_{i}\xi_{j}\leq C_{1}|\xi|^{2}$

Let

:
Let
$b$

,

$x$

,

$\xi\in R^{d}$be a bounded measurable function.
be a second order linear diﬀerential operator in.

$R^{2d}\rightarrow R^{d}$$L_{0}$$R^{2d}$given by

$L_{0}=\sum_{i=1}^{d}v^{i}\frac{\partial}{\partial x^{i}}+\frac{1}{2}\sum_{i,j=1}^{d}a^{ij}(x)\frac{\partial^{2}}{\partial v^{i}\partial v^{j}}+\sum_{i=1}^{d}b^{i}(x, v)\frac{\partial}{\partial v^{i}}$Then Theorem 1 is somehow equivalent to the following Theorem.
So we prove this Theorem.

240

S. Kusuoka, and suppose that
, $i,j=$Theorem 6. Let, , are smooth around. Then there exists a unique probability,
satisfying the following conditions.
measure over
(1) $\mu(w(0)=(x_{0}, v_{0}))=1$ .
(2) $\mu(w(t)\in D^{c}\times R^{d}, t\in[0, \infty))=1$ .
(3) For any, $\{f(w(t))-\int_{0}^{t}L_{0}f(w(s))ds;t\geq 0\}$ isa martingale under $\mu(dw)$ .
(4) $\mu(1_{\{0\}}(x^{1}(t))(v^{1}(t)+v^{1} (t-))=0, t\in[0, \infty))=1$ and$a^{ij}$$(x_{0}, v_{0})\in(\overline{D})^{c}\times R^{d}$

$d$

$1$

$x_{0}$$\ldots$$\tilde{W}^{d}$$\mu$$f\in C_{0}^{\infty}((\overline{D})^{c}\times R^{d})$

$\mu$

(

is continuous in $t\in[0,$$v^{i}(t)$Proof. Let$\tilde{a}^{ij}$:$R^{d}\rightarrow R$, , $j=1$ ,Let$\overline{b}^{i}$:$R^{2d}\rightarrow R$, $i=1$ ,$\ldots d$$\ldots d$$i$$\tilde{a}^{ij}(x)=a^{ij}(|x^{1}|, x^{2}, \ldots, x^{d})$$\infty$,), $i=2$ ,$\ldots$, $d)=1$ .

, be given by$x=(x^{1}, x^{2}, \ldots, x^{d})\in R^{d}$.

, be given by$\tilde{b}^{1}(x)=sgn(x^{1})b^{1}(|x^{1}|, x^{2}, \ldots, x^{d})$,and$\tilde{b}^{i}(x)=b^{i}(|x^{1}|, x^{2}, \ldots, x^{d})$for $x=(x^{1}, x^{2}, \ldots, x^{d})\in R^{d}$ . Let
given byoperators in$\tilde{L}_{0}$, $i=2$ ,$\ldots$,$d$be second order linear diﬀerential$R^{2d}$$\tilde{L}_{0}=\sum_{i=1}^{d}v^{i}\frac{\partial}{\partial x^{i}}+\frac{1}{2}\sum_{i,j=1}^{d}\tilde{a}^{ij}(x)\frac{\partial^{2}}{\partial v^{i}\partial v^{j}}+\sum_{i=1}^{d}\tilde{b}^{i}(x, v)\frac{\partial}{\partial v^{i}}$.

Then by transformation of drift (cf. Ikeda-Watanabe[l]), we see thaton $C([0, \infty);R^{2d})$ such that
there is a unique probability measure$\nu(w(0)=(x_{0}, v_{0}))=1$ and that $\{f(w(t))-\int_{0}^{t}\tilde{L}_{0}f(w(s))ds;t\geq 0\}$ is a.
martingale under $\nu(dw)$ for any$\tilde{\xi}(w)=\inf\{t>0;x^{1}(t)=0,v^{1}(t-)=0\}$. Then by Corollary 5
Let$\nu(\overline{\xi}(w)=\infty)=1$. Let
and Girsanov’s transformation, we see that$\nu$

$f\in C_{0}^{\infty}(R^{2d})$$X(t, w)=(|x^{1}(t)|, x^{2}(t),$$\ldots$, $x^{d}(t))$ ,$t\in[0, \infty)$,and$V(t, w)=\frac{d^{+}}{dt}X(t, w)$,$t\in[0, \infty)$.

is the probability law of $(X(\cdot, w)$ , $V(\cdot, w))$ under . Then we seesatisfies the conditions $(1)-(4)$ . So we see the existence.
Now let us prove the uniqueness. Let be a probability measure asin Theorem. Let $\xi(w)=\inf\{t>0;x^{1}(t)=0, v^{1} (t-)=0\}$ . Also, let us

Let
that

$\nu$

$\mu$

$\mu$

$\mu$

241

Stochastic Newton equation with reflecting boundary condition

define stopping times$\tau_{0}(w)=0$ and$\tau_{k}$:, $k=0,1$ , 2,$\tilde{W}_{0}\rightarrow[0, \infty]$$\tau_{k+1}(w)=\inf\{t>\tau_{k}(w);x^{1}(t)=0\}$,$\ldots$$w\in\tilde{W}^{d}$, inductively by, $k=0,1$ ,$\ldots$.

Then we see ﬀom the assumption (4) that if $\tau_{k}(w)<\xi(w)$ , then $\tau_{k}(w)<$$\tau_{k+1}(w)$ for-a.s.w. Also, it is easy to see that $\xi(w)\leq\sup_{k}\tau_{k}(w)$ ,.
, let
For $any\in>0$ and $k=0,1$ , 2,$\mu$

$w\in\tilde{W}^{d}$$\ldots$$\sigma_{k}^{0}(w)=\inf\{t>\tau_{k}(w);x^{1}(t)>\in\}$,and$\sigma_{k}^{1}(w)=\inf\{t>\sigma_{k}^{0}(w);x^{1}(t)<\in/2\}$,$w\in\tilde{W}^{d}$, $k=0,1$ ,$\ldots$.

Then we see ﬀom the assumption (3) that$f(x(t\wedge\sigma_{k}^{1}), v(t\wedge\sigma_{k}^{1}))-f(x(t\wedge\sigma_{k}^{0}), v(t\wedge\sigma_{k}^{0}))-\int_{t\wedge\sigma_{k}^{O}}^{t\wedge\sigma_{k}^{1}}L_{0}f(x(s), v(s))ds$

is a bounded continuous martingale for any
Now let$f\in C_{0}^{\infty}(R^{2d})$.

$\tilde{X}(t, w)$$x(t)$$=\{$$(-x^{1}(t), x^{2}(t)$, $x^{d}(t))$ ,, if$t\in[\tau_{k}(w),$ $\tau_{k+1}(w))$ , if

, $v^{d}(t))$ ,, if$t\in[\tau_{k}(w),$ $\tau_{k+1}(w))$ , if,,$t\in[\tau_{k}(w),$ $\tau_{k+1}(w))$$\ldots$$k$$k$

is even,is odd,$\tilde{V}(t, w)$$v(t)$$=\{$$(-v^{1}(t), v^{2}(t)$,,$t\in[\tau_{k}(w),$ $\tau_{k+1}(w))$$\ldots$

Then we can see that
Also, we see that$(\tilde{X}(t\wedge\xi),\overline{V}(t\wedge\xi))$

is continuous in

$t$

$k$

$k$

is even,is odd.

for $\mu-a.s.w$ .

$f(\overline{X}(t\wedge\sigma_{k}^{1}),\tilde{V}(t\wedge\sigma_{k}^{1}))-f(\tilde{X}(t\wedge\sigma_{k}^{0}),\tilde{V}(t\wedge\sigma_{k}^{0}))-\int_{t\wedge\sigma_{k}^{0}}^{t\wedge\sigma_{k}^{1}}\tilde{L}_{0}f(\tilde{X}(s),\tilde{V}(s))ds$

is a continuous martingale for any
Therefore we see that$f\in C_{0}^{\infty}(R^{2d})$.

$f(\tilde{X}(t\wedge\tau_{k+1}),\tilde{V}(t\wedge\tau_{k+1}))-f(\tilde{X}(t\wedge\tau_{k}),\tilde{V}(t\wedge\tau_{k}))$$-\int_{t\wedge\tau_{k}}^{t\wedge\tau_{k+1}}\tilde{L}_{0}f(\tilde{X}(s),\tilde{V}(s))ds$

242

S. Kusuoka

is a continuous martingale for any
that$f\in C_{0}^{\infty}(R^{2d})$. So we can conclude$f(\tilde{X}(t\wedge\xi),\tilde{V}(t\wedge\xi))-\int_{0}^{t\wedge\xi}\tilde{L}_{0}f(\tilde{X}(s),\tilde{V}(s))ds$.
is a continuous martingale for any
Therefore we see that the probability law of
under$shift$is the same ofunder , by the argument ofof drift andthe fact that a strong solution of stochastic diﬀerential equation with
Lipschitz continuous coeﬃcients is unique. So we see that $\mu(\xi(w)=$$\infty)=1$ . Since we see that$f\in C_{0}^{\infty}(R^{2d})$$(\tilde{X}(\cdot\wedge\xi),\tilde{V}(\cdot\wedge\xi))$$w(\cdot\wedge\tilde{\xi})$

$\mu$

$\nu$

$x(t)=(|\tilde{X}^{1}(t)|,\tilde{X}^{2}(t),$$\ldots,\tilde{X}^{d}(t))$,$t\in[0, \xi)$,and$v(t)=(\frac{d^{+}}{dt}|\overline{X}^{1}(t)|,\tilde{V}^{2}(t),$$\ldots,\overline{V}^{d}(t))$,$t\in[0, \xi)$,

we see the uniqueness.
This completes the proof.

\S 4.

Proof of Theorem 2

We will make some preparations to prove Theorem 2.

Proposition 7. Let $T>0$ . Let
be the set of $w\in D([0, T);R)$
for which $w(0)=0$ , $w(T-)\leq 1$ , and $w(t)$ is non-decreasing in . Thenis compact in $L^{p}((0, T)$ , $dt)$ , $p\in(1, \infty)$ , and its cluster points are in$A_{0}$$t$$A_{0}$$D([0, T);R)$ .

Suppose that $w_{n}\in A_{0}$ , $n=1$ , 2,
. Then we see that$w_{n}(t)\in[0,1]$ , $t\in[0, T)$ , $n\geq 1$ . So taking subsequence if necessary, we
is convergent for any $r\in[0, T)\cap Q$ . Let
may assume that
, $r\in Q$ , and let, $t\in[0, T)$ , and$w(T)$ be arbitrary such that $\sup_{t\in[0,T)}w(t)\leq w(T)\leq 1$ . Then we seethat $w\in D([0, T);R)$ and $w$ is non-decreasing, and that $w_{n}(t)\rightarrow w(t)$ ,$t\in[0, T)$ , if is a continuous point of $w$ . So we see that,,in If $((0, T)$ , $dt)$ .
This completes the proof.
We have the following as an easy consequence of Proposition 7.

Proof.

$\ldots$$\{w_{n}(r)\}_{n=1}^{\infty}$$\tilde{w}(r)=\lim_{n\rightarrow\infty}w_{n}(r)$$t$$w(t)=\lim_{r\downarrow t}\tilde{w}(r)$$w_{n}\rightarrow w$$ n\rightarrow\infty$$I$

Corollary 8. Let $T>0$ . Let $A$ be the set of $w\in D([0, T);R^{d})$ forwhich $w(0)=0$ and the total variation of $w$ is less than 1. Then $A$ iscompact in $IP((0, T);R^{d}$ , $dt)$ , $p\in(1, \infty)$ , and its cluster points are in$D([0, T);R^{d})$ .

Stochastic Newton equation with reflecting boundary condition243

Now let us prove Theorem 2. Let$H_{t}^{\lambda}=\lambda U(X_{t}^{\lambda})+\frac{1}{2}|V_{t}^{\lambda}|^{2}$,$t\geq 0$.

Then we have
$H_{t}^{\lambda}=\frac{1}{2}|v_{0}|^{2}+\int_{0}^{t}V_{s}^{\lambda}\cdot\sigma(X_{s}^{\lambda})dB_{s}+\int_{0}^{t}V_{s}^{\lambda}\cdot b(X_{s}^{\lambda}, V_{s}^{\lambda})ds$$+\frac{1}{2}\int_{0}^{t}trace(\sigma(X_{s}^{\lambda})^{*}\sigma(X_{s}^{\lambda}))ds$.

So we see that for any $p\in[2, \infty)$ there is a constant $C$ independentof such that$\lambda$$E[\sup_{t\in[0,T]}(H_{t}^{\lambda})^{p}]\leq C(|v_{0}|^{2p}+1+E[\int_{0}^{T}|V_{t}^{\lambda}|^{p}dt])$$\leq C(|v_{0}|^{2p}+1+2^{p/2}TE[\sup(H_{t}^{\lambda})^{p}]^{1/2})$.

$t\in[0,T]$

So we see that(1)$\sup E[\sup(H_{t}^{\lambda})^{p}]<\infty$$\lambda>0$,$p\in[1, \infty)$.

,$p\in[1, \infty)$.

$t\in[0,T]$

Therefore we see that$\sup_{\lambda>0}E[\sup_{t\in[0,T]}|V_{t}^{\lambda}|^{p}]<\infty$

So we see that
Moreover, we see that$\{H_{t}^{\lambda}\}_{t\in[0,\infty)}$

(2)

, and

$E[\sup_{t\in[0,T]}U(X_{t}^{\lambda})^{p}]\rightarrow 0$

Let us take an$\in\in(0, \in_{0})$,$\{X_{t}^{\lambda}\}_{t\in[0,\infty)}$$\lambda\rightarrow\infty$,,$\lambda\geq 0$, are tight in $C$ .

$p\in[1, \infty)$.

such that$ C_{0}=\sup\{|\nabla U_{0}(x)|^{-1} ; dis(x, \partial D)\leq\in\}<\infty$.

, such that $0\leq\varphi\leq 1$ , $\varphi(x)=1$ , if $dis(x, \partial D)<\in/3$ ,Let$dzs(z;\partial D)$ $>\in/2$ .
and $\varphi(x)=0$ , if
Let $D_{0}=\{x\in D;dis(x, \partial D)>\in/4\}$ ,$\tau=\tau^{\lambda}=\inf\{t>0;X_{t}^{\lambda}\in
D_{0}\}$ . Then we see by Equation (2)and letthat,,$\varphi\in C_{0}^{\infty}(R^{d})$$P(\tau^{\lambda}<T)\rightarrow 0$$\lambda\rightarrow\infty$

244

S. Kusuoka

for any $T>0$ . Let
given by$A_{t}^{\lambda}$, $t\geq 0$ be a non-decreasing continuous process$A_{t}^{\lambda}=-\lambda\int_{0}^{t\wedge\tau^{\lambda}}\varphi(X_{s}^{\lambda})\rho’(U_{0}(X_{s}^{\lambda}))|\nabla U_{0}(X_{s}^{\lambda})|^{2}ds$

Note that$A_{0}^{\lambda}=0$,.

$t\geq 0$. Since we have$\varphi(X_{t\wedge\tau^{\lambda}}^{\lambda})(\nabla U_{0}(X_{t\wedge\tau^{\lambda}}^{\lambda})\cdot V_{t\wedge\tau^{\lambda}}^{\lambda})-\varphi(X_{0}^{\lambda})(\nabla U_{0}(X_{0}^{\lambda})\cdot V_{0}^{\lambda})$$=A_{t}^{\lambda}+\int_{0}^{t\wedge\tau^{\lambda}}\varphi(X_{s}^{\lambda})\nabla^{2}U_{0}(X_{s}^{\lambda})(V_{s}^{\lambda}, V_{s}^{\lambda})ds$$+\int_{0}^{t\wedge\tau^{\lambda}}\varphi(X_{s}^{\lambda})(\nabla U_{0}(X_{s}^{\lambda})\cdot b(X_{s}^{\lambda}, V_{s}^{\lambda}))ds$$+\int_{0}^{t\wedge\tau^{\lambda}}\varphi(X_{s}^{\lambda})(\nabla U_{0}(X_{s}^{\lambda}))^{*}\sigma(X_{s}^{\lambda})dB_{s}$$+\int_{0}^{t\wedge\tau^{\lambda}}(\nabla\varphi(X_{s}^{\lambda})\cdot V_{s}^{\lambda})(\nabla U_{0}(X_{s}^{\lambda})\cdot V_{s}^{\lambda})ds$,

we see that$\sup_{\lambda>0}E[(A_{T}^{\lambda})^{p}]<\infty$,$p\in[1, \infty)$.

Since we have$\int_{0}^{T\wedge\tau^{\lambda}}\lambda U(X_{t}^{\lambda})dt=\int_{0}^{T\wedge\tau^{\lambda}},\frac{\rho(U_{0}(X_{t}^{\lambda}))}{|\rho(U_{0}(X_{t}^{\lambda}))|}|\nabla U_{0}(X_{t}^{\lambda})|^{-2}dA_{t}^{\lambda}$

we see that
$P(\int_{0}^{T\wedge\tau^{\lambda}}\lambda U(X_{t}^{\lambda})dt>\delta)$$\leq P(\sup_{t\in[0,T]}U(X_{t}^{\lambda})>\eta)+P(C_{0}^{2}A_{T}^{\lambda}\sup_{\rho^{-1}(\eta)\leq s<0}\frac{\rho(s)}{|\rho(s)|},>\delta)$

for any , $\eta>0$ . So we see that$\delta$(3)$P(\int_{0}^{T\wedge\tau^{\lambda}}|H_{t}^{\lambda}-\frac{1}{2}|V_{t}^{\lambda}|^{2}|dt>\delta)\rightarrow 0$

for any $\delta>0$ .
Also, we see that
$V_{t\wedge\tau^{\lambda}}^{\lambda}=v_{0}+V_{t}^{\lambda,0}+V_{t}^{\lambda,1}$,,$\lambda\rightarrow\infty$

,

245

Stochastic Newton equation with reflecting boundary condition

where$V_{t}^{\lambda,0}=+\int_{0}^{t\wedge\tau^{\lambda}}|\nabla U_{0}(X_{s}^{\lambda}))|^{-2}\nabla U_{0}(X_{s}^{\lambda})dA_{s}^{\lambda}$,

and$V_{t}^{\lambda,1}=\int_{0}^{t\wedge\tau^{\lambda}}\sigma(X_{s}^{\lambda})dB_{s}+\int_{0}^{t\wedge\tau}b(X_{s}^{\lambda}, V_{s}^{\lambda})ds$.

, $t\in[0, T]$ , is dominated by
So we see that the total variation of. Also,is tight in $C$ .
is tight in
Then by Corollary 8 it is easy to see that$L^{p}((0, T);R^{d})$ and its limit process is in $D([0, T);R^{d})$ with probability
one for any $T>0$ and $p\in(1, \infty)$ .
Let $F\in C^{\infty}(R^{d}\times R^{d};R^{d})$ be given by$V_{t}^{\lambda,0}$$C_{0}A_{T}^{\lambda}$$\{V_{t}^{\lambda,0}\}_{t\in[0,\infty)}$$\{V_{t}^{\lambda}\}_{t\in[0,T)}$$F(x, v)=\varphi(x)(v-|\nabla U_{0}(x)|^{-2}(\nabla U_{0}(x)\cdot v)\nabla U_{0}(x))$,$(x, v)\in R^{d}\times R^{d}$.

,Then by It\^o’s lemma it is easy to see thatis a
, is tight in $C$ , and that.
andcontinuous martingale for any
andSo we see that there are stochastic processes, such that,,and a subsequence
converges in law to.
inwith respect the metric function$\{f(X_{t}, V_{t})-\int_{0}^{t}L_{0}f(X_{s}, V_{s})ds\}_{t\in[0,\infty)}$
is a continuThen we see that, and thatous martingale for any
is a continuous process. Also, we see by Equation (3) that$\{F(X_{t}^{\lambda}, V_{t}^{\lambda})\}_{t\in[0,\infty)}$$\lambda\in$$\{f(X_{t}^{\lambda}, V_{t}^{\lambda})-\int_{0}^{t}L_{0}f(X_{s}^{\lambda}, V_{s}^{\lambda})ds\}$$(0, \infty)$$f\in C_{0}^{\infty}((\overline{D})^{c}\times R^{d})$$\lambda\in(0, \infty)$$\{(X_{t}, V_{t})\}_{t\in[0,\infty)}$$\lambda_{n}\rightarrow\infty$$\{\lambda_{n}\}_{n=1}^{\infty}$$\{H_{t}\}_{t\in[0,\infty)}$$\{((X_{t}^{\lambda_{n}}, V_{t}^{\lambda_{n}}), H_{t}^{\lambda_{n}})\}_{t\in[0,\infty)}$$ n\rightarrow\infty$$\{((X_{t}, V_{t}), H_{t})\}_{t\in[0,\infty)}$$\tilde{W}^{d}\times C$$d\tilde{i}s+dis_{C}$$f\in C_{0}^{\infty}((\overline{D})^{c}\times R^{d})$$\int_{0}^{T}|H_{t}-\frac{1}{2}|V_{t}|^{2}|dt=0$

for any $T>0$ . So we see that
Therefore we have$\{|V_{t}|^{2}\}_{t\in[0,\infty)}$$\{F(X_{t}, V_{t})\}_{t\in[0,\infty)}$

$a.s$

.

is a continuous process.

$P(1_{\partial D}(X_{t})(V_{t}-V_{t-}-2(n(X_{t})\cdot V_{t-})n(X_{t}))=0, t\in[0, \infty))=1$ .

So we see that the probability law ofTheorem 1.
This complets the proof of Theorem 2$\{(X_{t}, V_{t})\}_{t\in[0,\infty)}$in$\tilde{W}$

is

$\mu$

in

246

S. Kusuoka

References[1] Ikeda, S., and S. Watanabe, Stochstic Diﬀerential Equations and Diﬀusion
processes, 2nd Edition, North-Holland/Kodansha, 1989.
[2] Kunita, H., Stochastic flows and stochastic diﬀerential equations, Cambridge University Press, 1990.
[3] Kusuoka, S., and D.W. Stroock, Applications of Malliavin Calculus II, J.
Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1985), 1-76.
Graduate School of Mathematical Sciences
The University of Tokyo
Komaba 3-8-1, Meguro-ku
Tokyo 153-8914, Japan

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 247-255

Cubic Schr\"odinger: The Petit Canonical Ensemble
Henry P. McKean\S 1.

Introduction

This report describes some aspects of the Gibbsian petit canonicalensemble for the cubic Schr\"odinger equation in the space of functionsof period 1, say. \S 2-5 (defocussing case) represent joint work with K.
Vaninskyl). \S 6 is a brief report on the much more diﬃcult focussing
case. The original hope, that the petit ensemble might provide a pictureof the typical solution, is far ﬀom being achieved.
1.1. $Preliminaries^{2)}$The mechanical state is a pair $QP$ of nice functions of period 1,
moving according to the defocussing flow:$\frac{\partial Q}{\partial t}=-\frac{\partial^{2}P}{\partial x^{2}}+(Q^{2}+P^{2})P=\frac{\partial H_{3}}{\partial P}$$\frac{\partial P}{\partial t}=+\frac{\partial^{2}Q}{\partial x^{2}}-(Q^{2}+P^{2})Q=-\frac{\partial H_{3}}{\partial Q}$

This is a Hamiltonian system, relative to the classical bracket in function
space, with Hamiltonian$H_{3}=\frac{1}{2}\int_{0}^{1}[(Q^{/})^{2}+(P’)^{2}]+\frac{1}{4}\int_{0}^{1}(Q^{2}+P^{2})$.

It is integrable in the full technical sense of the word, having an infinite, $H_{2}=$series of (commuting) constants of motion,, and so on. The flow is integrated with the help of the Diracequation$H_{1}=\frac{1}{2}\int_{0}^{1}(Q^{2}+P^{2})$$\int_{0}^{1}Q^{l}P$$H_{3}$$M^{/}=[\left(\begin{array}{ll}Q & P\\P & -Q\end{array}\right)$$+\frac{\lambda}{2}$$\left(\begin{array}{ll}0 & -1\\1 & 0\end{array}\right)]M$

Received November 4, 2002.
McKean-Vaninsky [1997]
2) Manakov et al. [1984] $and/or$ McKean-Vaninsky [1997]
1)

248

H. P. McKean

for the 2

monodromy matrix $M=[m_{ij} : 1<i,j\leq 2]$ with $M(x=$$O)=I$ . Introduce the “discriminant” $\triangle(\lambda)=\frac{1}{2}spM(x=1)$ and the$\times 2$

with points $p=[\lambda,$
associated “Dirac curve”
latter is a double cover of the complex plane where
the roots$\mathfrak{M}$. Thelives, ramified over$\sqrt{\triangle^{2}(\lambda)-1}]$$\lambda$$\ldots\lambda_{-1}^{-}\leq\lambda_{-1}^{+}<\lambda_{-1}^{-}\leq\lambda_{-1}^{+}<\lambda_{0}^{-}\leq\lambda_{0}^{+}<\lambda_{1}^{-}\leq\lambda_{1}^{+}<\ldots$,$\lambda_{n}^{\pm}\simeq 2\pi n$

etc.

of
indicated in the figure. These comprise the periodic/antiperiodic spectrum of the Dirac equation and may be interpreted as a$\triangle(\lambda)=\pm 1$$a_{-2}$$a_{-1}$$a_{0}$$a_{1}$$\bigcap_{\cup}^{Po}$$\dot{\wedge p_{-1}}$,$\cap\bigcup_{P-2}$$\bigwedge_{\cup,p_{1}}$$\Delta=+1$$\lambda$$\Delta=-1$

complete list of constants of motion, commuting among themselves and,,, etc. The cycles:
with the prior constants,
seenin the upper part of the figure are the “real ovals” ofcovering the, these being all open for $QP$ in general position, as is“gaps”
mostly assumed below. $QP$ is encoded into a divisorofhaving 1 point on each real oval: the numbers
are the roots of $m_{12}(\mu)=0^{3)}$ and the radical
is declared. The mapto beis 1 : 1 or to the product$H_{1}$$H_{2}$$H_{3}$$a_{n}$$n\in \mathbb{Z}$$\mathfrak{M}$$[\lambda_{n}^{-}, \lambda_{n}^{+}]$$\mathfrak{P}=[p_{n} :$\mathfrak{M}$n\in \mathbb{Z}]$$\lambda(p_{n})\equiv\mu_{n}\in[\lambda_{n}^{-}, \lambda_{n}^{+}]$$\sqrt{\triangle^{2}-1}(p_{n})$$\frac{1}{2}(m_{11}-m_{12})(\mu_{n})^{4)}$3) $m_{12}(\lambda)$$QP\rightarrow \mathfrak{P}$looks much like $-\sin(\lambda/2)$ .
$\det M(1)=1$ so $m_{11}m_{12}=1$ if $m_{12}=0$ and $ m_{11}+m_{12}=2\triangle$ always,
whence this possibility.
4)

249

Cubic Schr\"odinger: The petit canonical ensembleof all the ovals. The next actor in the play is the “Abel map” of thedivisor into the (real) Jacobi variety Jac of , determined as follows.
are of the form
DFK= the “diﬀerentials of the first kind” ofwith certain entire functions , and a basis may
is now
be chosen so thator 0 according as $i=j$ or $not.5$ ))construed mod
mapped to Jac via the “angles”$\mathfrak{M}$$\mathfrak{M}$$\omega=$$\phi_{n}(\lambda)d\lambda/\sqrt{\triangle^{2}(\lambda)-1}$$\phi$$ a_{i}(\omega_{j})=2\pi$$\mathfrak{P}$$2\pi,6$$\theta_{n}=\sum_{k\in \mathbb{Z}}\int_{o_{k}}^{p_{k}}\omega_{n}$

$i.e$

.
$\sigma p$$\rightarrow\Theta=[\theta_{n} :n\in \mathbb{Z}]\in(\mathbb{R}/2\pi \mathbb{Z})^{\infty}=Jac$,and this map likewise is 1 : 1 and onto. Now you have the composite
map $QP\rightarrow divisor\rightarrow Jac$ , the point of the whole exercise being thatthe (complicated) flow of $QP$ is converted thereby into (simple) straightline motion at constant speed in Jac which may be mapped back to theoriginal (mechanical) variables with the help of a Riemann-like “theta”
function. In this way, the flow is “integrated”.

Petit Ensemble at Levels 1 &3

\S 2.

Level 1 is a warm-up for “level 3” to be described below. Introand note the $trace$
duce the “level 1 actions”. The petit $ensemb1e^{8)}$formula$I_{n}=\frac{1}{4\pi}a_{n}(ch^{-1}\triangle, d\lambda)^{7)}$$H_{1}=\frac{1}{2}\int(Q^{2}+P^{2})=\sum_{\mathbb{Z}}I_{n}$$e^{-H_{1}}d^{\infty}Qd^{\infty}P=\frac{e^{-\frac{1}{2}\int_{0}^{1}Q^{2}}}{(2\pi/0_{+})^{\infty/2}}d^{\infty}Q\times\frac{e^{-\frac{1}{2}\int_{0}^{1}P^{2}}}{(2\pi/0_{+})^{\infty/2}}d^{\infty}P$$=\prod_{\mathbb{Z}}e^{-I_{n}}dI_{n}\times\prod_{\mathbb{Z}}d\theta_{n}/2\pi$:

is descriptive of 2 independent copies of white noise; line 2 comes fromthe $trace$ formula plus the formal identification of the volume elements
prompted by the fact that actions&angles
are canonically conjugate and together form a full coordinate system in$QP$ -space. Naturally, line 2 requires proof as does the invariance of theensemble under the flow, for which see McKean-Vaninsky [1997].
$ d^{\infty}Qd^{\infty}P\& d^{\infty}Id^{\infty}\theta/2\pi$5) Ishould say diﬀerentials of the third kind as they have simple poles at the
2 points ofcovering , but as they play the role of classical DFK, I keep
. with 1 root,
divided by
the name.
looks much likeleft out.
6), some such choice being necessary for the convergence of thesum.
7)
The name will be justified in \S 4.
8)
Here and below, I will be ﬀee and easy with possibly infinite norming
constants.
$\mathfrak{M}$$\phi_{n}(\lambda)$$0_{k}=[\lambda_{k}^{-}, 0]$$\infty$$m_{12}(\lambda)$$\lambda-\mu$$i.e$H. P. McKean

250

Level 3. The petit ensemble at “level 3” :$e^{-H_{3}}d^{\infty}Qd^{\infty}P=\frac{e^{-\frac{1}{2}\int_{0}^{1}(Q’)^{2}}}{(2\pi 0_{+})^{\infty/2}}d^{\infty}Q\frac{e^{-\frac{1}{2}\int_{0}^{1}(P’)^{2}}}{(2\pi 0_{+})^{\infty/2}}d^{\infty}P\times e^{-\frac{1}{4}\int_{0}^{1}(Q^{2}+P^{2})^{2}}$is descriptive of 2 independent “circular” Brownian $motions^{9)}$ coupledby the third factor; it is invariant under the flow as for level 1. To
describe it in $action/angle$ language requires a revision: DFK at level 3:
normalized as inis as before (level 1) but with a new basis
or 0 according as $i=j$ or not. The level 3 actions are,and you have the $trace$ formula
whence$\omega_{n}’$$n\in \mathbb{Z}$$ a_{i}(\lambda^{2}\omega_{j}^{/})=2\pi$$H_{3}=\sum_{\mathbb{Z}}I_{n}’$$I_{n}’=\frac{1}{q_{\pi}}a_{n}(\lambda^{2}ch^{-1}\triangle d\lambda)$$e^{-H_{3}}d^{\infty}Qd^{\infty}P=\prod_{\mathbb{Z}}e^{-I_{n}’}\times[d^{\infty}Qd^{\infty}P=d^{\infty}Id^{\infty}\frac{d\theta}{2\pi}$at level

$1]$

, ,

$=\prod_{\mathbb{Z}}e^{-I_{n}’}dI_{n}^{/}\prod_{\mathbb{Z}}d\frac{\theta_{n}}{2\pi}\times\det\frac{\partial I}{\partial I}$

in which the third (Jacobian) factor is still to be understood. The level 3,
actions are canonically paired to the level 3
so$ang1es^{10)}\theta_{n}^{J}=\sum_{r\in \mathbb{Z}}\int_{o_{k}}^{\mathfrak{p}_{k}}\omega_{n}^{/}$$\det\frac{\partial I}{\partial I},$$=\det\frac{\partial\theta^{/}}{\partial\theta}$$\det[\omega_{i}^{/}/d\lambda(p_{j})]$$=\overline{\det[\omega_{i}/d\lambda(p_{j})]}$$\times\int\frac{\det\prod_{i>j}(\mu_{i}-\mu_{j})}{\prod_{\mathbb{Z}}\sqrt{\triangle^{2}-1(p_{n})}}d^{\infty}\mu$divided by$\int\prod_{\mathbb{Z}}\mu^{2}\times$

the same “volume element”.

This rather fanciful expression comes from level 2 in case all but $N$
gaps are closed and making
with an (unpardonable) disregardof normalizing factors. Now the “volume element” seen in line 3 is
$ N\uparrow\infty$

9)

CBM is standard Brownian motion, conditioned to end where it began,with this common displacement distributed over by flat Lebesgue measure.
The coupling holds down the total mass so that normalization is possible.
10)
These must be construed, not mod , but relative to another, pretty
complicated lattice of periods.
$\mathbb{R}$$ 2\pi$

251

Cubic Schr\"odinger: The petit canonical ensemble

nothing but an un-normalized expression of the flat (level 1) volume
on Jac, written out in the language of the divisor;
element
precisely; and so it is analso
educated guess that, after proper normalization, the Jacobian
ought to be the reciprocal of $ N=\int_{Jac}m_{12}^{2}(0)d^{\infty}\theta/2\pi$ .
This is correct as far as it $goes^{11)}$ , but what does $N$ really look like?
It is a function of actions alone, so the levell angles are still independentof them, with the same flat distribution as before. There are 10 integrals
of products of 2 entries of $M(1)$ , and I know 9 relations among them
, but the value of $N$ is notandinvolving the constants of motion
revealed by these. Too bad! Crude estimates of $N$ can be had but do
not help to describe how the actions couple. I leave the subject in thisunsatisfactory state.
$ d^{\infty}\theta/2\pi$$m_{12}(\lambda)=\frac{1}{2}(\mu_{0}-\lambda)\prod_{\mathbb{Z}}(2\pi n)^{-1}(\mu_{n}-\lambda)$$\det\partial I/\partial I^{J}$$\triangle$\S 3.

$\triangle.$

Some Tricks

I record here 3 amusing examples of averaging over Jac with re, but first a general principle. Think of the (still to be
spect tonormalized) expression$ d^{\infty}\theta/2\pi$$ d^{\infty}\frac{\theta}{2\pi}=\prod_{i>j}(\mu_{i}-\mu_{j})d^{\infty}\mu$

encountered in \S 3.
proportional todivided by$\prod_{\mathbb{Z}}\sqrt{\triangle^{2}-1}(p_{n})$The top, considered as a function of, so $yu$ have the “splitting rule at$mi_{2}(\mu_{n})$$d^{\infty}\frac{\theta}{2\pi}=\frac{mi_{2}(\mu_{n})}{\sqrt{\triangle^{2}-1}(p_{n})}$$\times a$on the oval$\mu_{n}$, say, is$n\in \mathbb{Z}$”:$a_{n}$

volume element on the product of all the other ovals.

This principle is now applied in 3 ways:.

like
Example 1.
looks like $-\sin(\lambda/2)$ andso you may expect
to
be
of
1
lower”
than“degree$-m_{12}$that Lagrange interpolation would apply. This is correct:$m_{12}(\lambda)$$ 2\triangle$$\triangle(\lambda)$$2\triangle.(\lambda)-m_{12}(\lambda)=12)\sum_{\mathbb{Z}}\frac{2\triangle(\mu_{n})}{mi_{2}(\mu_{n})}.\frac{m_{12}(\lambda)}{\lambda-\mu_{n}}$11)McKean-Vaninsky [1997].

12) $m_{12}(\mu_{n})=0$ of course.

,and$\cos(\lambda/2)$$m_{12}$

H. P. McKean252

Now average over Jac, exchange sum and average, and split the volume
atto produce$n\in \mathbb{Z}$$2$$\frac{m_{12}(\lambda)}{\lambda-\mu_{n}}\int_{a_{n}}\frac{\triangle d\mu_{n}}{\sqrt{\triangle^{2}-1}}.=13)0$$2\triangle.(\lambda)-\int_{Jac}m_{12}(\lambda)\frac{d^{\infty}\theta}{2\pi}=\sum_{\mathbb{Z}}$,$k\neq n\int_{\times a_{k}},$

$i.e$

.

$ 2\triangle$.

$=$

average$m_{12}$.

DFK at level 1 looks likeof
Example 2. The numerator
with 1 root factored out, so it, too, should be capable of interpolation:$\phi_{n}$$\omega_{n}\in$$m_{12}$$\phi_{n}(\lambda)=\sum_{i\in \mathbb{Z}}\frac{\phi(\mu_{i})}{mi_{2}(\mu_{i})}\frac{mi_{2}(\lambda)}{\lambda-\mu_{i}}$But this object has nothing to do with angles, so an average over Jac
does it no harm, and proceeding as in ex. 1, you find$\phi_{n}(\lambda)=\sum_{j}\int_{a_{j}}i\in \mathbb{Z}_{\times,,\neq i}\frac{m_{12}(\lambda)}{\lambda-\mu_{i}}\int_{a_{i}}\frac{\phi_{n}(\mu_{i})d\mu_{i}}{\sqrt{\triangle^{2}-1}(p_{i})}$$=$$\frac{m_{12}(\lambda)}{\lambda-\mu_{n}}\times 2\pi$$j\neq n\int_{\times a_{j}},$$\frac{m_{12}(\lambda)}{mi_{2}(\mu_{n})(\lambda-\mu_{n})}d^{\infty}\frac{\theta}{2\pi}$$=\int_{\times a_{j}=Jac}$.

divided by$\frac{1}{2\pi}\int_{a_{n}}\frac{d\mu}{\sqrt{\triangle^{2}-1}}$

$i.e$

,

.

$\omega_{n}=$average$\frac{m_{12}(\lambda)}{mi_{2}(\mu_{n})(\lambda-\mu_{n})}$normalized to have massThis seems to be a new way of writing DFK.
13) $\triangle\cdot/\sqrt{\triangle^{2}-1}=dch^{-1}\triangle$ .

$ 2\pi$on$a_{n}$

Cubic Schr\"odinger: The petit canonical ensemble

253with a true mechanical
Example 3 identifies
action, as promised at start of \S 3.14) The physical actions are $A_{n}=$$(2\pi)^{-1}a_{n}(PdQ)$ :. To implement their evaluation, take the flow
once about its private cyclewith Hamiltonian
which carries
with, leaving the rest of the divisor fixed, and equate
in time$I_{n}=\frac{1}{4\pi}a_{n}(ch^{-1}\triangle d\lambda)$$n\in \mathbb{Z}$$e^{tX}$$I_{n}$$p_{n}$$ 2\pi$$a_{n}$$A_{n}$$\frac{1}{2\pi}\int_{0}^{2\pi}e^{tX}[\int_{0}^{1}P(x)XQ(x)dx]dt$.

Now
has nothing to do with angles, so you can average over Jac, exchange this average with the time-average, and use the invariance of theflat volume under the present flow and the flow of translation produced
by $H_{2}=\int_{0}^{1}Q’P$ to reduce the previous display to $\int_{Jac}P(0)XQ(0)d^{\infty}\theta/2\pi$ .
Here,$A_{n}$$XQ(0)=\frac{1}{4\pi}\int_{a_{n}}(1/2)(m_{12}+m_{21})\frac{d\lambda}{\sqrt{\triangle^{2}-1}}$

,

.

.

is invariant under the “phase flow” $Q$ $=P$ and $P$ $=-Q$, and the average of $P(0)$ under thisproduced byflow is 0, permitting a further reduction to$m_{12}-m_{21}$$H_{1}=\frac{1}{2}\int_{0}^{1}(Q^{2}+P^{2})$$A_{n}=\frac{1}{4\pi}\int_{a_{n}}\frac{d\lambda}{\sqrt{\triangle^{2}-1}}\int_{Jac}P(0)[m_{12}(\lambda)-2\triangle.(\lambda)]d^{\infty}\frac{\theta}{2\pi}$.

The $trace$ formula $P(0)=15)_{\frac{1}{2}\sum}(\mu_{i}-\lambda_{i})$ and the interpolation of$-m_{12}$ from example 1 are now inserted under the average, sums
and avarage are exchanged, and the volume elementis split at$ 2\triangle$$ d^{\infty}\theta/2\pi$

14)

The level 3 actions have also a mechanical interpretation, but I do not go
into it here.
15):.
are the roots of.

$\lambda_{n}$$n\in \mathbb{Z}$$\triangle.(\lambda)=0$

H. P. McKean254$j\in \mathbb{Z}$, with the result that

$A_{n}=\frac{1}{4\pi}\int_{a_{n}}\frac{d\lambda}{\sqrt{\triangle^{2}-1}}\sum_{i\in \mathbb{Z}}\sum_{j\in \mathbb{Z}}$$\frac{m_{12}}{\lambda-\mu_{j}}\int_{a_{j}}$$(\lambda_{i}.

$k\neq j\int_{\times a_{k}},$-\mu_{i})dch^{-1}\triangle(p_{j})$$=\frac{1}{4\pi}\int_{a_{n}}\frac{d\lambda}{\sqrt{\triangle^{2}-1}}\sum_{k}\int_{i}i\in \mathbb{Z}_{X}\neq^{a_{k}}\frac{m_{12}}{\lambda-\mu_{i}}\int_{a_{n}}ch^{-1}\triangle d\mu_{i}$$\frac{m_{12}}{mi_{2}(\mu_{i})(\lambda-\mu_{i})}d^{\infty}\frac{\theta}{2\pi}$$\int_{\times a_{k}=Jac}$.

divided by$\frac{1}{2\pi}\int_{a_{i}}\frac{d\mu_{i}}{\sqrt{\triangle^{2}-1}}$$=\frac{1}{2\pi}\int_{a_{n}}\frac{d\lambda}{\sqrt{\triangle^{2}-1}}\sum_{i\in \mathbb{Z}}\{$

multiplied by$\frac{1}{4\pi}\int_{a_{i}}ch^{-1}\triangle d\mu_{i}$$=\frac{1}{2\pi}\int_{a_{n}}\frac{d\lambda}{\sqrt{\triangle^{2}-1}}\sum_{i\in \mathbb{Z}}\phi_{i}I_{i}$$=I_{n}$,as advertised.

\S 4.

Thermodynamic Limit.
Now let Q&P have period and take the large volume limit? The answer is nice
What happens to the petit ensemble. Then theinand simple. Let be the ground state
tend (in law) to the stationary
mechanical variables $[Q(x), P(x)]$ :. This is
diﬀusion with infinitessimal operator
even easy to prove.
$L$$ L\uparrow\infty$$e^{-H_{3}}d^{\infty}Qd^{\infty}P$$of-\frac{1}{2}\triangle+\frac{1}{4}r^{4}$$\psi$$x$$\mathbb{R}^{2}$$\in \mathbb{R}$$\frac{1}{2}\triangle+(grad\ell n\psi)\cdot grad$\S 5.

Focussing Case

This is much harder. The Hamiltonian is changed tominus
and the associated petit ensemble has total mass
. This prompted Lebowitz-Rose-Speer [1989] to introduce the microcanonical ensemble obtained by conditioning upon the value $N$ of the$\frac{1}{2}\int[(Q^{J})^{2}+(P^{J})^{2}]$$\frac{1}{4}\int(Q^{2}+P^{2})$$+\infty$

Cubic Schr\"odinger.. The petit canonical ensemble

255

constant of motion $H_{1}=\frac{1}{2}\int(Q^{2}+P^{2}).16)$ Their interest was in thethermodynamic limit: with fixed “density” $D$ , “particle number” $N=$
$DL$ , and, they found by numerical simulation, that the temperature dependent ensemble $e^{-H_{3}/T}d^{\infty}Qd^{\infty}P$ favors “solitons” /“radiation”. some kind of phase change takes place.
at $low/high$ temperatures,
Chorin [private communication] used a more sophisticated simulation of
the Brownian motion and found the opposite: no phase change. This
made me curious and, subsequentlyl7), I claimed to prove that the thermodynamical limit does not exist, explaining (as I thought) the discrepancy just described. But alas, all the big boys were wrong: in fact,
my student B. $Rider^{18)}$ proved that, at any values of temperature anddensity, the whole ensemble collapses onto $Q\equiv 0\& P\equiv 0$ . A pity.
$ L\uparrow\infty$$i.e$Bibliography
LEBOWITZ, J., H. A. ROSE, and E. R. SPEER: Statistical mechanics ofthe nonlinear Schr\"odinger equation(2). Mean field approximation. J. Stat.
Phys. 54 (1989), 17-56.
MANKOV, S.V., S. NOVIKOV, L. P. PITAEVSKII, and V. E. ZAKAROV:
Theory of Solitons. The Inverse Scattering Method. Contemporary Soviet
Math., Consultants Bureau, New York-London, 1984.
McKEAN, H. P. : Statistical mechanics of nonlinear wave equations (4):cubic Schr\"odinger. Comm. Math. Phys. 168 (1995), 479-491; erratum:
173 (1995) 675.
–,H. P. : A Martin boundary connected with the -volume limitof the focussing cubic Schr\"odinger equation. It\^o’s Stochastic Calculus andProbability Theory, ed. Ikeda et al., pp. 251-260, Springer-Verlag, Tokyo,
1995.
McKEAN, H. P. and K. VANINSKY: Action-angle variables for the cubic
Schr\"odinger equation. CPAM 50 (1997), 489-562.
–: Cubic Schr\"odinger: the petit canonical ensemble in action-angle
variables. CPAM 50 (1997), 593-622.
RIDER, B. : On the -volume limit of the focussing cubic Schr\"odinger equation. CPAM 55 (2002), 1231-1248.
$\infty$$\infty$

CIMS, 251 Mercer Street, New York, NY 10012
16) The total mass is now finite, so the ensemble can be normalized.

For aprobabilistic proof of the existence of the flow and for the invariance of the
micro-canonical ensemble under it, see McKean [1995].
17)
McKean [1995]
18)
Rider [2002]

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 257-278

Risk-sensitive Portfolio Optimization with Full andPartial Information
Hideo Nagai
Abstract.
We discuss an application of risk-sensitive control to portfolio optimization problems for a general factor model, which is considered avariation of Merton’s intertemporal capital asset pricing model ([18]).
In the model the instantaneous mean returns as well as volatilities ofthe security prices are aﬀected by economic factors and the security
prices. The economic factors are assumed to satisfy stocahstic differential equations whose coeﬃcients depend on the security prices
as well as themselves. In such general incomplete market models under Markovian setting we consider constructing optimal strategies forrisk-sensitive portfolio optimization problems on a finite time horizon. We study the Bellman equations of parabolic type corresponding to the optimization problems. Through analysis of the Bellman
equations we construct optimal strategies from the solution of theequation. We further discuss the problem with partial information.
We shall obtain a necessary condition for optimality using backward
stochastic partial diﬀerential equations.

Introduction\S 1.

Let us consider a market model with $m+1$ securities. Hereand factors
standsfor transposed matrix of . We assume that the set of securities includes
one bond, whose price is defined by the ordinary diﬀerential equation:$(S_{t}^{0}, S_{t}):=$$(S_{t}^{0}, S_{t}^{1}, \ldots, S_{t}^{m})^{*}$$S^{*}$$X_{t}=(X_{t}^{1}, X_{t}^{2}, \ldots, X_{t}^{n})^{*}$

$n$

$S$

(1.1)

$dS^{0}(t)=r(X_{t}, S_{t})S^{0}(t)dt$,$S^{0}(0)=s^{0}$,where $r(x, s)$ is a nonnegative function on $R^{n+m}$ . The other secutity
prices , $i=1,2$ , , $m$ , and the factors
are assumed to satsN thefollowing stochastic diﬀerential equations:$X_{t}$$S_{t}^{i}$$\ldots$

Received February 28, 2003.

H. Nagai

258$dS^{i}(t)=S^{i}(t)\{g^{i}(X_{t}, S_{t})dt+\sum_{k=1}^{n+m}\sigma_{k}^{i}(X_{t}, S_{t})dW_{t}^{k}\}$

,

(1.2)
$S^{i}(0)=s^{i}$,$i=1$ ,$\ldots$

, $m$

and
$dX_{t}=b(X_{t}, St)dt+\lambda(X_{t}, S_{t})dW_{t}$

,

(1.3)
$X(0)=x\in R^{n}$ ,where $W_{t}=(W_{t}^{k})_{k=1,..,(n+m)}$ is an $m+n$ dimensional standard Brownian. Heremotion process defined on a filtered probability space$n\times(m+n)$$m\times(m+n)$respectively
,
and
are
matrix valued
functions. Set$(\Omega, F, P, \mathcal{F}_{t})$$\lambda$$\sigma$$\mathcal{G}_{t}=\sigma(S(u), X(u);u\leq t)$,byand let us denote investment strategy to -th security$(i=0,1, \ldots, m)$ representing portfolio proportion of the amount of the-th security to the total wealththat the investor possesses, which isdefined as follows:$S^{i}(t)$

$i$

$i$

$h^{i}(t)$$V_{t}$

Definition 1.1. $(h^{0}(t), h(t))\equiv(h^{0}(t), (h^{1}(t), h^{2}(t),$ , ’ $h^{m}(t))^{*})$ is
said to be an invetment strategy if the following conditions are satisfiedi) $h(t)$ is anvalued progressively measurable stochastic process such that$\ldots$$R^{m}$$\mathcal{G}_{t}$$\sum_{i=1}^{m}h^{i}(t)+h^{0}(t)=1$$ii)$

and that$P(\int_{0}^{T}|h(s)|^{2}ds<\infty)=1$.

The set of all investment strategies will be denoted by $H(T)$ . When
we will often write $h\in H(T)$ for simplicity.
In what follows we always assume that$(h^{0}(t), h(t)^{*})_{0\leq t\leq T}\in H(T)$(1.4)$\sigma\sigma^{*}>0$.

For given $h\in\gamma\{(T)$ the wealth process $V_{t}=V_{t}(h)$ satisfies$\frac{dV_{t}}{V_{t}}$$=$$\sum_{i=0}^{m}h^{i}(t)\frac{dS^{i}(t)}{S^{i}(t)}$$=$$h^{0}(t)r(X_{t}, S_{t})dt+\sum_{i=1}^{m}h^{i}(t)\{g^{i}(X_{t}, S_{t})dt$$+\sum_{k=1}^{m+n}\sigma_{k}^{i}(X_{t}, S_{t})dW_{t}^{k}\}$$V_{0}$$=$

259

Risk-sensitive portfolio optimization

under the assumption of the self-financing condition. Then, taking i)
above into account it turns out to be the solution of$\frac{dV_{t}}{V_{t}}=r(X_{t}, S_{t})dt+h(t)^{*}(g(X_{t}, S_{t})-r(X_{t}, S_{t})1)dt+h(t)^{*}\sigma(X_{t}, S_{t})dW_{t}$$V_{0}=v$,,

where $1=(1,1, \ldots, 1)^{*}$ .
We first consider the following problem. For a given constant $\mu<$$\mu\neq
0$ maximize the following risk-sensitized expected growth rate up,
to time horizon $T$ :
$1$

(1.5)

$J(v, x;h;T)=\frac{1}{\mu}\log E[e^{\mu\log V_{T}(h)}]=\frac{1}{\mu}\log E[V_{T}(h)^{\mu}]$,

where ranges over the set $A(T)$ of all admissible strategies definedlater. The meaning of the maximization is well understood by looking:
at the asymptotics of the criterion as$h$$\mu\rightarrow 0$$\frac{1}{\mu}\log E[e^{\mu\log V_{T}(h)}]\sim E[\log V_{T}(h)]+\frac{\mu}{2}$Var[log $V_{T}(h)$ ] $+O(\mu^{2})$ .

Maximizing (1.5) is a risk-sensitive counterpart of the problem maximizing the expected growth rate of the investor’s wealth. The case where$\mu<0$ is called risk averse and $\mu>0$ risk seeking. Concerning thisproblem we introduce the Bellman equation corresponding to the valuefunction and we present the results constructing an optimal strategy
ﬀom the solution to the equation through its analysis in section 2. Notethat the problem maximizing the criterion $J(v, x;h;T)$ is equivalent to
HARA utility maximization:$\sup_{h}E[\frac{1}{\mu}V_{T}(h)^{\mu}]=\sup_{h}E[\frac{1}{\mu}e^{\mu\log V_{T}(h)}]$,$\mu<1$.

The problems on infinite time horizon maximizing(1.6)$\lim\inf\underline{1}\log E[e^{\mu\log V_{T}(h)}]$$T\rightarrow\infty\mu T$

have been considered by several authors e.g. in $[8],[9],[10],[14]$ , in thecase of linear Gaussian factor models since the work by Fleming [7], or
under more general setting in [5] with the assumption that randomnessof security price processes and that of factor processes are independent.
In [23] we have discussed the problem under rather general setting for socalled Merton’s ICAPM ([18]), and the results in section 2 of the present

H. Nagai

260

paper are its genaralization in the case of finite time horizon. Here by
Merton’s ICAPM, we mean the case that$r(x, s)=r_{1}(x)$ ,$\sigma(x, s)=\sigma_{1}(x)$$g(x, s)=g_{1}(x)$ ,,$\lambda(x, s)=\lambda_{1}(x)$$b(x, s)=b_{1}(x)$ ,

.

If ,
and
are linear functions andandare constant matrices
the models are said to be linear Gaussian. In that case the solutions of
the Bellman equations are expressed explicitly as the quadratic functionsof whose coeﬃcients are determined as the solutions of the matrix
Riccati diﬀerential equations and linear diﬀerential equations.
We then consider the maximization problem with partial inﬀomation. In the above investment strategies are defined asprogressively
measurable processes. However, it is not always realistic since economic
factors
are to be considered implicit and so it might be better toselect our strategies without using all past informations of securities. Our strategies may be well selected by using only inand factors
formations of security prices. Rishel [24] has considered the problem ona finite time horizon in such a way in a particular case, namely for alinear Gaussian model of one factor and one risky and one riskless assets
under the assumtion that randomness of the factor process and that of
the risky asset are independent. We have also considered the problem
for general linear Gaussian factor models [21] on a finite time horizon
and, by solving two kinds of Riccati diﬀerential equations, constructed
an optimal strategy. The results are extended to the case of infinite
time horizon in [22] by studying asymptotics of the solutions of inhomogeneous (time dependent) Riccati diﬀerential equations as time horizon
goes to infinity. In the present paper we shall consider the maximization
problem in section 3 under more general setting, namely the case where
coeﬃcients of security prices are nonlinearly depend on economic factors.
In that case we don’t have explicit expression of the optimal strategies
but study necessity of optimality. We introduce backward stochastic
partial diﬀerential equations (BSPDEs), which are considered to be adjoint equations of the problems, and find the necessary condition for
optimality by using the solutions of the BSPDEs under suitable conditions. Such necessary condition is a kind of maximum principle and it
has been studied by A. Bensoussan for stochastic control problems forpartially observed diﬀusion processes (cf. [1], [2], [26]).
$r_{1}$$b_{1}$$g_{1}$$\lambda_{1}$$\sigma_{1}$$x$$\mathcal{G}_{t}$$X_{t}$$S_{t}$$X_{t}$\S 2.

Full information caseLet us set$Y_{t}^{i}=\log S_{t}^{i}$,$i=0,1$ , 2,$\ldots$, $m$ ,

Risk-sensitive portfolio optimization

and$Y_{t}=(Y_{t}^{1}, Y_{t}^{2}, \ldots, Y_{t}^{m})^{*}$$e^{Y}=(e^{Y^{1}}, \ldots, e^{Y^{m}})^{*}$

261. Then$dY_{t}^{0}=r(X_{t}, e^{Y_{t}})dt$

and
(2.1)

$dY_{t}=E(Xt, Yt)dt+\Sigma(X_{t}, Y_{t})dW_{t}$

,

where
$F^{i}(x, y)$$=$$g^{i}(x, e^{y})-\frac{1}{2}(\sigma\sigma^{*})^{ii}(x, e^{y})$$\Sigma_{k}^{i}(x, y)$$=$$\sigma_{k}^{i}(x, e^{y})$,.

In the same way, set$B(x, y)=b(x, e^{y})$ ,$\Lambda(x, y)=\lambda(x, e^{y})$.

Then the factor prosess is described as(2.2)$dX_{t}=B(X_{t}, Y_{t})dt+\Lambda(X_{t}, Yt)dWu$

So, by setting $Z_{t}=(X_{t}, Y_{t})^{*}$ and$\beta(z)=(B(x, y),$ $F(x, y))^{*}$ ,$\alpha(z)=(\Lambda(x, y),$ $\Sigma(x, y))^{*}$,

we have(2.3)$dZ_{t}=\beta(Z_{t})dt+\alpha(Z_{t})dW_{t}$Furthermore, by setting $\tilde{g}(z)=g(x, e^{y}),\tilde{r}(z)=r(x, e^{y})$ for simplicity
we have$\frac{dV_{t}}{V_{t}}=\tilde{r}(Z_{t})dt+h_{t}^{*}(\overline{g}(Z_{t})-\overline{r}(Z_{t})1)dt+h_{t}^{*}\Sigma(Z_{t})dW_{t}$and so,$V_{t}^{\mu}=v^{\mu}$$\exp\{-\mu\int_{0}^{t}\eta(Z_{s}, h_{s})ds+\mu\int_{0}^{t}h_{s}^{*}\Sigma(Z_{s})dW_{s}$$-\frac{\mu^{2}}{2}\int_{0}^{t}h_{s}^{*}\Sigma\Sigma^{*}(Z_{s})h_{s}ds\}$

,

where
$\eta(z, h)=\frac{1-\mu}{2}h^{*}\Sigma\Sigma^{*}(z)h-\tilde{r}(z)-h^{*}(\tilde{g}(z)-\tilde{r}(z)1)$

262

H. NagaiIf a given investment strategy
(2.4)

$h$

satisfies$E[e^{\mu}\int^{T}oh_{s}^{*}\Sigma^{*}(Z_{s})dW_{s}-\mu_{-\int_{0}^{T}h_{S}^{*}\Sigma\Sigma^{*}(Z_{s})h_{s}ds}^{2}2]=1$

then we can introduce a probability measure$P^{h}$,

given by$P^{h}(A)=E[e\mu\int o^{T}h_{S}^{*}\Sigma(Z_{s})dW_{s}-L^{2}2\int_{0}^{T}h_{S}^{*}\Sigma\Sigma^{*}(Z_{s})h_{s}ds;A]$for $A\in F_{T}$ , $T>0$ . By the probability measure$J(v, x;h;T)$ can be written as follows:(2.5)

our criterion$P^{h}$$J(v, x;h, T)=\log v+\frac{1}{\mu}\log E^{h}[e^{-\mu\int_{0}^{T}\eta(Z_{s},h_{s})ds}]$.

On the other hand, under the probability measure$W_{t}^{h}$$=$$W_{t}-\langle W., \mu\int_{\dot{0}}h^{*}(s)\Sigma(Z_{s})dW_{s}\rangle_{t}$$=$$W_{t}-\mu\int_{0}^{t}\Sigma^{*}(Z_{s})h(s)ds$

is a standard Brownian motion process, and therefore the factor processsatisfies the following stochastic diﬀerential equation$X_{t}$$dX_{s}$$=$$(B(X_{s}, Y_{s})+\mu\Lambda\Sigma^{*}(X_{s}, Y_{s})h_{s})ds+\Lambda(X_{s}, Y_{s})dW_{s}^{h}$$dY_{s}$$=$$(F(X_{s}, Y_{s})+\mu\Sigma\Sigma^{*}(X_{s}, Y_{s})h_{s})ds+\Sigma(X_{s}, Y_{s})dW_{s}^{h}$

(2.6)
.

And so,(2.7)$dZ_{t}=\beta_{\mu}(Z_{t}, h_{t})dt+\alpha(Z_{t})dW_{t}^{h}$

,

where
$\beta_{\mu}(z, h)=\beta(z)+\mu\alpha\Sigma^{*}(z)h$.

We regard (2.7) as a stochastic diﬀerential equation controlled by
the criterion function is written by
as follows:$P^{h}$(2.8)$J(v, x;h;T-t)=\log v+\frac{1}{\mu}\log E^{h}[e^{-\mu\int_{0}^{T-t}\eta(Z_{s},h(s))ds}]$

and the value function(2.9)$u(t, z)=$$\sup$$h\in A(T-t)$$J(v, z;h;T-t)$ ,$0\leq t\leq T$

.

$h$

and

Risk-sensitive portfolio optimization

263

Here we denote by $A(T)$ the set of all investment strategies satisfying(2.4). Then, according to Bellman’s dynamic programming principle, it
should satisfy the following Bellman equation$\underline{\partial u}+\sup L^{h}u=0$,$\partial t$(2.10)$h\in R^{m}$$u(T, z)=\log v$ ,where$L^{h}$

is defined by$L^{h}u(t, z)=\frac{1}{2}tr(\alpha\alpha^{*}(z)D^{2}u)+\beta_{\mu}(z, h)Du+\frac{\mu}{2}(Du)^{*}\alpha\alpha^{*}(z)Du-\eta(z, h)$Note that $\sup_{h\in R^{m}}L^{h}u$ can be written as$\sup_{h\in R^{m}}L^{h}u(t, z)=\frac{1}{2}tr(\alpha\alpha^{*}(z)D^{2}u)+\beta(z)^{*}Du+\frac{\mu}{2}(Du)^{*}\alpha\alpha^{*}Du+\tilde{r}$$+\sup_{h}\{\mu h^{*}\Sigma\alpha^{*}Du+h^{*}(\tilde{g}-\tilde{r}1)-\frac{1-\mu}{2}h^{*}\Sigma\Sigma^{*}h\}$$=\frac{1}{2}tr(\alpha\alpha^{*}(z)D^{2}u)+\beta(z)^{*}Du+\frac{\mu}{1-\mu}(\tilde{g}-\tilde{r}1)^{*}(\Sigma\Sigma^{*})^{-1}\Sigma\alpha^{*}Du$$+\frac{\mu}{2}(Du)^{*}\alpha(I+\overline{1}-A\overline{\mu}\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma)\alpha^{*}Du$$+\frac{1}{2(1-\mu)}(\overline{g}-\tilde{r}1)^{*}(\Sigma\Sigma^{*})^{-1}(\tilde{g}-\tilde{r}1)+\tilde{r}$Therefore our Bellman equation (2.10) is written as follows:(2.11)$\frac{\partial u}{\partial t}+\frac{1}{2}tr(\alpha\alpha^{*}D^{2}u)+\hat{\beta}_{\mu}^{*}Du+(Du)^{*}\alpha N^{-1}\alpha^{*}Du+U(z)=0$$u(T, z)=\log v$ ,where$\hat{\beta}_{\mu}(z)=\beta(z)+\frac{\mu}{1-\mu}\alpha\Sigma^{*}(\Sigma\Sigma^{*})^{-1}(\tilde{g}-\overline{r}1)$(2.12)$N^{-1}(z)=\frac{\mu}{2}(I+\frac{\mu}{1-\mu}\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma(z))$$U(z)=\frac{1}{2(1-\mu)}(\tilde{g}-\tilde{r}1)^{*}(\Sigma\Sigma^{*})^{-1}(\tilde{g}-\overline{r}1)+\tilde{r}(z)$

.

,

.

H. Nagai

264

As for (2.11) we note that if $\mu<0$ , then$\frac{\mu}{2(1-\mu)}I\leq N^{-1}\leq\frac{\mu}{2}I$and therefore we have$\frac{\mu}{2(1-\mu)}\alpha\alpha^{*}\leq\alpha N^{-1}\alpha^{*}\leq\frac{\mu}{2}\alpha\alpha^{*}<0$.

On the other hand if $0<\mu<1$ , then$\frac{\mu}{2}I\leq N^{-1}\leq\frac{\mu}{2(1-\mu)}I$

and therefore we have$0<\frac{\mu}{2}\alpha\alpha^{*}\leq\alpha N^{-1}\alpha^{*}\leq\frac{\mu}{2(1-\mu)}\alpha\alpha^{*}$.

In what follows we assume that$B$

, $F$,$\Lambda$,$\Sigma$are locally Lipshitz and that

(2.11)$\frac{1}{2}||\Sigma\Sigma^{*}||+\frac{1}{2}||\Lambda\Lambda^{*}||+\beta^{*}z\leq c(1+|z|^{2})$then we have a solution$S_{t}^{i}=e^{Y_{t}^{i}}$$(X_{t}, Y_{t})$, $i=1,2$ ,$\ldots$

we have a market model
have the following theorem.

, $m$ ,,of (2.1) and (2.2), and so setting$S_{t}^{0}=e^{Y_{t}^{0}}=e^{\log s^{0}+\int_{0}^{t}r(X_{s},e^{Y_{S}})ds}$$(S_{t}^{0}, S_{t})$satisfying (1.2) and (1.3). Then we

Theorem 2.1. Let $u\in C^{1,2}([0, T)\times R^{N})$ be a solution of (2.11).

Define$\hat{h}_{t}=\hat{h}(t, Z_{t})$$\hat{h}(t, z)=\frac{1}{1-\mu}(\Sigma\Sigma^{*})^{-1}(\tilde{g}-\tilde{r}1+\mu\Sigma\alpha^{*}Du)(t, z)$where(2.14)$Z_{t}$,

is the solution of (2.3), then, under the assumption that$E[e^{-\int_{0}^{T}(2N^{-1}\alpha^{*}Du+2\mu K)^{*}dW_{s}-\frac{1}{2}\int_{0}^{T}(2N^{-1}\alpha^{*}Du+2\mu K)^{*}(2N^{-1}\alpha^{*}Du+2\mu K)ds}]=1$

,

265

Risk-sensitive portfolio optimization

with$K=\frac{1}{2(1-\mu)}\Sigma^{*}(\Sigma\Sigma^{*})^{-1}(\overline{g}-\tilde{r}1)$

is an optimal strategy for the portfolio optimization problem ofmaximizing the criterion (1.5).

$\hat{h}_{t}\in A_{T}$The proof of this theorem is similar to that of Proposition 2.1 in [23]and we omit it here.
We then consider equation (2.11). Such kinds of equations have beenstudied in [20], or [3] in relation to risk-sensitive control problems undermore general settings in the case of $\mu<0$ and in [4] in the case where$\mu>0$ . Here we consider the case where $\mu<0$ and obtain the following
result along the line [3], Theorem 5.1 with refinement on estimate (2.16).
It is a generalization of Theorem 2.1 in [23].
Theorem 2.2. )$i$(2.15)$If_{f}$in addition to (2.13), $\mu<0$ and$\iota/_{r}|\xi|^{2}\leq\xi^{*}\alpha\alpha^{*}(z)\xi\leq\nu_{r}^{/}|\xi|^{2}$,$r=|z|$ ,$lJ_{r}$,$lJ_{r}^{J}>0$,

then we have a solution of (2.11) such that$u$,$\frac{\partial u}{\partial t}$$\frac{\partial^{2}u}{\partial t^{2}}$,, $D_{k}u$ , $D_{kj}u\in L^{p}(0, T;L_{loc}^{p}(R^{n+m}))$ ,$\frac{\partial D_{k}u}{\partial t}$$u\geq\log v$,,$\frac{\partial D_{k}u}{\partial t}$$\frac{\partial u}{\partial t}\leq 0$, $D_{kjl}u\in L^{p}(0, T;L_{loc}^{p}(R^{n+m}))$ ,.
$ 1<\forall p<\infty$

Furthermore we have the estimate$|\nabla u|^{2}(t, z)-\frac{c_{0}}{\nu_{r}}\frac{\partial u}{\partial t}(t, z)\leq c_{r}(|\nabla Q|_{2r}^{2}+|Q|_{2r}^{2}$(2.16)$+|\nabla(\alpha\alpha^{*})|_{2r}^{2}+|\nabla\beta_{\mu}|_{2r}+|\beta_{\mu}|_{2r}^{2}$$+|U|_{2r}+|\nabla U|_{2r}^{2}+1)$,$z\in B_{r}$,$t\in[0, T)$

where$Q=\alpha N^{-1}\alpha^{*}$,$c_{0}=\frac{4(1+c)(1-\mu)}{-\mu}$$|\cdot|_{2r}=||\cdot||_{L^{\infty}(B_{2r})}$, $c>0$

H. Nagai

266

and

$c_{r}$$ii)$

is a positive constant depending on , ,
$n$

$r$

$\nu_{r}$,$\nu_{r}^{/}$

and .
$c$

If, in addition to the above conditions,$\inf_{|z|\geq r}U(z)$,$r^{2}\frac{1}{\nu_{r}^{/}}\inf_{|z|\geq r}U(z)$,$ r\inf\underline{U(z)}\rightarrow\infty$$|z|\geq r|\beta_{\mu}(z)|$as$ r\rightarrow\infty$, then the above solution$\inf$$u$satisfies$ u(z, t)\rightarrow\infty$,as$ r\rightarrow\infty$.

$|z|\geq r,t\in(0,T)$

Moreover, there exists at most one such solution in $L^{\infty}(0, T;W_{loc}^{1,\infty}(R^{n+m}))$Remark. If$\underline{1}$(2.17), $\nu_{r}^{/}\leq M(1+r^{m})$ ,$\exists m>0$,$\nu_{r}$then we have$c_{r}\leq M^{/}(1+r^{m’})$,$\exists m’$in estimete (2.16). In particular, if $m=0$ , thenpendent of .

$c_{r}$

can be taken inde-$r$

Corollary 2.1. Condition (2.14) is valid if$c_{1}|\xi|^{2}\leq\xi^{*}\alpha\alpha^{*}(x)\xi\leq c_{2}|\xi|^{2}$,$c_{1}$, $c_{2}>0$(2.18)$B$, $F$,$\Lambda$,$\Sigma$are globally Lipshitz.

The proofs of Theorem 2.2 and Corollary 2.1 are similar to thoseof Theorem 2.1 and Proposition 2.1 (ii) in [23] and we omit them here.
Instead, we illustrate an example.

Example (Generalized linear Gaussian factor model)
Let us consider the case where, and are all linear functionsof and and are constant matrices, namely$B,\tilde{g}$$z$$\Lambda$$\tilde{r}$$\Sigma$$\beta(z)=\left(\begin{array}{l}B(x,y)\\F(x,y)\end{array}\right)$$=(A_{1}x+A_{2}y+a-\frac{1}{2}\overline{(\Sigma\Sigma*)}B_{1}x+B_{2}y+b)$$\overline{g}(x, y)=A_{1}x+A_{2}y+a$,$\tilde{r}(x, y)=R_{1}^{*}x+R_{2}^{*}y+r$

,

267

Risk-sensitive portfolio optimization

where$\overline{(\Sigma\Sigma^{*})}=((\Sigma\Sigma^{*})^{ii})\in R^{m}$$\hat{\beta}_{\mu}(z)$. Then$=$$\beta(z)+\frac{\mu}{1-\mu}(\Lambda\Sigma^{*}(\Sigma\sum_{\tilde{g}}*)^{-1}(\tilde{g}-r1)-r1)$$=$$K_{1}z+L_{1}$

where$K_{1}=$$(^{B_{1}}+\frac{\mu}{1-}\Sigma*(\sum_{-\frac{\mu 1\Lambda}{1-\mu}A_{1_{\overline{1}-\overline{\mu}}}^{A}1R_{1}^{*}}\Sigma*)^{-1}(A_{1}-1R_{1}^{*})B_{2}+\frac{\mu}{1-}\Sigma*(\sum_{-\frac{\mu 1\Lambda}{1-\mu}A_{2_{\overline{1}-\overline{\mu}}}^{\lrcorner i}1R_{2}^{*}}\Sigma*)^{-1}(A_{2}-1R_{2}^{*}))$$L_{1}=(b+\frac{\mu}{1\mu\frac{-1}{2}}(\sum_{+a-\frac{\Lambda\Sigma^{*}}{(\Sigma\Sigma^{*})}}\Sigma*)^{-1}(a-r1)\overline{1}-A\overline{\mu}(a-r1))$.

Furthermore$\alpha N^{-1}\alpha^{*}=\frac{\mu}{2}$($\Lambda(I+\frac{\mu}{1-}(\sum_{\Sigma\frac{\mu\sum_{1}*}{1-\mu}\Lambda^{*}}\Sigma*)^{-1}\Sigma)\Lambda^{*}$$\frac{\frac{1}{1-1\mu}}{1-\mu}\Sigma\Sigma^{*}\Lambda\Sigma*)\equiv\frac{1}{2}K_{0}$$U(z)=\frac{1}{2}z^{*}K_{2}z+L_{2}z+r+\frac{1}{2(1-\mu)}(a-r1)^{*}(\Sigma\Sigma^{*})^{-1}$,(a-rl),

where$K_{2}=$$\frac{1}{1-\mu}(_{(A_{2}-1R_{2}^{*})^{*}}^{(A_{1}-1R_{1}^{*})^{*}(\Sigma\sum_{*})^{-1}(A_{1}-1R_{1}^{*})}(\Sigma\Sigma*)^{-1}(A_{1}-1R_{1}^{*})(A_{2}-1R_{2}^{*})^{*}(A_{1}-1R_{1}^{*})^{*}(\sum_{(\Sigma}\sum_{\Sigma^{*}}*)^{-1}(A_{2}-1R_{2}^{*}))^{-1}(A_{2}-1R_{2}^{*}))$

and$L_{2}=\frac{1}{1-\mu}((A_{2}(A_{1}--1R_{2}^{*})^{*}()^{-1}(a1R_{1}^{*})^{*}(\sum_{\Sigma}\sum_{\Sigma^{*}}*)^{-1}(a--r1)r1))$.

In this case the solution to (2.11) has an explicit form such that$u(t, z)=\frac{1}{2}z^{*}P(t)z+q(t)^{*}z+k(t)$ ,provided that equation (2.19) below has a solution. Here $P(t)$ , $q(t)$ and$k(t)$ are the solutions to the following ordinary diﬀerential equations:(2.19)$\dot{P}(t)+K_{1}^{*}P(t)+P(t)K_{1}+P(t)K_{0}P(t)+K_{2}=0$ ,$P(T)=0$ ,(2.20)$\dot{q}(t)+K_{1}q(t)+P(t)L_{1}+P(t)K_{0}q(t)+$$\left(\begin{array}{l}R_{1}\\R_{2}\end{array}\right)$$+L_{2}=0$ ,$q(T)=0$268

H. Nagai

and(2.21)$\dot{k}(t)+\frac{1}{2}tr(\alpha\alpha^{*}P(t))+L_{1}^{*}q(t)+r+\frac{1}{2(1-\mu)}(a-r1)^{*}(\Sigma\Sigma^{*})^{-1}$(a-rl) $=0$ ,$k(T)=\log v$ .

Note that if $\mu<0$ , then (2.19) has a unique solution and so do (2.20)
and (2.21).

\S 3.

Partial information caseNow we consider a partial information case. Namely, the case where
portfolio strategies are selected by using only past information of secuis consideredrity prices. In this case the economic factor process
unobservable and so we cannot use the information about it to choose
defined before by (2.2) may be
our strategies. Thus the factor process
reformulated as the solution with the initial condition $X_{0}=x_{0}$ , where. We
is a random variable having a disrtibution density $\pi(x)$ on
then introduce$X_{t}$$X_{t}$$R^{n}$$x_{0}$$\tilde{\mathcal{G}}_{t}=\sigma(S(u);u\leq t)$

measurable. In thisand the admissible strategies are assumed to be
above,
namely we assume
thespecific
one thancase we consider more
(2.18) and that $\sigma(x, S)=\sigma(S)$ .
Then we consider the problem maximising the criterion (1.5) byselecting portfolio stratgies which aremeasurable.
Let us set$\tilde{\mathcal{G}}_{t}$$\tilde{\mathcal{G}}_{t}$(3.1)$I(v;h;T)=E[e^{\mu\log V_{T}(h)}]$ ,and reformulate the problem as the one of partially observable stochastic
control. Recall thatis a solution of$Y_{t}$$(2.1)^{/}$$dY_{t}=F(X_{t}, Y_{t})dt+X(Yt)dWt$ ,in the present case and we regard it as the SDE defining the observation
process. On the other hand,defined by (2.2) with the initial condition$X_{0}=x_{0}$ is regarded as a system process. System noise,;are correlated in general.
and observation noise$t)=\sigma(S(u);u\leq t)$ holds sinceastrictlyfunction,so ourisincreasingproblem is to minimize (or maximize) the criterion (3.1) while looking at
measurable
the observation process
and choosing astrategy $h(t)$ . Though there is no control in SDE (2.2) defining the$X_{t}$$\Lambda(X_{t}, Y_{t})dW_{t}$$\Sigma(Y_{t})dW_{t}$$\sigma(Y_{u}$$\log$$Y_{t}$$\sigma(Y_{u}, ; u\leq t)=\tilde{\mathcal{G}}_{t}$$ u\leq$269

Risk-sensitive portfolio optimization

system process
the criterion $I(v;h;T)$ is defined as a functional of thestrategy $h(t)$ measurable with respect to observation and the problem isthe one of stochstic control with partial observation.
In what follows we consider the case where $F(x, y)=F(x)$ , $\Sigma(y)=$constant, $B(x, y)=B(x)$ , $\Lambda(x, y)=\Lambda(x)$ , $\tilde{r}(x, y)=r(x)$ for
simplicity. Similar arguments are possible for general case as long as
does not depend on . Now let us introduce a new probability measuredefined byon$X_{t}$$\Sigma\equiv$$\Sigma$$x$$\hat{P}$$(\Omega, F)$$\frac{d\hat{P}}{dP}|_{F_{T}}=\rho_{T}$

,

where
$\rho_{t}=\exp\{-\int_{0}^{t}F(X_{s})^{*}(\Sigma\Sigma^{*})^{-1}\Sigma dW_{s}$(3.2)$-\frac{1}{2}\int_{0}^{t}F(X_{s})^{*}(\Sigma\Sigma^{*})^{-1}F(X_{s})ds\}$.

We see thatis a probability measure since it can be seen by standard arguments (cf. [1]) thatis a martingale and $E[\rho\tau]=1$ underassumption (2.18). Moreover, according to Girsanov theorem,$\hat{P}$$\rho_{t}$(3.3)$\hat{W}_{t}=W_{t}+\int_{0}^{t}\Sigma^{*}(\Sigma\Sigma^{*})^{-1}F(X_{s})ds$

turns out to be a standard Brownian motion process under the probability measure
and we have$\hat{P}$

(3.4)

(3.5)

$dY_{t}=\Sigma d\hat{W}_{t}$$dX_{t}=\{B(X_{t})-\Lambda(X_{t})\Sigma^{*}(\Sigma\Sigma^{*})^{-1}F(X_{t})\}dt+\Lambda(X_{t})d\hat{W}_{t}$We rewrite our criterion $I(v;h;T)$ by new probability measure(3.3)$I(v;h;T)=v^{\mu}\hat{E}$[ [exp{$ E\wedge$$-\mu\int_{0}^{T}\eta(X_{s},$$h_{s})ds\}\Psi_{T}|\tilde{\mathcal{G}}_{t}]$$\hat{P}$

.

.

],

where
$\Psi_{t}=\exp\{\int_{0}^{t}Q(X_{s}, h_{s})^{*}dY_{s}-\frac{1}{2}\int_{0}^{t}Q(X_{s}, h_{s})^{*}(\Sigma\Sigma^{*})Q(X_{s}, h_{s})ds\}$and$Q(x, h)^{*}$$=$$(\Sigma\Sigma^{*})^{-1}F(x)+\mu h$$=$$(\Sigma\Sigma^{*})^{-1}\{F(x)+\mu(\Sigma\Sigma^{*})h\}$

.

270

H. Nagai

Set
(3.7)

$q^{h}(t)(\varphi(t))=\hat{E}$[exp{$-\mu\int_{0}^{t}\eta(X_{s},$ $h_{s}))ds\}\Psi_{t}\varphi(t$,$X_{t})|\mathcal{G}_{t}$].

Then (3.6) reads(3.8)$I(v;h;T)=v^{\mu}\hat{E}[q^{h}(T)(1)]$Hence, if $\mu<0$ (resp. $1>\mu>0$ ) our problem is reduced to minimize(resp. maximize) I of (3.8) when taking over $H(T)$ . Let us set
$h$

(3.8)

$ L\varphi=\frac{1}{2}(\Lambda\Lambda^{*})^{ij}(x)D_{ij}\varphi+B(x)^{i}D_{i}\varphi$.

Here and in what follows we utilize summation convention. Then, we cansee thatsatisfies a so called modified Zakai equation in a similar
way to deducing Zakai equations as for conditional expectations of diffusion processes with respect to unnormalized conditional probabilities(cf. [2], [13], [21]). We actually have the following proposition.
$q^{h}(t)$Proposition 3.1. Assume (2.18), Then $q(t)(\varphi(t))\equiv q^{h}(t)(\varphi(t))$ satisfies the following stochastic partial diﬀerential equation (SPDE):(3.10)$q(t)(\varphi(t))=q(0)(\varphi(0))+\int_{0}^{t}q(s)(\frac{\partial\varphi}{\partial t}(s, \cdot)+L\varphi(s, \cdot)+\mu h_{s}^{*}\Sigma\Lambda^{*}(\cdot)D\varphi(s, \cdot)$$-\mu\eta_{s}(\cdot)\varphi(s, \cdot))ds+\int_{0}^{t}q(s)(\varphi(s, \cdot)Q(\cdot, h_{s}))dY_{s}$$+\int_{0}^{t}q(s)((D\varphi)^{*}(s, \cdot)\Lambda(\cdot)\Sigma^{*}(\Sigma\Sigma^{*})^{-1})dY_{s}$where$\eta_{s}(\cdot)=\eta(\cdot, h_{s}))$,.

Let us introduce some notations and describe a strong form ofstochstic partial diﬀerential equation (3.10). Set$L^{0}\varphi=\frac{1}{2}D_{i}(\Lambda\Lambda^{*}(x)^{ij}D_{j}\varphi)$,$\tilde{B}(x)^{i}=B(x)^{i}-\frac{1}{2}D_{j}(\Lambda\Lambda^{*})^{ji}$Then$ L\varphi=L^{0}\varphi+\tilde{B}(x)^{*}D\varphi$.

and its formal adjoint$L^{*}\varphi=L^{0}\varphi-D_{i}(\tilde{B}(x)^{i}\phi)$$L^{*}$

is written as.

We set$ G(h)\varphi=-D_{i}(\tilde{B}^{i}(\cdot)\varphi)-\mu h_{s}^{i}D_{j}((\Sigma\Lambda^{*})^{ij}\varphi)-\mu\eta(\cdot, h_{s})\varphi$

271

Risk-sensitive portfolio optimization

and$M(h)_{j}\varphi=Q_{j}(\cdot, h_{s})\varphi-D_{i}([\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}\varphi)$

We define$\mathcal{L}_{Y}^{2}(0, T;H^{1}(R^{n}))$$=$$\{v\in L^{2}(\Omega,$ $F,\hat{P};L^{2}(0, T;H^{1}(R^{n}))$$v(t)\in L^{2}(\Omega,\tilde{\mathcal{G}}_{t},\hat{P};H^{1}(R^{n}))a.e$

.

,

$t$

}

Then we consider the following stochastic partial diﬀerential equation.
which has a solution $q(t)$ such that$q_{t}e^{\delta\sqrt{1+|x|^{2}}}\in \mathcal{L}_{Y}([0, T];H^{1})$(3.11)$dq_{t}=(L^{0}q_{t}+G(h)q_{t})dt+M(h)_{j}q_{t}dY_{t}^{j}$.

Furthermore we assume(3.12)$\Lambda$,$ D\Lambda$

are bounded, $B$ , $DB$ , $F$,

is defined as the totality ofand the set of admissible strategies
measurable strategy satisfying the condition i) of definition 2.1 and. Take a positive constant,for some convex compact$\delta>0$ .
Then we have the following theorem.
$\tilde{\mathcal{G}}_{t}$$A_{T}$$h$$ h_{t}\in\Gamma$$\Gamma\subset R^{m}$$\forall t$

Proposition 3.2. Le us assume (2.18), (3.12), and. Then for each addmissible strategy $h(3.11)$ has a unique solu;
tion $q_{t}=q(t, x)$ such that
$C(0, T;L^{2}(R^{n}))$ and that $ q_{0}=\pi$ . Furthermore we have $\int q(T,x)\psi(x)dx=$$q(T)(\psi)$
for all bounded Borel function .
$\pi e^{\delta\sqrt{1+|x|^{2}}}\in$$H^{1}$$q_{t}e^{\delta\sqrt{1+|x|^{2}}}\in \mathcal{L}_{Y}^{2}(0, T;H^{1}(R^{n}))\cap L^{2}(\Omega,$$\mathcal{F},\hat{P}$$\psi$For the proof of this proposition we prepare the following lemma.
Lemma 3.1. Under assumption (2.18)$\Lambda(I_{n+m}-\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma)\Lambda^{*}\geq c_{1}I_{n}$Proof. Note that
(3.11)$(\xi_{1}^{*}, \xi_{2}^{*})$$\left(\begin{array}{ll}\Lambda\Lambda^{*} & \Lambda\Sigma^{*}\\\Sigma\Lambda^{*} & \Sigma\Sigma^{*}\end{array}\right)\left(\begin{array}{l}\xi_{1}\\\xi_{2}\end{array}\right)\geq c_{1}|\xi|^{2}$

under assumtion (2.18). Therefore, setting, we see that,$\forall\xi=\left(\begin{array}{l}\xi_{1}\\\xi_{2}\end{array}\right)$$\zeta=-(\Sigma\Sigma^{*})^{-1}\Sigma\Lambda^{*}\xi_{1}$$\xi_{1}\in R^{n}$$\xi_{1}^{*}\Lambda(I_{n+m}-\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma)\Lambda^{*}\xi_{1}$$=\xi_{1}^{*}\Lambda\Lambda^{*}\xi_{1}+\xi_{1}^{*}\Lambda\Sigma^{*}\zeta+\zeta\Sigma\Lambda^{*}\xi_{1}+\zeta\Sigma\Sigma^{*}\zeta$$\geq c_{1}(|\xi_{1}|^{2}+|\zeta|^{2})\geq c_{1}|\xi_{1}|^{2}$

.

for

272

H. Nagai$\square $Set
Proof of Proposition 3.2.
$\nu(x)=\delta\sqrt{1+|x|^{2}}$ . Then (3.11) can be written as$\tilde{q}_{t}=q_{t}e^{\delta\sqrt{1+|x|^{2}}}$(3.14)$d\tilde{q}_{t}=(L^{0}\overline{q}_{t}+\tilde{G}(h)\tilde{q}_{t})dt+\tilde{M}(h)_{j}\overline{q}_{t}dY_{t}^{j}$, $\delta>0$ and

,

where
$\tilde{G}(h)\tilde{q}=G(h)\tilde{q}+(\Lambda\Lambda)^{ij}D_{j}\nu D_{i}\tilde{q}$$+\{\frac{1}{2}D_{i}((\Lambda\Lambda)^{ij}D_{j}\nu)+\frac{1}{2}(\Lambda\Lambda^{*})^{ij}D_{i}\nu D_{j}\nu-(\tilde{B}^{i}D_{i}\nu+\mu h^{i}(\Sigma\Lambda^{*})^{ij}D_{j}\nu)\}\tilde{q}$,$\tilde{M}(h)_{j}\overline{q}=\tilde{Q}_{j}\tilde{q}-D_{i}([\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}\tilde{q})$and$\tilde{Q}_{j}=Q_{j}-D_{i}\nu[\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}$.

It suﬃces to check the coercivity condition for (3.14) because of general
theory of stochastic partial diﬀerential equarions ([2], [13], [25]):$-2\langle L^{0}q, q\rangle-2\langle\tilde{G}(h)q, q\rangle+c_{1}||q||_{L^{2}}^{2}\geq c_{0}||q||_{H^{1}}^{2}+\langle\tilde{M}(h)_{j}q, (\Sigma\Sigma^{*})^{jk}\tilde{M}(h)_{k}q\rangle$

Indeed, setting$\hat{Q}_{j}=\tilde{Q}_{j}-D_{i}([\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i})$we see that$-2\langle L^{0}q, q\rangle-\langle\tilde{M}(h)_{j}q, (\Sigma\Sigma^{*})^{jk}\tilde{M}(h)_{k}q\rangle$$=\int(Dq)^{*}\Lambda\Lambda^{*}Dqdx-\int(Dq)^{*}\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma\Lambda^{*}Dqdx$$-\int(\hat{Q}^{*}\Sigma\Sigma^{*}\hat{Q})q^{2}dx-2\int(Dq)^{*}\Lambda\Sigma^{*}\hat{Q}qdx$$\geq\int(Dq)^{*}\Lambda(I-\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma)\Lambda^{*}Dqdx-\int\hat{Q}(\Sigma\Sigma^{*})\hat{Q}q^{2}dx$$-\epsilon\int|Dq|^{2}dx-\frac{1}{\epsilon}\int|\Lambda\Sigma^{*}\hat{Q}|^{2}q^{2}dx$$\geq c_{2}\int|Dq|^{2}dx-\epsilon\int|Dq|^{2}dx-c_{3}\int|q|^{2}dx$

for some , $c_{3}>0$ and suﬃciently small
lemma. Since$c_{2}$$\epsilon>0$

by using the above$|\langle\tilde{G}q, q\rangle|\leq\epsilon_{1}\int|Dq|^{2}dx+(\frac{c_{4}}{\epsilon_{1}}+c_{5})\int|q|^{2}dx$

we can easily see the coercivity condition holds for (3.14).
$\square $273

Risk-sensitive portfolio optimization

Lemma 3.2. Let us assume the assumptions of the above proposition and ,be admissible strategies, then
$h$

(315)

$k$

$\lim_{\theta\rightarrow 0}.\frac{I(v,h.+\theta k.,T)-I(v\ovalbox{\tt\small REJECT} h.\cdot T)}{\theta}.,=v^{\mu}\hat{E}[<\zeta_{T}(k), 1>]$

where
ential equation(3.16)$\zeta_{t}=\zeta_{t}(k)$,is a solution of the following stochastic partial diﬀer-$d\zeta_{t}=(L^{0}\zeta_{t}+G(h)\zeta_{t}+k_{t}^{i}G_{h^{i}}(h)q_{t})dt+(M(h)_{j}\zeta_{t}+k_{t}^{i}M_{h^{i}}(h)_{j}q_{t})dY_{t}^{j}$$\zeta_{0}=0$

where$q_{t}$,
is a solution to (3.11),$G_{h^{i}}(h)q$$=$$-\mu D_{j}((\Sigma\Lambda^{*})^{ij}q)-\mu\frac{\partial\eta}{\partial h^{i}}q$$=$$-\mu D_{j}((\Sigma\Lambda^{*})^{ij}q)-\mu[(1-\mu)(\Sigma\Sigma^{*}h)^{i}-(g-r1)^{i}]$and$M_{h^{i}}(h)_{j}q=\frac{\partial Q_{j}(\cdot,h)}{\partial h^{i}}q=\mu\delta_{ij}q$.

Proof. Note that we can see that (3.14) has a unique solution such$\mathcal{F},\hat{P};C(0, T;L^{2}(R^{n}))$
in a
that
similar way to the proof of the above proposition. Let us set$q_{t}e^{\delta\sqrt{1+|x|^{2}}}\in \mathcal{L}_{Y}^{2}(0, T;H^{1}(R^{n}))\cap L^{2}(\Omega,$$\overline{q}_{\theta}(t)=\frac{q_{\theta}(t)-q(t)}{\theta}-\zeta$

where(3.17)$q_{\theta}(t)$,is the solution to :$dq_{\theta}(t)=\{L^{0}q_{\theta}(t)+G(h+\theta k)q_{\theta}(t)\}dt+M(h+\theta k)_{j}q_{\theta}(t)dY_{t}^{j}$$ q_{\theta}(0)=\pi$

We define in the same way as above$\tilde{q}_{\theta}(t)=q_{\theta}(t)e^{\nu(x)}$,$\tilde{\zeta}=\zeta e^{\nu(x)}$.

Then in a similar way to $getting_{-}(3.14)$ , we have stochastic partial differential equations forand . We set$\tilde{q}_{\theta}(t)$$\zeta$$q_{\theta}(\simeq t)=\frac{\tilde{q}_{\theta}(t)-\tilde{q}(t)}{\theta}-\tilde{\zeta}$.

Then we can see that$\sup_{0\leq t\leq T}E[||q_{\theta}(\simeq t)||_{L^{2}}^{2}]\rightarrow 0$

274

H. Nagaiasby using the energy equality for$\theta\rightarrow 0$$q_{\theta}(\simeq t)$. Since$.\frac{I(v,h.+\theta k.,T)-I(v,h.,T)}{\theta}...-v^{\mu}\hat{E}[\langle\zeta_{T}, 1\rangle]=v^{\mu}\hat{E}[\langle q_{\theta}(\simeq T), e^{-\nu(x)}\rangle]$

we obtain the present proposition.
$\square $Let us introduce the following backward stochastic partial diﬀerential equation.
(3.18)$-d\gamma_{t}$$\gamma_{T}$$=$$(L^{0}\gamma_{t}+\hat{G}(h)\gamma_{t}+\hat{M}(h)R_{t})dt-R_{t}^{*}(\Sigma\Sigma^{*})^{-1}dY_{t}$

$=$

1

where
$\hat{G}(h)\varphi=\tilde{B}^{*}D_{\varphi}+\mu h^{*}\Sigma\Lambda^{*}D\varphi-\mu\eta(\cdot, h)\varphi$$\hat{M}(h)R=R^{j}Q_{j}(\cdot, h)+[\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}D_{i}R^{j}$.

Set$\check{\gamma}_{t}=e^{-\nu(x)}\gamma_{t}$,$\check{R}_{t}=e^{-\nu(x)}R_{t}$.

Then we have the following backward SPDE$-d\check{\gamma}_{t}$(3.19)$\check{\gamma}_{T}$

$=$
$=$

$(L^{0}\check{\gamma}_{t}+\check{G}(h)\check{\gamma}_{t}+\check{M}(h)\check{R}_{t})dt-\check{R}_{t}^{*}(\Sigma\Sigma^{*})^{-1}dY_{t}$$e^{-\nu(x)}$

,

where
$\check{G}(h)\varphi=\{(D\nu)^{*}\Lambda\Lambda^{*}+\tilde{B}^{*}+\mu h^{*}\Sigma\Lambda^{*}\}D\varphi$$+\{L^{0}\nu+\frac{1}{2}(D\nu)^{*}\Lambda\Lambda^{*}D\nu+(\tilde{B}^{*}+\mu h^{*}\Sigma\Lambda^{*})D\nu-\mu\eta(\cdot, h)\}\varphi$$\equiv G_{1}^{*}D\varphi+G_{2}\varphi$

and$\check{M}(h)U=\sum_{i,j}[\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}D_{i}U^{j}$$+\sum_{j}\{Q_{j}(\cdot, h)+\sum_{i}[\Lambda\Sigma^{*}(\Sigma\Sigma^{*})^{-1}]_{j}^{i}D_{i}\nu\}U^{j}$$\equiv\sum_{i,j}(M_{1})_{j}^{i}D_{i}U^{j}+\sum_{j}(M_{2})_{j}U^{j}$

.

275

Risk-sensitive portfolio optimizationLet
be a solution to (3.19) with the terminal condition
set $(M_{2}^{J})_{j}=(M_{2})_{j}+D_{j}(G_{1})^{j}$ . We have by It\^o’s formula(3.20)$\check{\gamma}_{t}$$\check{\gamma}_{T}=0$

and$\hat{E}[||\check{\gamma}_{t}||_{L^{2}}^{2}]$$=\hat{E}[\int_{t}^{T}\{2\langle L^{0}\check{\gamma}_{s}+\check{G}(h)\check{\gamma}_{s}+\check{M}(h)\check{R}_{s},\check{\gamma}_{s}\rangle-(\check{R}_{s}, (\Sigma\Sigma^{*})^{-1}\check{R}_{s})\}ds]$$=\hat{E}[\int_{t}^{T}\int\{-(D\check{\gamma}_{s})^{*}\Lambda\Lambda^{*}D\check{\gamma}_{s}+G_{1}^{*}D(\check{\gamma}_{s}^{2})+2G_{2\check{\gamma}_{s}}^{2}$$+2(M_{2}^{J})_{j}\check{R}_{s}^{j}\check{\gamma}_{s}-2\check{R}_{s}^{j}(M_{1})_{j}^{i}D_{i}\check{\gamma}_{s}-\check{R}_{s}^{*}(\Sigma\Sigma^{*})^{-1}\check{R}_{s}\}dxds$$=\hat{E}[\int_{t}^{T}\int\{-(D\check{\gamma}_{s})\Lambda(I-\Sigma^{*}(\Sigma\Sigma^{*})^{-1}\Sigma)\Lambda^{*}D\check{\gamma}_{s}$$-[(\Sigma\Sigma^{*})^{-1}\check{R}_{s}+M_{1}^{*}D\check{\gamma}_{s}-(M_{2}^{J})^{*}\gamma_{s}]^{*}(\Sigma\Sigma^{*})[(\Sigma\Sigma^{*})^{-1}\check{R}_{s}$$+M_{1}^{*}D\check{\gamma}_{s}-(M_{2}’)^{*}\gamma_{s}]$$+[M_{2}^{/}\Sigma\Sigma^{*}(M_{2}^{J})^{*}+2G_{2}-\sum_{j}D_{j}(G_{1}+M_{1}\Sigma\Sigma^{*}(M_{2}’)^{*})^{j}]\check{\gamma}_{s}^{2}\}dxds$$\leq C\int_{t}^{T}\hat{E}[||\check{\gamma}_{s}||_{L^{2}}^{2}]ds$

for some constant $C>0$ . By using (3.20) we can obtain the following
lemma.

Lemma 3.3. Under the assumptions of Proposition 3.2 the solution
to (3.18) such that$\mathcal{F},\hat{P};C(0, T;L^{2}(R^{n}))$and$F,\hat{P};C(0, T;L^{2}(R^{n}))$ , $i=1,2$ ,, $m$ is unique.
$(\gamma_{t}, R_{t})$$ e^{-\delta\sqrt{1+|x|^{2}}}\gamma_{t}\in \mathcal{L}_{Y}^{2}(0, T;H^{1}(R^{n}))\cap$$ e^{-\delta\sqrt{1+|x|^{2}}}R^{i}\in \mathcal{L}_{Y}^{2}(0, T;H^{1}(R^{n}))\cap$$L^{2}(\Omega,$$L^{2}(\Omega,$$\ldots$

We can also see the existence of the solution to (3.18) in a similar
way to Theorem 8.2.3 [2] through aproximation procedure, or directly
thanks to Chapter 5, Theorem 2.2 in [16].
Lemma 3.4. Under the assumptions of Proposition 3. 2$\hat{E}[<\zeta_{T}, 1>]=\hat{E}[\int_{0}^{T}\{\langle\gamma_{t}, k_{t}^{i}G_{h^{i}}(h)q_{t}>+<R_{t}^{j}, k_{t}^{i}M_{h^{i}}(h)_{j}q_{t}>\}dt]$.

Proof. From (3.16) and (3.18) we obtain$d\langle\zeta_{t}, \gamma_{t}\rangle=\{\langle k_{t}^{i}G_{h^{i}}(h)q_{t}, \gamma_{t}\rangle+\langle k_{t}^{i}M_{h^{i}}(h)_{j}q_{t}, R_{t}^{j}\rangle\}dt$$+\{\langle M(h)_{j}\zeta_{t}, \gamma_{t}\rangle+\langle k_{t}^{i}M_{h^{i}}(h)_{j}q_{t}, \gamma_{t}\rangle+\langle M(h)_{j}\zeta_{t}, \gamma_{t}\rangle\}dY_{t}^{j}$and we have the present lemma.
$\square $276

H. NagaiFinally we have the following theorem

Theorem 3.1. We assume the assumptions of Proposition 3.2. If
is optimal, then it satisfies(3.21)$h$$(k-h_{t})^{*}\{-(1-\mu)(\Sigma\Sigma^{*})h_{t}<\gamma_{t}$, $q_{t}>+<\Sigma\Lambda^{*}D\gamma_{t}$ , $q_{t})>$,,$+<(g-r1)\gamma_{t}$ $q_{t}>+<R_{t}$ $q_{t}>\}\leq 0$$a.e$. $ta.s$ .

$\forall k\in\Gamma$,.

Proof. Let ,be admissible atrategies andis an optimal one.
$h+\theta(k-h)=(1-\theta)h+\theta
k\in\Gamma$. Thus, for ,
Since is convex
we have$h_{t}$$k_{t}$$h_{t}$$h$$\Gamma$$I(v;h.

; T)\geq I(v;h.

+\theta(k.

-h.);T)$ ,$ k\in\Gamma$$0\leq\forall\theta\leq 1$

if $\mu>0$ . Therefore, because of Lemma 3.2$\hat{E}[\langle\zeta_{T}(k-h), 1\rangle]\leq 0$,

which implies that
(3.22)$\hat{E}[\int_{0}^{T}\{\langle\gamma_{t}, (k_{t}^{i}-h_{t}^{i})G_{h^{i}}(h)q_{t}\rangle+\langle R_{t}^{j}, (k_{t}^{i}-h_{t}^{i})M_{h^{i}}(h)_{j}q_{t}\rangle\}dt]\leq 0$

for all admissible strategy$k_{t}$by Lemma 3.4. Set.

$(U_{t})_{i}=\langle\gamma_{t}, G_{h^{i}}(h)q_{t}\rangle+\langle R_{t}^{j}, M_{h^{i}}(h)_{j}q_{t}\rangle$

For each $t_{0}\in[0, T]$ ,, we define$\epsilon>0$, $M>0$ andmeasurable random variable$\tilde{G}_{t_{0}}$$k_{t_{0}}$$kt_{0}1\{|U_{t}|\leq M\}+h_{t}1_{\{|U_{t}|>M\}}$$k_{t}=\{$$h_{t}$,,$ t_{0}\leq t\leq t_{0}+\epsilon$$t\in[t_{0}, t_{0}+\epsilon]^{c}$Then, through limiting procedure as(3.22) by , we see that$\epsilon\rightarrow 0$

and $M$$\rightarrow\infty$.

after multilying$\frac{1}{\epsilon}$$\hat{E}[(k_{t_{0}}-h_{t_{0}})^{i}(U_{t_{0}})_{i}]\leq 0$

for each$t_{0}$

and$\tilde{G}_{t_{0}}$measurable random variable$k_{t_{O}}$, which implies that$(k-h_{t})^{i}\{\langle\gamma_{t}, G_{h^{i}}(h)q_{t}\rangle+\langle R_{t}^{j}, M_{h^{i}}(h)_{j}q\rangle\}\leq 0$$a.e$. $ta.s$ .

277

Risk-sensitive portfolio optimizationfor each(3.23)$ k\in\Gamma$. Hence$\mu(k-h_{t})^{*}\{-(1-\mu)(\Sigma\Sigma^{*})h_{t}<\gamma_{t}$,$q_{t}>+<\Sigma\Lambda^{*}D\gamma_{t}$, $q_{t})>$$+<(g-r1)\gamma_{t}$ , $q_{t}>+<R_{t}$ , $q_{t}>\}\leq 0$.

Since $\mu>0$ we have (3.21).
If $\mu<0$ we obtain the converse inquality of (3.23) and we conclude
the present theorem.
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diﬀerential equations, Prob. Theory Rel. Fields, 113 (1999) 135-170[16] J. Ma and J. Yong, Forward-Backward Stochastic Diﬀerential equations
and Their Applications, Lect. Notes in Math. 1702, Springer(1999)[17] R.C. Merton, Optimum Consumption and Portfolio Rules in aContinuous-Time Model, Journal of Economic Theory 3 (1971) 373413[18] R.C. Merton, An Intertemporal Capital Asset Pricing Model, Econometrica (1973) 867-887[19] R.C. Merton, Continuous Time Finance, Blackwell, Malden (1990)[20] H. Nagai, Bellman equations of risk-sensitive control, SIAM J. Control
and Optimization. Vol. 34, No. 1 (1996) 74-101.
[21] H. Nagai, Risk-senstive dynamic asset management with partial information, Stochastics in finite and infinite dimension, a volume in honor of
G. Kallianpur, Eds. Rajput et al., Birkh\"auser (2000) 321-340[22] H. Nagai and S. Peng, Risk-sensitive dynamic portfolio optimization withpartial information on infinite time horizon, Annals of Applied Probability, vol. 12, No. 1 (2002) 173-195[23] H. Nagai, Optimal strategies for risk-sensitive portfolo optimization problems for general factor models, to appear in SIAM J. Control and Optim.
[24] R. Rishel, Optimal Portfolio Management with Partial Observations andPower Utility Function, Stochastic Analysis, Control, Optimization
and Applications, Birkh\"auser (1999) 605-620[25] B.L. Rozovski, Stochastic Evolution Systems, Kluwer Academic Publishers (1990)[26] X.Y. Zhou, On the necessary conditions of optimal controls for stochstic
partial diﬀerential equations, SIAM J. Control Optim. 31,(1993) 14621478
Department of Mathematical Science
Osaka University
Toyonaka, 560-8531
JapanAdvanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 279-299

An Approximation for Exponential Hedging
Jun Sekine
Abstract.
An optimization problem in mathematical finance, called theexponential hedging problem is addressed. First, the relations between the problem and the backward stochastic diﬀerential equation(abbreviated to BSDE) having a quadratic growth term in the drift
are reviewed. Next, the asymptotic analysis by Davis (2000) for the
problem and the motivation of this paper are stated. Further, withsome extensions, his analysis is reinterpreted by using the asymptotic
expansion of the BSDE with respect to a small parameter, which suggests an alternative approach to the analysis, and the result on an
approximated optimizer is obtained.

\S 1.

Introduction

In [7], Rouge and El Karoui treated the following optimization problem of mathematical finance. For a fixed $T>0$ , let $S:=(S_{t})_{t\in[0,T]}$ ,be the price process of -risky assets defined by thestochastic diﬀerential equation:$S_{t}:=(S_{t}^{1}, \ldots S_{t}^{n})^{/}$

diag

$n$

$dS_{t}$$=$$(S_{t})$$:=$$diag(S_{t})(\sigma_{t}dw_{t}+\mu_{t}dt)$,$S_{0}\in R_{+}^{n}$,$\left(\begin{array}{llll}S_{t}^{1} & 0 & \cdots & 0\\0 & S_{t}^{2} & \cdots & 0\\0 & 0 & \cdots & 0\\0 & 0 & \cdots & S_{t}^{n}\end{array}\right)$

on the probability space $(\Omega, F, P)$ with a $d(\geq n)$ -dimensional Brownian$:=(w_{t})_{t\in[0,T]}$ on it and the augmented Brownian filtration
motion. Here, is an $n\times d$-matrix-valued left-continuous adapted
process such thatand thatis invertible$w$$(F_{t})_{t\in[0,T]}$$\sigma$$\sigma\sigma’\in L^{\infty}([0, T]\times\Omega, R^{n\times n})$$\sigma_{t}\sigma_{t}^{/}$

Received March 3, 2003.
Revised September 1, 2003.
The author is grateful to the anonymous referees for their careful reading,helpful suggestions, and pointing out errors in the original version.

280

J. Sekine

for all $\in[0, T]$ (denotes the transpose of a matrix or a vector),
is an -dimensional predictable process, and $:=\sigma^{/}(\sigma\sigma’)^{-1}(\mu-r1)$ is, where $r(>0)$ is the constant interest
an element ofrate and 1 $:=(1, \ldots, 1)^{/}\in R^{n}$ . On the other hand, let
be the payoﬀ of a derivative security maturing at time $T$ and consider
a seller of the derivative security, who trades the assets continuously
in self-financing way on the time-interval $[0, T]$ to control the terminal
wealth. The value process of the self-financing portfolio is given by$t$$(\cdot)^{/}$$\mu$$\lambda$$n$$L^{\infty}([0, T]\times\Omega, R^{d})$$F\in L^{\infty}(\Omega, F_{T})$$dX_{t}^{x,\pi}=\pi_{t}^{/}(diag(S_{t}))^{-1}dS_{t}+(X_{t}^{x,\pi}-\pi_{t}’1)rdt$,$X_{0}^{x,\pi}=x$,or equivalently,$X_{t}^{x,\pi}:=e^{rt}\{x+\int_{0}^{t}\pi_{u}^{/}\sigma_{u}(dw_{t}+\lambda_{t}dt)\}$where is the initial capital and an -dimensional predictable process
is the asset holding strategy. To optimize the terminal wealth $-F+X_{T}^{x,\pi}$of the seller, the utility maximization problem (called the exponential
hedging problem in this paper, following Delbaen et. ; 2002, [2]
$x$

$n$

$\pi$

$a1$

(P)

$)$

$V(x):=\sup_{\pi\in A}E[U_{\gamma}(-F+X_{T}^{x,\pi})]$with respect to the exponential utility function:$U_{\gamma}(x):=-\frac{e^{-\gamma x}}{\gamma}$,$(\gamma>0)$

over an appropriately chosen space $A$ of admissible strategies is considered.
The importance of this problem is, ﬀom a viewpoint of mathematicalfinance, that it relates to the pricing and hedging problems of derivative
securities in incomplete markets: the quantity called utility indiﬀerence
price,(1)$p(x, F):=\inf_{y\in R}\{V(x+y)\geq\sup_{\pi\in A}E[U_{\gamma}(X_{T}^{x,\pi})]\}$,

is proposed as a coherent price of the derivative security in Davis (2000),[1] and [7], and the optimizer of the problem (P) is focused and studied
to control (hedge) the “risk” of the seller in [1], [2] and [7].
Duality argument is well established for utility maximization (cf.,Karatzas and Shreve; 1998, [5], for example) and is often used to attack

281

An approximation for exponential hedging

this problem, as follows. For example, let us employ the space$A_{2}:=\{\pi\in \mathcal{L}_{T}^{2,n}$;$\pi_{t}\in C$

for all$t$$\in[0, T]$, $E[\int_{0}^{T}|\pi_{t}|^{2}dt]<\infty\}$ ,is a fixed closed
as the set of admissible strategies, $A$ , whereconvex cone andis the totality of the -dimensional predictable.
,processes on the time-interval $[0, T]$ such that$x\inR$
, and $y>0$ , denote
For ,$C\subset R^{n}$$\mathcal{L}_{T}^{2,n}$$n$$\int_{0}^{T}|\pi_{t}|^{2}dt<\infty$

$\pi$

$a.s$

$f$

$u_{\gamma}(x;y, f):=U_{\gamma}(-f+x)-yx$

and$I_{\gamma}(y):=(U_{\gamma}’)^{-1}(y)=-\frac{1}{\gamma}\log(y)$to see the relation.

$\sup_{x\in R}u_{\gamma}(x;y, f)=u_{\gamma}(f+I_{\gamma}(y);y, f)=-y(f-\frac{1+\log y}{\gamma})$

Define$Z^{1/}:=\mathcal{E}(-(\lambda-\iota/)\cdot w)$

where$\iota/is$$\overline{(\sigma_{t}^{J}C)}$$\overline{Z}_{t}^{\nu}:=e^{-rt}Z_{t}^{\nu}$,an element of$D:=\{\iota/\in \mathcal{L}_{T}^{2,d}$

and

and

; bounded and$\iota/_{t}\in\overline{(\sigma_{t}’C)}$

for all$t$

is the notation for the negative polar cone of$\overline{(\sigma_{t}^{/}C)}(\omega):=$For $\pi\in A:=A_{2}$ and{$y\in R^{d};xy\leq 0$$\iota/\in D$

for all$\in[0, T]\}$$\sigma_{t}’C$$x\in\sigma_{t}^{J}(\omega)C$, i.e.,}.

, observe that$e^{-rt}X_{t}^{x,\pi}\leq x+\int_{0}^{t}\pi_{u}^{J}\sigma_{u}\{dw_{u}+(\lambda_{u}-\iota/_{u})du\}$and that $Z^{\nu}\int\pi^{/}\sigma\{dw+(\lambda-\nu)du\}$ is a martingale since$<\infty$ and since$E[\sup_{t\in[0,T]}|Z_{t}^{\nu}|^{2}]$$ E[\sup_{t\in[0,T]}|\int_{0}^{t}\pi_{u}^{/}\sigma_{u}\{dw_{u}+(\lambda_{u}-\iota/_{u})du\}|^{2}]\leq C_{1}E[\int_{0}^{T}|\pi_{u}|^{2}du]<\infty$

ﬀom Doob’s inequality and the boundedness assumptions of. Therefore, the relation$lJ$$E[\overline{Z}_{T}^{\mathfrak{l}J}X_{T}^{x,\pi}]\leq x$$\sigma\sigma^{J}$,$\lambda$

and

282

J. Sekine

follows. Based on the relation, for $\pi\in A$ and $x\in R$ , $y>0$ , we observe
the inequalities(2)$E[U_{\gamma}(-F+X_{T}^{x,\pi})]-yx$$\leq$$\inf_{\nu\in D}E[U_{\gamma}(-F+X_{T}^{x,\pi})-y\overline{Z}_{T}^{\nu}X_{T}^{x,\pi}]$$\leq$$\inf_{\nu\in D}\sup_{\pi\in A}E[u_{\gamma}$$\leq$$\inf_{\nu\in D}E[u_{\gamma}$(($X_{T}^{x,\pi}$;$y\overline{Z}_{T}^{\nu}$

, $F$

)

$]$

$F+I_{\gamma}(y\overline{Z}_{T}^{\nu});y\overline{Z}_{T}^{\nu}$

, $F$

) .
$]$

The minimization problem(D)$\hat{V}(y):=\inf_{\nu\in D}E[u_{\gamma}($ $F+I_{\gamma}(y\overline{Z}_{T}^{\nu})$;$y\overline{Z}_{T}^{\nu}$, $F)]$

is called the dual problem of the primal problem (P), and the inequality(3)$V(x)\leq\inf_{y>0}(\hat{V}(y)+yx)$

is deduced from (2). Indeed, the equality can be established in (3) (i.e.,there is no “duality-gap”) and the following expression is obtained.
$A_{2}$

Theorem 1. (Theorem 2.1 of Rouge and $El$ Karoui, [7]) For $A:=$, it holds that(4)

$V(x)=U_{\gamma}(e^{rT}x-\frac{1}{\gamma}\sup_{\nu\in D}\{E^{\nu}[\gamma F]-H(P^{\nu}|P)\})$

where
sure

,

denotes the expectation with respect to the probability meaon
defined by$E^{\nu}[\cdot]$$P^{\nu}$$(\Omega, \mathcal{F}_{T})$$\frac{dP^{\nu}}{dP}|_{F_{t}}:=Z_{t}^{\nu}$and$E[\frac{dQ}{dP}\log\frac{dQ}{dP}]$

if $Q<<P$ ,$+\infty$

otherwise$H(Q|P):=\{$

is the relative entropy of $Q$ with respect to $P$ .

Remark 1. The duality relations similar to (4) have been obtained
for more general semimartingale
and for other choices of the set of$A$
by Delbaen et. al. in [2]. Also, the work byadmissible strategies
Kabanov and Strieker (2002), [4], should be referred.
For the computations of the value $V(x)$ and the optimizer, one cansolve the BSDE for the value process of the dual problem, described asfollows.
$S$283

An approximation for exponential hedging

Theorem 2$Z_{t,T}^{\nu}:=Z_{T}^{\nu}/Z_{t}^{\nu}$,(Theorem 4 1-2 of Rouge and $El$ Karoui, [7]) Denote, and $\tau:=T-t$ for $0\leq t\leq T$ . Let$\cdot$$\cdot$$Z_{t,T}^{\nu}:=\overline{Z}_{T}^{\nu}/\overline{Z}_{t}^{\nu}$$ess\inf_{\nu\in D}E[u_{\gamma}($ $F+I_{\gamma}(y\overline{Z}_{t,T}^{\nu});y\overline{Z}_{t,T}^{\nu}$

$=$

$=$

:

, $F)|F_{t}]$$\frac{ye^{-r\tau}}{\gamma}\{-ess\sup_{\nu\in D}E^{\nu}[\gamma F-\log Z_{t,T}^{\nu}|F_{t}]+(1+\log y-r\tau)\}$$\frac{ye^{-r7^{-}}}{\gamma}\{-Y_{t}+(1+\log y-r\tau)\}$.

There exists — $\in H_{T}^{2,d}:=\{f\in \mathcal{L}_{T}^{2,d};E[\int_{0}^{T}|f_{t}|^{2}dt]<\infty\}$ such that$(Y,---)$

satisfies

(5)
where

,

$=$

$f(t, ---t)dt+---/tdw_{t}$$f(t, \xi)$$:=$$\lambda_{t}’\Pi_{\sigma_{\acute{t}}C}(\xi+\lambda_{t})-\frac{1}{2}|\xi-\Pi_{\sigma_{t}C},(\xi+\lambda_{t})|^{2}$

and:operator onto the closed convex cone
satisfying
In particular,$\Pi_{\sigma_{t}’(\omega)C}$,$dY_{t}$$Y_{T}=\gamma F$$R^{d}\ni x\mapsto\square _{\sigma_{t}’(\omega)C}x\in\sigma_{t}’(\omega)C(\subset R^{d})$$\sigma_{t}’(\omega)C$is the projection.

$\pi^{*}\in A_{2}$(6)$\sigma_{t}^{/}\pi_{t}^{*}:=\frac{e^{-rT}}{\gamma}\Pi_{\sigma_{t}C},(_{-t}^{-}-+\lambda_{t})$for all$t$$\in[0, T]$

is an optimizer of the primal problem (P) with $A:=A_{2}$ , and $\nu^{*}:=$
attains the infimum of the dual problem (D). Further,$(I -\Pi_{\sigma’C})(_{-}^{-}-+\lambda)$(7)$V(x)=U_{\gamma}(e^{rT}x-\frac{Y_{0}}{\gamma})$holds.
Remark 2. The existence and the uniqueness of the solution, whereof the quadratic BSDE (5) in the space([0,); predictable} is ensured by the work of Kobylanski{(2000), [6]. Further, utilizing the dynamic programming principle andthe comparison theorems between linear BSDEs and between quadratic
BSDEs in [6], the above theorem is established.
On the other hand, if the model has a Markovian structure, one cansolve a dynamic programming equation to compute the value, which issuggested in Delbaen et(2002), [2], and is employed and studied inDavis (2000), [1], In particular, in [1], a special but a typical situationis addressed, which can be stated as follows in our setting.
$(Y^{ },---)$$H_{T}^{\infty}\times H_{T}^{2,d}$$f\in L^{\infty}$$ T]\times\Omega$$a1$$H_{T}^{\infty}:=$284

J. Sekine(i) Let $d=n=2$ .

(8)$\sigma$is the following constant matrix$\sigma:=(\sigma_{2}\sqrt{1-\epsilon^{2}}\sigma_{1}$with , $\sigma_{2}>0$ , $\epsilon\in[-1,1]$ .
,vector. Further,andare closely correlated:$\sigma_{1}$$\epsilon\neq 0$$\sigma_{2}\epsilon 0)$

is also a constantis assumed, i.e., two assets$\mu=(\mu_{1}, \mu_{2})^{J}$$\epsilon<<1$$S^{1}$$S^{2}$$\frac{d\langle S^{1},S^{2}\rangle}{\sqrt{d\langle S^{1}\rangle d\langle S^{2}\rangle}}=\sqrt{1-\epsilon^{2}}\approx 1$.

(ii) $F:=h(S_{T}^{1})$ with continuous, piecewise linear: $R_{+}\mapsto R$
bounded ﬀom above.
(iii) The constraint of the asset-holding strategy is given by$C:=\{0\}\times R$ : onlyis tradable, and the derivative security.
is written on the untradable asset
Recall, in the situation, that the expressions$h$$\pi_{t}\in$$\pi$$S^{2}$$S^{1}$$\sigma^{/}C=\{kd_{\epsilon}; k\in R\}$,

$D=\{\eta d_{\epsilon}^{\perp};$$\eta\in \mathcal{L}_{T}^{2,1}$, $bounded\}$ ,and$\lambda^{\epsilon}:=\sigma^{-1}(\mu-r1)=\frac{1}{\epsilon\sigma_{1}\sigma_{2}}(-\sigma_{2}\sqrt{1-\epsilon^{2}}\epsilon\sigma_{2}$$\sigma_{1}0)$$\left(\begin{array}{l}\mu_{1}-r\\\mu_{2}-r\end{array}\right)$

hold, where we denote$d_{\epsilon}:=(\sqrt{1-\epsilon^{2}}$,$\epsilon$)

and$d_{\epsilon}^{\perp}:=(\epsilon,$$-\sqrt{1-\epsilon^{2}})’$The dual problem is now, rewritten as$\inf_{\nu\in D}E[(-y\overline{Z}_{T}^{\nu})\{h(S_{T}^{1})-\frac{1}{\gamma}(1+\log y+\log\overline{Z}_{T}^{\nu})\}]$$=$$\frac{ye^{-rT}}{\gamma}\{-\sup_{\nu\in D}E^{\nu}[\gamma h(S_{T}^{1})-\log Z_{T}^{\nu}]+(1+\log y-rT)\}$

.

Since
$\log Z_{T}^{\nu}=-\int_{0}^{T}(\lambda^{\epsilon}-\nu_{t})’dw_{t}^{\nu}+\frac{1}{2}\int_{0}^{T}|\lambda^{\epsilon}-\nu_{t}|^{2}dt$

,

,
where
is a 2-dimensional-Brownian motion, it is equivalent to solve the following:$w^{\nu}:=(w_{1}^{\nu}, w_{2}^{\nu})^{/}$$w_{t}^{\nu}:=w_{t}+\int_{0}^{t}(\lambda^{\epsilon}-\nu_{u})du$$P^{\nu}$$\sup_{\nu\in D}E^{\nu}[\gamma h(S_{T}^{1})-\frac{1}{2}\int_{0}^{T}|\lambda^{\epsilon}-\nu_{t}|^{2}dt]$

,

285

An approximation for exponential hedging

in which the process$dS_{t}^{1}$

where we denotevalue function$S^{1}$

has the dynamics:$=$$S_{t}^{1}[\sigma_{1}dw_{1}^{\nu}(t)+\{\mu_{1}-\sigma_{1}(\overline{\lambda}^{\epsilon}-\nu_{t}^{1})\}dt]$$=$$S_{t}^{1}\{\sigma_{1}dw_{1}^{\nu}(t)+(r-\epsilon\sigma_{1}\eta_{t})dt\}$$\nu$$:=\eta d_{\epsilon}^{\perp}$,

with some bounded predictable$\eta$. For the$v^{\epsilon}(t, y):=ess\sup_{\nu\in D}E^{\nu}[\gamma h(S_{T}^{1})-\frac{1}{2}\int_{t}^{T}|\lambda^{\epsilon}-\nu_{t}|^{2}dt|S_{t}^{1}=y]$,a dynamic-programming equation is derived and the existence of its
smooth solution is checked in the setting of [1]. Moreover, the following
expressions are obtained.
Theorem 3. (Theorem 6.1, 6.4 and 7.3 of Davis, [I])1. An optimal strategy of the problem (P) is given by$\pi_{t}^{*}$$=$$\frac{e^{-rT}}{\gamma}(\sigma^{/})^{-1}\Pi_{\sigma’C}\{$$=$$(\frac{e^{-rT}}{\gamma}\{\frac{\mu_{2}-r}{\sigma_{2}^{2}}-\sqrt{1-\epsilon^{2}}^{\sigma}\overline{\sigma}_{2}\perp\partial_{x}v^{\epsilon}(t, S_{t}^{1})S_{t}^{1}0\})$$\left(\begin{array}{l}-\partial_{x}v^{\epsilon}(t,S_{t}^{1})S_{t}^{1}\sigma_{1}\\0\end{array}\right)$$+\lambda_{t}^{\epsilon}\}$.

2. For the utility indiﬀerence price defined by (1),$p(x, F)=\frac{e^{-rT}}{\gamma}\{v^{\epsilon}(0, S_{0}^{1})+\frac{T}{2}(\frac{\mu_{2}-r}{\sigma_{2}})^{2}\}$holds for any $x\in R$ .
, the value function has the expansion
3. As$\epsilon\downarrow 0$(9)$v^{\epsilon}(t, y)$$=$$\gamma E[h(A_{T})|A_{t}=y]-\frac{T-t}{2}(\frac{\mu_{2}-r}{\sigma_{2}})^{2}$$+\epsilon^{2}\frac{\gamma^{2}}{2}Var[h(A_{T})|A_{t}=y]+O(\epsilon^{4})$where $Var[*|\cdot]:=E[(*)^{2}|\cdot]-(E[*|\cdot])^{2}$ ,and the process $A$ is defined by$O(\epsilon^{4})$,depends on the value $(t, y)$ ,$dA_{t}=A_{t}[\sigma_{1}dw_{1}(t)+\{\mu_{1}-\sqrt{1-\epsilon^{2}}\frac{\sigma_{1}(\mu_{2}-r)}{\sigma_{2}}\}dt]$,$A_{0}=S_{0}^{1}$.

J. Sekine286

In particular, we are interested in the expansion (9). From a practical viewpoint, it is an eﬀective and useful expansion: it gives nice approximations of the value of the problem (P) and the utility indiﬀerence
price. By using the relation (7),$\log V^{\epsilon}(x)-\log U_{\gamma}(e^{rT}x-\gamma E[h(A_{T})]-\frac{T}{2}(\frac{\mu_{2}-r}{\sigma_{2}})^{2}$$-\epsilon^{2}\frac{\gamma^{2}}{2}$

is observed, where we denote the value by$V^{\epsilon}(x)$Var $[h(A_{T})])=O(\epsilon^{4})$emphasizing , and$\epsilon$$p(x, F)=e^{-rT}\{E[h(A_{T})]+\epsilon^{2}\frac{\gamma}{2}Var[h(A_{T})]\}+O(\epsilon^{4})$holds for any $ x\in$ R. Also, both quantities $E[h(A_{T})|A_{t}=y]$ and
Var $[h(A_{T})|A_{t}=y]$ are fairly “computable”. In [1], it is derived ﬀom aclever observation, however, the reason why the second term has
and the error term has
seems to be obscure. To see its intrinsic
reason is one of our motivations.
Further, we are interested in the approximation of the optimal strategy (optimizer), which is not mentioned in [1]. It looks natural to deduce
the strategy
anddefined by$O(\epsilon^{2})$$O(\epsilon^{4})$$\check{\pi}^{1}\equiv 0$$\check{\pi}:=(\check{\pi}^{1},\check{\pi}^{2})^{/}$$\check{\pi}_{t}^{2}$$:=$$\frac{e^{-rT}}{\gamma}[\frac{\mu_{2}-r}{\sigma_{2}^{2}}-\sqrt{1-\epsilon^{2}}\frac{\sigma_{1}}{\sigma_{2}}S_{t}^{1}$$\times\partial_{y}(\gamma E[h(A_{T})|A_{t}=y]+\epsilon^{2}\frac{\gamma^{2}}{2}Var[h(A_{T})|A_{t}=y])|_{y=S_{t}^{1}}]$and expect the approximation such that(10)$\log V^{\epsilon}(x)-\log E[U_{\gamma}(-F+X_{T}^{x,\check{\pi}})]=O(\epsilon^{4})$,for example.
In the next section, using the BSDE in Theorem 2 and its asymptotic
expansion with respect to (precisely saying, with respect to , cf., theBSDE (10) , we reconstruct the expansions (9-10), which yields an alternative approach to the above analysis. The main contribution of this paper is Theorem 4 in the next section, extensions of (9-10) under Assumption 1. It is also suggested that the -th derivatives(cf., Remark 4).
for odd numbers $i=1,3,5$ ,$\epsilon’$$\epsilon$

$)$

$i$

$\cdots$$(\partial_{\epsilon}i,Y0,\epsilon, \partial_{\epsilon}i,---0,\epsilon)\equiv 0$

287

An approximation for exponential hedging\S 2.

An approximated optimizer

In this section, the probability space is assumed to be the product of, where $\Omega_{i}:=C_{0}([0, T], R)$ ,Wiener spaces:$F^{i}:=B(\Omega_{i})$ andis the Wiener measure, the law of the -th canonical Brownian motion $w^{i}:=(w_{t}^{i})_{t\in[0,T]}$ . The filtrationis the augmented natural filtration. Sometimes a ranis identified with $Xoj_{1}$ on $(\Omega, F, P)$ ,dom variable $X$ on:
is the projection onto the firstwhere
probability space.
We now impose the following conditions.
(i)’ The volatility matrix of the process is given by (8). On the-predictable process,is a boundedother hand,
$[0,
T]$
i.e.,
:
is measurable
$[0,
T]$
.
with respect to the predictable -algebra on(ii)’ $F(\omega_{1})=h(S^{1}(\omega_{1}))$ with a bounded measurable function on$C([0, T], R_{+})$ .
(iii) The constraint of the strategy is given by $\pi_{t}\in C:=\{O\}\times R$ .
Remark 3. The condition (i)’ is considered as an extension of the constant case employed in (i) in the previous section, though Assumption
1.1 will be added later. On the other hand, the conditoin (ii)’ does notinclude the conditon (ii) in the previous section.
Further, we consider the problem (P) over the extended space: $A:=$, where$(\Omega, \mathcal{F}, P):=\prod_{i=1}^{2}(\Omega_{i}, F^{i}, P_{i})$$i$$P_{i}$$(\mathcal{F}_{t})_{t\in[0,T]}:=$$(F_{t}^{1}\times F_{t}^{2})_{t\in[0,T]}$$(\Omega_{1}, \mathcal{F}^{1}, P_{1})$$j_{1}$$\Omega\ni\omega:=(\omega_{1}, \omega_{2})\mapsto\omega_{1}\in\Omega_{1}$$S$$\mu=(\mu_{1}, \mu_{2})^{/}$$\mu$$F_{t}^{1}$$\times\Omega_{1}\ni(t, \omega_{1})\mapsto\mu(t, \omega_{1})\in R^{2}$$\times\Omega_{1}$$\sigma$

$h$

$\pi$

$\mu$

$A_{1}$$A_{1}$$:=$$\{\pi\in \mathcal{L}_{T}^{2,2}$;$\pi_{t}\in C$

for $\forall t\in[0, T]$ ,$ E[(\int_{0}^{T}|\pi_{t}|^{2}dt)^{q/2}]<\infty$

and construct an approximated optimizer inremark the following.

$A_{1}$

for $\exists q>1\}$, not in$A_{2}$. We first

Proposition 1. Let
be the process defined by the formulaand by the solutionof the BSDE (5). It is also an
optimizer of the problem (P) with $A:=A_{1}$ .
$\pi^{*}$$(\theta)$$(Y_{ },---)\in H_{T}^{\infty}\times H_{7}^{2}i^{2}$

Proof. We first observe thatand $x\in R$ . For the purpose, since$E[\overline{Z}_{T}^{\nu}X_{T}^{x,\pi}]\leq x$

for all $(\pi, \nu)\in A_{1}\times D$$e^{-rt}X_{t}^{x,\pi}=x+\int_{0}^{t}\pi_{u}^{J}\sigma_{u}dw_{u}^{\nu}$J. Sekine288

holds, to show the martingale property of the process
suﬃcient, which can be verified by checking$Z^{\nu}\int\pi^{/}\sigma dw^{\nu}$is$E[\sup_{t\in[0,T]}|Z_{t}^{\nu}\int_{0}^{t}\pi_{u}’\sigma_{u}dw_{t}^{\nu}|]$$\leq$$E[\sup_{t\in[0,T]}|Z_{t}^{\nu}\int_{0}^{t}\pi_{u}^{/}\sigma_{u}dw_{t}|]+E[\sup_{t\in[0,T]}|Z_{t}^{\nu}\int_{0}^{t}\pi_{u}’\sigma_{u}(\lambda_{u}-\nu_{u})du|]$$\leq$$E[\sup_{t\in[0,T]}(Z_{t}^{\nu})^{p}]^{1/p}\{E[\sup_{t\in[0,T]}|\int_{0}^{t}\pi_{u}^{/}\sigma_{u}dw_{u}|^{q}]1/q$$+E[\sup_{t\in[0,T]}|\int_{0}^{t}\pi_{u}’\sigma_{u}(\lambda_{u}-\nu_{u})du|^{q}]1/q\}$$\leq$$C_{1}E[\langle Z^{\nu}\rangle_{T}^{p/2}]1/p$$\times\{C_{2}E[(\int_{0}^{T}|\pi_{u}|^{2}du)^{q/2}]^{1/q}+C_{3}E[(\int_{0}^{T}|\pi_{u}|du)^{q}]^{1/q}\}$$\leq$$ C_{4}E[\langle Z^{\nu}\rangle_{T}^{p/2}]^{1/p}E[(\int_{0}^{T}|\pi_{u}|^{2}du)^{q/2}]1/q<\infty$

for , $q>1$ satisfying $1/p+1/q=1$ by using the H\"older inequality andthe Burkholder-Davis-Gundy inequality. In particular, the inequalities(2) and$p$(11)$E[U_{\gamma}(-F+X_{T}^{x,\pi})]-yx\leq\hat{V}(y)$are deduced for any $\pi\in A_{1}x\in R$ and $y>0$ .
Next, note that the pairdefined by (6) and the formulasatisfies the relation$(\pi^{*}, \nu^{*})$$\nu_{t}^{*}:=(I-\Pi_{\sigma_{t}’C})(_{-t}^{-\epsilon}-+\lambda_{t}^{\epsilon})$$F+I_{\gamma}(\mathcal{Y}(x)\overline{Z}_{T}^{\nu^{*}})=X_{T}^{x,\pi^{*}}$

(12)

$\{$

with

$\mathcal{Y}(x):=\exp(Y_{0}+rT-\gamma e^{rT}x)$

.

289

An approximation for exponential hedging

In fact, the solution of the BSDE satisfies
$F$

$=$

$=$

$\frac{1}{\gamma}\{Y_{0}+\int_{0}^{T}f(t, ---t)dt+\int_{0}^{T}(_{-t}^{-}-)^{/}dw_{t}\}$$\frac{Y_{0}}{\gamma}+\int_{0}^{T}\{e^{rT}(\pi_{t}^{*})^{J}\sigma_{t}\lambda_{t}-\frac{1}{2\gamma}|\lambda_{t}-\nu_{t}^{*}|^{2}\}dt$$+\int_{0}^{T}\{e^{rT}\sigma_{t}^{/}\pi_{t}^{*}-\frac{1}{\gamma}(\lambda_{t}-\nu_{t}^{*})\}^{/}dw_{t}$$=$$\frac{Y_{0}}{\gamma}+X_{T}^{0,\pi^{*}}-I_{\gamma}(Z_{T}^{\nu^{*}})$,

which is equivalent to (12). Using (12) and Theorem 2, we observe that
(13)

$E[\overline{Z}_{T}^{\nu^{*}}X_{T}^{x,\pi^{*}}]$

Finally, replacing
deduce that

$y$

by

$=$

$E[\overline{Z}_{T}^{\nu^{*}}($ $F+I_{\gamma}(\mathcal{Y}(x)\overline{Z}_{T}^{\nu^{*}}))]$$=$$\frac{e^{-rT}}{\gamma}\{Y_{0}-\log(\mathcal{Y}(x)e^{-rT})\}=x$$\mathcal{Y}(x)$.

in (11) and using Theorem 2 and (12-3), we$E[U_{\gamma}(-F+X_{T}^{x,\pi})]-\mathcal{Y}(x)x$($\leq$$E[u_{\gamma}$$=$$E[U_{\gamma}(-F+X_{T}^{x,\pi^{*}})]-\mathcal{Y}(x)x$$F+I_{\gamma}(\mathcal{Y}(x)\overline{Z}_{T}^{\nu^{*}});\mathcal{Y}(x)\overline{Z}_{T}^{\nu^{*}}$

, $F$

)

$]$

, which implies the optimality of . 1
for all
The BSDE (5) for the optimizer is now rewritten as, in the situationof this section,$\pi\in A_{1}$$\pi^{*}$$dY_{t}^{\epsilon}$where$(\cdot, \cdot)$$f(t, \xi, \epsilon)$,$=$$f(t,---t’\epsilon\epsilon)dt+(_{-t}^{-\epsilon}-)^{J}dw_{t}$$:=$$\frac{1}{2}\{\overline{\lambda}_{t}^{2}-(\xi, d_{\epsilon}^{\perp})^{2}\}+\overline{\lambda}_{t}(\xi, d_{\epsilon})$denotes the standard inner-product inand$R^{2}$$\overline{\lambda}:=(\lambda^{\epsilon}, d_{\epsilon})=\frac{\mu_{2}-r}{\sigma_{2}}$,$Y_{T}^{\epsilon}=\gamma F$.

Denote$\overline{w}_{t}^{\epsilon}=(\overline{w}_{1}^{\epsilon}(t), \overline{w}_{2}^{\epsilon}(t))^{J}:=w_{t}+(\int_{0}^{t}\overline{\lambda}_{u}du)d_{\epsilon}$

,

290

J. Sekine

to reexpress the solution$(Y^{\epsilon-\epsilon},--)=(Y^{\epsilon,\epsilon-\epsilon,\epsilon},--)$(14)$g(t,$$dY_{t}^{\epsilon’,\epsilon}$

where

$=$

$g(t, \xi, \epsilon^{J})$$:=$by using the BSDE:$---t\epsilon’,\epsilon,/\epsilon)dt+(_{-t}^{-\epsilon’,\epsilon}-)^{/}d\overline{w}_{t}^{\epsilon}$$\frac{1}{2}\{\overline{\lambda}_{t}^{2}-(\xi, d_{\epsilon}^{\perp},)^{2}\}$,$Y_{T}^{\epsilon’,\epsilon}=\gamma F$,.

We consider the asymptotic expansion ofwith respect toat 0. Letand introduce the BSDEs:$(Y\epsilon’,\epsilon, ---\epsilon’,\epsilon)$$\epsilon^{/}$$(\partial_{\epsilon}0,Y0,\epsilon, \partial_{\epsilon}0,---0,\epsilon):=(Y0,\epsilon,---0,\epsilon)$$d(\partial_{\epsilon}^{i},Y_{t}^{0,\epsilon})=g_{i}(t,$(15)$\partial_{\epsilon}^{i},Y_{T}^{0,\epsilon}=0$using the functions,$0)dt+(\partial_{\epsilon}^{i},---t)^{J}0,\epsilon Fw_{t}^{\epsilon}$,,defined inductively$g_{i}$$g_{0}(t, \xi^{0}, \epsilon’)$and$(\partial_{\epsilon t}^{j-0,\epsilon},--)_{j=0,..,i}$$\{$$g_{i}$$(t, (\xi^{j})_{j=0,.,i}$,$\epsilon^{/})$$:=$$g(t, \xi^{0}, \epsilon^{/})$$:=$$\sum_{j=0}^{i-1}(\partial_{\xi^{j}}g_{i-1}(t, (\xi^{k})_{k=0,i-1}\ldots,,$$+\partial_{\epsilon’}g_{i-1}$

Formally, it is expected that$(t, (\xi^{k})_{k=0,..,i-1}$$(\partial_{\epsilon}i,Y0,\epsilon, \partial_{\epsilon}i,---0,\epsilon)$,$\epsilon’)$$\epsilon^{/})$,$\xi^{j+1})$.

is the -th derivative of$i$, although we have not beenwith respect toatable to show the property. The standard results on the diﬀerentiation of
the solution of BSDE with respect to a parameter (cf., El Karoui et.
; 1997, [3], for example) cannot be applied to our quadratic BSDE (14).
byDefine the probability measureon$(Y\epsilon’,\epsilon,---\epsilon’,\epsilon)$$\epsilon^{/}$$\epsilon^{/}=0$$a1$$\overline{P}^{\epsilon}$$(\Omega, F_{T})$$\frac{d\overline{P}^{\epsilon}}{dP}|_{F_{t}}$$:=$$\mathcal{E}_{t}(-\int\overline{\lambda}d_{\epsilon}’dw)=:\overline{Z}_{t}^{\epsilon}$$=$$\mathcal{E}_{t}(-\sqrt{1-\epsilon^{2}}\int\overline{\lambda}dw_{1})\mathcal{E}_{t}(-\epsilon\int\overline{\lambda}dw_{2})=:\overline{Z}_{1}^{\epsilon}(t)\overline{Z}_{2}^{\epsilon}(t)$;and the space
to obtain theexpressions for the solution of (15) for $i=0,1$ , 2, 3, as follows.
$H_{T}^{2,2,\epsilon}:=\{f\in \mathcal{L}_{T}^{2,2}$$\int_{0}^{T}|f_{t}|^{2}dt\in L^{1}(\overline{P}^{\epsilon})\}$

291

An approximation for exponential hedging

Lemma 1. 1. The solution
the expressions:$(Y^{0,\epsilon},---0,\epsilon)$in the space$H_{T}^{\infty}\times H_{T}^{2,2}$$Y_{t}^{0,\epsilon}$$=$$\overline{E}^{\epsilon}[\gamma F-\frac{1}{2}\int_{t}^{T}\overline{\lambda}_{u}^{2}du|F_{t}]$,$Y_{0}^{0,\epsilon}+\int_{0}^{t}-^{0,\epsilon}--1(u)cT\overline{w}_{1}^{\epsilon}(u)$$=$$\overline{E}^{\epsilon}[\gamma F-\frac{1}{2}\int_{0}^{T}\overline{\lambda}_{u}^{2}du|F_{t}]$,and $---02’\epsilon(t)=0$ for all $t\in[0, T]$ , where.
with respect to the probability measure2.
for $i=1,3$ .
3. A solution of (15) with $i=2$ exists in$\overline{E}^{\epsilon}[\cdot]$

hasdenotes the expectation$\overline{P}^{\epsilon}$$(\partial_{\epsilon}^{i},Y0,\epsilon, \partial_{\epsilon}i,---0,\epsilon)\equiv 0$$H_{T}^{\infty}\times H_{T}^{2,2,\epsilon}$$\partial_{\epsilon}^{2},Y_{t}^{0,\epsilon}$$\partial_{\epsilon}^{2},Y_{0}^{0,\epsilon}+\int_{0}^{t}\partial_{\epsilon}^{2},---0,\epsilon(1u)T\overline{w}_{1}^{\epsilon}(u)$and$\partial_{\epsilon 2}^{2-0,\epsilon},--(t)=0$

and is given by
du$|F_{t}]$$=$$\overline{E}^{\epsilon}[\int_{t}^{T}(_{-1}^{-0,\epsilon}-(u))^{2}$$=$$\overline{Var}^{\epsilon}[\gamma F-\frac{1}{2}\int_{t}^{T}\overline{\lambda}_{u}^{2}du|F_{t}]$$=$$\overline{E}^{\epsilon}[\int_{0}^{T}(_{-1}^{-0,\epsilon}-(u))^{2}$for all $t\in[0, T]$ , $w/iere$ we denote

du$|F_{t}]$,,$\overline{Var}^{\epsilon}[\cdot|F_{t}]:=$$\overline{E}^{\epsilon}[(\cdot)^{2}|F_{t}]-(\overline{E}^{\epsilon}[\cdot|\mathcal{F}_{t}])^{2}$

Proof. 1. $Suppose---02’\epsilon\equiv 0$ , then$dY_{t}^{0,\epsilon}=\frac{1}{2}\overline{\lambda}_{t}^{2}dt+--^{0,\epsilon}-1(t)\Gamma w_{1}^{\epsilon}(t)$,$Y_{T}^{0,\epsilon}=\gamma F$

is observed. 1 is now a consequence of the standard result of linear
BSDE (cf., El Karoui et. , [3]) and the result on the uniqueness of thequadratic BSDE studied in Kobylanski (2000), [6].
2-3. Observe that$a1$$d_{\epsilon}^{\perp}$

$=$

$=$

where

:

$+\frac{\epsilon^{2}}{2}$$\left(\begin{array}{l}0\\-1\end{array}\right)$$+\epsilon$$\left(\begin{array}{l}1\\0\end{array}\right)$$d_{0}^{\perp}+\sum_{i=1}^{3}\frac{\epsilon^{i}}{i!}\partial_{\epsilon}^{i},d_{0}^{\perp}+O(\epsilon^{4})$$o(\epsilon^{4})\in R^{2}$$+\frac{\epsilon^{3}}{3!}$$\left(\begin{array}{l}0\\1\end{array}\right)$$\left(\begin{array}{l}0\\0\end{array}\right)$,

is a vector with the norm$|O(\epsilon^{4})|\sim\epsilon^{4}$.

$+O(\epsilon^{4})$292

J. Sekine(i) Noting that$g_{1}(t, (\xi^{j})_{j=0,1},0)=-(\xi^{0}, d_{0}^{\perp})\{(\xi^{1}, d_{0}^{\perp})+(\xi^{0}, \partial_{\epsilon’}d_{0}^{\perp})\}$and that$---02\equiv 0$, we can deduce$d(\partial_{\epsilon’}Y_{t}^{0,\epsilon})=\partial_{\epsilon’-t}^{-0,\epsilon}-\Gamma w_{t}^{\epsilon}$and(ii) Observing that$(\partial_{\epsilon’}Y0,\epsilon, \partial_{\epsilon},--\cup)0,\epsilon\equiv 0$,$\partial_{\epsilon’}Y_{T}^{0,\epsilon}\equiv 0$.

$g_{2}(t, (\xi^{j})_{j=0,1,2},0)$$=$$-(\xi^{1}, d_{0}^{\perp})\{(\xi^{1}, d_{0}^{\perp})+(\xi^{0}, \partial_{\epsilon’}d_{0}^{\perp})\}$$-(\xi^{0}, d_{0}^{\perp})\{(\xi^{2}, d_{0}^{\perp})+(\xi^{1}, \partial_{\epsilon’}d_{0}^{\perp})\}$$-(\xi^{0}, \partial_{\epsilon’}d_{0}^{\perp})\{(\xi^{1}, d_{0}^{\perp})+(\xi^{0}, \partial_{\epsilon’}d_{0}^{\perp})\}$$-(\xi^{0}, d_{0}^{\perp})\{(\xi^{1}, \partial_{\epsilon’}d_{0}^{\perp})+(\xi^{0}, \partial_{\epsilon}^{2},d_{0}^{\perp})\}$

we rewrite the BSDE for$(\partial_{\epsilon}^{2},Y0,\epsilon, \partial_{\epsilon}2,---0,\epsilon)$as$d(\partial_{\epsilon}^{2},Y_{t}^{0,\epsilon})=-(_{-1}^{-0,\epsilon}-(t))^{2}dt+(\partial_{\epsilon}2,---t0,\epsilon)’\mathcal{F}\overline{w}_{t}^{\epsilon}$

sinceformulas$---02’\epsilon\equiv 0$and$\partial_{\epsilon^{!-}}^{-0,\epsilon}-\equiv 0$. Define,$\overline{P}^{\epsilon}$,$\partial_{\epsilon}^{2},Y_{T}^{0,\epsilon}\equiv 0$-martingales $M$ , $N$ by the$M_{t}:=\int_{0}^{t}---01’\epsilon(u)T\overline{w}_{1}^{\epsilon}(u):=-Y_{0}^{0,\epsilon}+\overline{E}^{\epsilon}[\gamma F-\frac{1}{2}\int_{0}^{T}\overline{\lambda}_{u}^{2}du|\mathcal{F}_{t}]$andbounded and that $N$ is$N_{t}:=\overline{E}^{\epsilon}[\langle M\rangle_{T}|F_{t}]$for $\in[0, T]$ , respectively. Note that $M$ is-square integrable:$t$$\overline{P}^{\epsilon}$$\overline{E}^{\epsilon}[N_{t}^{2}]\leq\overline{E}^{\epsilon}[\langle M\rangle_{t}^{2}]=\overline{E}^{\epsilon}[(M_{t}^{2}-M_{0}^{2}-2\int_{0}^{t}M_{u}dM_{u})^{2}]<\infty$

The martingale representation theorem implies that$=N_{0}+\int_{0}^{t}\phi_{u}dw_{1}(u)$ holds for all$\in[0, T]$ and for some
. Therefore,such that.

$H_{t}:=E[\overline{Z}_{1}^{\epsilon}(T)N_{T}|\mathcal{F}_{t}]$$t$$\phi$$\mathcal{F}_{t}^{1}$-predictable$\int_{0}^{T}\phi_{u}^{2}du<\infty$$N_{t}-N_{0}=\int_{0}^{t}d(\frac{\overline{Z}_{1}^{\epsilon}(u)N_{u}}{\overline{Z}_{1}^{\epsilon}(u)})=\int_{0}^{t}\frac{\phi_{u}+\sqrt{1-\epsilon^{2}}H_{u}\overline{\lambda}_{u}}{\overline{Z}_{1}^{\epsilon}(u)}c\Gamma\overline{w}_{1}^{\epsilon}(u)$293

An approximation for exponential hedging

is observed from the It\^o-formula. The solution is now constructed by
setting$\partial_{\epsilon}^{2},Y_{t}^{0,\epsilon}:=N_{t}-\int_{0}^{t}(_{-1}^{-0,\epsilon}-(u))^{2}$and(iii) Forthat$\partial_{\epsilon 2}^{2-0,\epsilon},--\equiv 0$du,$\partial_{\epsilon 1}^{2-0,\epsilon},--(t):=\frac{\phi_{t}+\sqrt{1-\epsilon^{2}}H_{t}\overline{\lambda}_{t}}{\overline{Z}_{1}^{\epsilon}(t)}$.

$(\xi^{j})_{j=0,1,2,3}$

such that$\xi_{2}^{0}=\xi_{2}^{2}=0$

and $\xi^{1}=0$ , we can check$g_{3}(t, (\xi^{j})_{j=0,1,2,3},0)=0$,

so the equation$d(\partial_{\epsilon}^{3},Y_{t}^{0,\epsilon})=\partial_{\epsilon t}^{3-0,\epsilon},--d\overline{w}_{t}^{\epsilon}$,$\partial_{\epsilon}^{3},Y_{T}^{0,\epsilon}\equiv 0$

are deduced.
We are now in the position to state our last theorem, an extensionof Theorem 3.3. We require the conditions:and$(\partial_{\epsilon}^{3},Y^{0,\epsilon}, \partial_{\epsilon}^{3},--^{0,\epsilon}-)\equiv 0$$I$

Assumption 1. 1. The process
is bounded and deterministic.
, finite measures
on $B([0, T])$ for each
2. There is a kernel, satisfying$\mu_{2}$$\partial F$$\partial F(\omega_{1}, \cdot)$$\omega_{1}\in\Omega_{1}$$\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\{F(\omega_{1}+\epsilon\phi)-F(\omega_{1})\}=\int_{0}^{T}\phi(t)\partial F(\omega_{1}, dt)$

for all$\phi\in C^{1}([0, T])$and the Clark formula:$F=E[F]+\int_{0}^{T}E[\partial F(\cdot, (t, T])|F_{t}]dw_{1}(t)$.

(For suﬃcient conditions on $F$ andto ensure the formula, cf., Ap$E$
pendix of [5], for example). Moreover,holds.
$\partial F$$\partial F(\cdot,$

For

$n$

$=1,2$ ,$\ldots$, define$\overline{Y}^{\epsilon,n}:=\sum_{i=0}^{n}\partial_{\epsilon}^{2i},Y_{t}^{0,\epsilon}\frac{\epsilon^{2i}}{(2i)!}$and$=--\epsilon,n:=\sum_{i=0}^{n}\partial_{\epsilon}2,i---t0,\epsilon\frac{\epsilon^{2i}}{(2i)!}$and introduce the approximated strategy
formula(16)$\overline{\pi}_{t}^{\epsilon,n}$$(\cdot, T])\in L^{\infty}(\Omega_{1}\times[0, T])$$\overline{\pi}^{\epsilon,n}:=(\overline{\pi}_{t}^{\epsilon,n})_{t\in[0,T]}$$:=$$\frac{e^{-rT}}{\gamma}(\sigma^{/})^{-1}\Pi_{\sigma’C}(_{-t}^{=}-\epsilon,n+\lambda^{\epsilon}(t))$

$=$

$\{$

by the$\frac{e^{-rT}}{\gamma}\{\sigma_{2}^{-2}(\mu_{2}(t)-r)+0\sqrt{1-\epsilon^{2}}\sigma_{21}^{-1=}--(\epsilon,nt)\})$

.

294

J. Sekine

Note that$\overline{\pi}^{\epsilon,1}\in A:=A_{1}$since$ E[\int_{0}^{T}|_{-t}^{-0,\epsilon}-|^{2}dt]<\infty$

and since$E[(\int_{0}^{T}|\partial_{\epsilon}2,---t0,\epsilon|^{2}dt)^{q/2}]$$\overline{E}^{\epsilon}[(\overline{Z}_{T}^{\epsilon})^{-1}(\int_{0}^{T}|\partial_{\epsilon}2,---t0,\epsilon|^{2}dt)^{q/2}]$$=$$\overline{E}^{\epsilon}[(\overline{Z}_{T}^{\epsilon})^{-\frac{2}{2-q}}]2\overline{E}^{\epsilon}\underline{2}-\Delta[\int_{0}^{T}|\partial_{\epsilon}2,---t0,\epsilon|^{2}dt]q/2<\infty$$\leq$

for $0<q<2$ . We obtain the following.

Theorem 4. Under Assumption 1, the relations

andhold as$\epsilon\downarrow 0$$||Y^{\epsilon}-\overline{Y}^{\epsilon,1}||_{L^{\infty}([0,T]\times\Omega)}$$=$$o(\epsilon^{4})$$\log V^{\epsilon}(x)-\log E[U_{\gamma}(-F+X_{T}^{x,\overline{\pi}^{\epsilon,1}})]$$=$$o(\epsilon^{4})$.

Proof. Denoteand the kernel:$\Lambda^{\epsilon}(t):=(\int_{0}^{t}\overline{\lambda}_{u}du)d_{\epsilon}$$G(\omega_{1}):=\gamma F(\omega_{1}-\Lambda_{1}^{\epsilon})-\frac{1}{2}\int_{0}^{T}\overline{\lambda}_{u}^{2}du$,and define the Wiener functional$\partial G(\omega_{1}, dt):=\gamma\partial F(\omega_{1}-\Lambda_{1}^{\epsilon}, dt)$.

First, we observe the following:$---01’\epsilon(t, \omega_{1})$$=$$E[\partial G(\cdot, (t, T])|\mathcal{F}_{t}](\omega_{1}+\Lambda_{1}^{\epsilon})$$\partial_{\epsilon 1}^{2-0,\epsilon},--(t, \omega_{1})$

$=$

2

,

$\{E[\partial G(\cdot, (t, T])G|\mathcal{F}_{t}]$$-E[\partial G(\cdot, (t, T])|\mathcal{F}_{t}]E[G|F_{t}]\}(\omega_{1}+\Lambda_{1}^{\epsilon})$.

In fact, the first expression is a consequence from the Clark formula,$G(\omega_{1})=E[G]+\int_{0}^{T}E[\partial G(\cdot, (t, T])|\mathcal{F}_{t}](\omega_{1})dw_{1}(t, \omega_{1})$the Cameron-Martin formula,,$\overline{P}^{\epsilon}(\cdot)=P(\cdot+\Lambda^{\epsilon})$,, and the relation$w_{t}(\omega+$$\Lambda^{\epsilon})=\overline{w}_{t}^{\epsilon}(\omega)$$G(\omega_{1}+\Lambda_{1}^{\epsilon})$$=$$\overline{E}^{\epsilon}[G(\omega_{1}+\Lambda_{1}^{\epsilon})]$$+\int_{0}^{T}E[\partial G(\cdot, (t, T])|F_{t}](\omega_{1}+\Lambda_{1}^{\epsilon})d\overline{w}_{1}^{\epsilon}(t, \omega_{1})$

.

295

An approximation for exponential hedging

The second expression is deduced ﬀom the relation$\int_{0}^{T}(_{-1}^{-0,\epsilon}-(t))^{2}dt$$=$$(\int_{0}^{T}--^{0,\epsilon}-1(t)d\overline{w}_{1}^{\epsilon}(t))^{2}-2\int_{0}^{T}(\int_{0}^{t}-^{0,\epsilon}--1(u)d\overline{w}_{1}^{\epsilon}(u))---01’(\epsilon t)\mathcal{F}\overline{w}_{1}^{\epsilon}(t)$$=$$(G-\overline{E}^{\epsilon}[G])^{2}-2\int_{0}^{T}\{E[G|F_{t}](\cdot+\Lambda^{\epsilon})-E[G]\}_{-1}^{-0,\epsilon}-(t)T\overline{w}_{1}^{\epsilon}(t)$$=$$G^{2}-(\overline{E}^{\epsilon}[G])^{2}-2\int_{0}^{T}(E[G|F_{t}]E[\partial G(\cdot, (t, T])|F_{t}])(\cdot+\Lambda_{1}^{\epsilon})d\overline{w}_{1}^{\epsilon}(t)$the Claxk formula, and the chain rule for diﬀerentiation. In particular, it:. Therefore, in the BSDE forholds $that---01$ , ,$\epsilon$$\partial_{\epsilon 1}^{2-0,\epsilon},-\cup\in H_{T}^{\infty}$(17)$d\overline{Y}_{t}^{\epsilon,n}$where$R_{t}^{\epsilon,n}$$=$$\{g(t,$ $=--t’\epsilon)\epsilon,n+R_{t}^{\epsilon,n}\}dt+_{-t}^{=}-\epsilon,nd\overline{w}_{t}^{\epsilon}$$:=$$\sum_{i=0}^{2n+1}g_{i}$

, and

(,
$t$

$(\theta_{\epsilon}?,---t0)_{j=0,,i}\ldots$

$0$

’

)

,

$\overline{Y}_{T}^{\epsilon,n}=\gamma F$$\frac{\epsilon^{i}}{i!}-g(t,=--t’\epsilon)\epsilon,n$is satisfied because of the boundedness of$||R^{\epsilon,1}||_{L^{\infty}([0,T],\Omega)}=O(\epsilon^{4})$$\partial_{\epsilon}^{i},d_{0}^{\perp}$$(\overline{Y}^{\epsilon,n},=--)n,\epsilon$$\partial_{\epsilon}^{i-0,\epsilon},--(i=0, \ldots, 3)$,,$\lambda^{\epsilon}$,.

Next, we introduce the linear BSDE for, described as$(\Delta Y^{\epsilon,n}, \Delta_{-}^{-\epsilon,n}-):=(Y^{\epsilon}-$$\overline{Y}^{\epsilon,n},---\epsilon-=--\epsilon,n)$$d\Delta Y_{t}^{\epsilon,n}=\{-\frac{1}{2}$$(_{-t}^{-\epsilon}-+_{-t}^{=}-\epsilon,n$,$d_{\epsilon}^{\perp}$

$\{$

)

$(\Delta_{-t}^{-\epsilon,n}-, d_{\epsilon}^{\perp})-R_{t}^{\epsilon,n}\}dt+\Delta_{-t}^{-\epsilon,n}-d\overline{w}_{t}^{\epsilon}$$\Delta Y_{T}^{\epsilon,n}\equiv 0$

to observe the expression:(18)$-\Gamma_{s}\Delta Y_{s}^{\epsilon,n}=-\Gamma_{t}\Delta Y_{t}^{\epsilon,n}-\int_{s}^{t}\Gamma_{u}R_{u}^{\epsilon,n}du+M_{t}-M_{s}$for $0\leq s\leq t\leq T$ , where$d\Gamma_{t}=\Gamma_{t}\{\frac{1}{2}$$\Gamma:=(\Gamma_{t})_{t\in[0,T]}$$(_{-t}^{-\epsilon}-+_{-t}^{=}-\epsilon,n$and $M:=(M_{t})_{t\in[0,T]}$ is the$\overline{P}^{\epsilon}$,$d_{\epsilon}^{\perp}$)

is the solution of the SDE:$\Gamma_{0}=1$$(d_{\epsilon}^{\perp})^{l}(\Gamma w_{t}^{\epsilon}\},$-local-martingale defined by$M_{t}:=\int_{0}^{t}\Gamma_{u}\{\Delta_{-u}^{-\epsilon,n}-+\frac{1}{2}\Delta Y_{u}^{\epsilon,n}$$(_{-u}^{-\epsilon}-+=--\epsilon u’ n$,$d_{\epsilon}^{\perp}$)$d_{\epsilon}^{\perp}\}^{J}d\overline{w}_{u}^{\epsilon}$

.

,

,

296

J. Sekine

Let $n=1$ . For a sequence of increasing stopping times
localizes the local martingale $M$ , we deduce the relation$(\tau_{m})_{m\in N}$, which$\Gamma_{t\wedge\tau_{m}}|\Delta Y_{t\wedge\tau_{m}}^{\epsilon,1}|\leq\overline{E}^{\epsilon}[\Gamma_{T\wedge 7_{m}^{-}}|\Delta Y_{T\wedge\tau_{m}}^{\epsilon,1}|+\epsilon^{4}C_{1}\int_{t\wedge\tau_{m}}^{T\wedge\tau_{m}}\Gamma_{u}du|F_{t\wedge\tau_{m}}]$with some constant $C_{1}>0$ ﬀom (18). The first term of the right-handside is$\leq\overline{E}^{\epsilon}[\Gamma_{T\wedge\tau_{m}}|F_{t\wedge\tau_{m}}]||\Delta Y_{T\wedge\tau_{m}}^{\epsilon,1}||_{L^{\infty}(\Omega)}\leq\Gamma_{t\wedge\tau_{m}}||\Delta Y_{T\wedge\tau_{m}}^{\epsilon,1}||_{L^{\infty}(\Omega)}\rightarrow 0$

by using the optional stopping theorem, and the second termasof the right-hand-side is$ m\rightarrow\infty$$=\epsilon^{4}C_{1}\overline{E}^{\epsilon}[\int_{t\wedge\tau_{m}}^{T\wedge\tau_{m}}\Gamma_{u}du|\mathcal{F}_{t\wedge\tau_{m}}]\rightarrow\epsilon^{4}C_{1}\overline{E}^{\epsilon}[\int_{t}^{T}\Gamma_{u}du|F_{t}]\leq\epsilon^{4}C_{1}T\Gamma_{t}$asfor a continuous version ofmonotone convergence theorem. Therefore,
follows.
Finally, define the process$\overline{E}^{\epsilon}[\int^{T}.\Gamma_{u}du|\mathcal{F}.]$$ m\rightarrow\infty$$\overline{\nu}^{\epsilon,n}:=(\overline{\nu}^{\epsilon,n})_{t\in[0,T]}$(19)

by using the$||\Delta Y^{\epsilon,1}||_{L^{\infty}([0,T]\times\Omega)}=O(\epsilon^{4})$by$\overline{\nu}_{t}^{\epsilon,n}:=(I-\Pi_{\sigma’C})(_{-t}^{=}-^{\epsilon,n}+\lambda_{t}^{\epsilon})$to deduce$\gamma F$$=$,$\overline{Y}_{0}^{\epsilon,n}+\int_{0}^{T}(e^{rT}\gamma\sigma^{J}\overline{\pi}_{t}^{\epsilon,n}-\lambda_{t}^{\epsilon}+\overline{\nu}_{t}^{\epsilon,n})^{/}T\overline{w}_{t}^{\epsilon}$$+\int_{0}^{T}(\frac{|\lambda_{t}^{\epsilon}|^{2}-|\overline{\nu}_{t}^{\epsilon,n}|^{2}}{2}+R_{t}^{\epsilon,n})dt$

from (16-7) and (19). Therefore, for $x\in R$ , we obtain that

$F+I_{\gamma}(\overline{\mathcal{Y}}^{\epsilon,n}(x)Z_{T}^{\overline{\nu}^{\epsilon,n}})$where$\overline{\mathcal{Y}}^{\epsilon,n}(x)$$=$$X_{T}^{x,\overline{\pi}^{\epsilon,n}}+\int_{0}^{T}R_{t}^{\epsilon,n}dt$,$=$$\exp(\overline{Y}_{0}^{\epsilon,n}-\gamma e^{rT}x)$

,

297

An approximation for exponential hedging

which implies$\log E[U_{\gamma}(-F+X_{T}^{x,\overline{\pi}^{\epsilon,1}})]$$=$$\log E[U_{\gamma}(I_{\gamma}(\overline{\mathcal{Y}}^{\epsilon,1}(x)Z_{T}^{\overline{\nu}^{\epsilon,1}})-\int_{0}^{T}R_{t}^{\epsilon,1}dt)]$$=$$-\frac{1}{\gamma}\overline{\mathcal{Y}}^{\epsilon,1}(x)+O(\epsilon^{4})$$=$$\log U_{\gamma}(e^{rT}x-\frac{\overline {}Y_{0}^{\epsilon,1}}{\gamma})+O(\epsilon^{4})$$=$$\log U_{\gamma}(e^{rT}x-\frac{Y_{0}^{\epsilon}}{\gamma})+O(\epsilon^{4})$as$\epsilon\downarrow 0.1$

Remark 4. For the higher order terms, the following is observed, forexample:$\partial_{\epsilon}^{4}Y_{t}^{0,\epsilon}=-12\overline{E}^{\epsilon}[\int_{t}^{T}---0,\epsilon(u)\partial_{\epsilon}2,--1-0,\epsilon 1(u)du|F_{t}]$,$\partial_{\epsilon}^{4},Y_{0}^{0,\epsilon}-\int_{0}^{t}\partial_{\epsilon}4,---0,\epsilon(1u)f\overline{w}_{1}^{\epsilon}(u)$$=-12\overline{E}^{\epsilon}[\int_{0}^{T}---01’(u)\partial_{\epsilon}2\epsilon,---0,\epsilon 1(u)du|\mathcal{F}_{t}]$$\partial_{\epsilon 2}^{4-0,\epsilon},--(t)=0$$\partial_{\epsilon}^{4-0,\epsilon},--\in H_{T}^{\infty}$

and

, and

$\partial_{\epsilon}^{5-0,\epsilon},--(t)=0$,

for all $t\in[0, T]$ . So, if we assume, then$||Y^{\epsilon}-\overline{Y}^{\epsilon,2}||_{L^{\infty}([0,T]\times\Omega)}$$=$$O(\epsilon^{6})$$\log V^{\epsilon}(x)-\log E[U_{\gamma}(-F+X_{T}^{x,\overline{\pi}^{\epsilon,2}})]$$=$$O(\epsilon^{6})$are deduced as$\epsilon\downarrow 0$.

,
be constant andExample: European put option case. Let$=(k-S_{T}^{1})^{+}(k>0)$. Then, Assumption 1 is satisfied, and weset F$\mu_{1}$$\mu_{2}$J. Sekine

298

have that$\gamma\{k\Phi(-c_{t}^{-})-e^{\eta^{\epsilon}(T-t)}S_{t}^{1}\Phi(-c_{t}^{+})\}-\frac{\overline{\lambda}^{2}(T-t)}{2}$$Y_{t}^{0,\epsilon}$$=$$\partial_{\epsilon}^{2},Y_{t}^{0,\epsilon}$$=$,$\gamma^{2}\{k^{2}\Phi(-c_{t}^{-})-2ke^{\eta^{\epsilon}(T-t)}S_{t}^{1}\Phi(-c_{t}^{+})$$+e^{(2\eta^{\epsilon}+\sigma_{1}^{2})(T-t)}(S_{t}^{1})^{2}\Phi(-c_{t}^{++})\}$,,$---10,\epsilon(t)$$=$$-\gamma\sigma_{1}e^{\eta^{\epsilon}(T-t)}\Phi(-c_{t}^{+})S_{t}^{1}$$\partial_{\epsilon 1}^{2-0,\epsilon},--(t)$$=$$2\gamma^{2}\sigma_{1}\{-ke^{\eta^{\epsilon}(T-t)}S_{t}^{1}\Phi(-c_{t}^{+})$$+e^{(2\eta^{\epsilon}+\sigma_{1}^{2})(T-t)}(S_{t}^{1})^{2}\Phi(-c_{t}^{++})\}$$+2\gamma\sigma_{1}e^{\eta^{\epsilon}(T-t)}\Phi(-c_{t}^{+})S_{t}^{1}$$\times[\gamma\{k\Phi(-c_{t}^{-})-e^{\eta^{\epsilon}(T-t)}S_{t}^{1}\Phi(-c_{t}^{+})\}-\frac{\overline{\lambda}^{2}(T-t)}{2}]$

,

where
$\eta^{\epsilon}$$:=$$\mu_{1}-\sqrt{1-\epsilon^{2}}\sigma_{1}\sigma_{2}^{-1}(\mu_{2}-r)$,,$\Phi(d)$$:=$$\int_{-\infty}^{d}\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}dx$$c_{t}^{-}$$:=$$\frac{1}{\sigma_{1}\sqrt{T-t}}\{\log(\frac{S_{t}^{1}}{k})+(\eta^{\epsilon}-\frac{\sigma_{1}^{2}}{2})(T-t)\}$$:=$$c_{t}^{-}+\sigma_{1}\sqrt{T-t}$$c_{t}^{+}$

,

and

,

$c_{t}^{++}:=c_{t}^{-}+2\sigma_{1}\sqrt{T-t}$.

References[1] DAVIS, M.H.A. (2000). Optimal Hedging with Basis Risk, preprint, Imperial College, London, URL: http: //www. ma.ic.ac.uk/\tilde mdavis/[2] DELBAEN, F., P. GRANDITS, T. RHEINL\"ANDER, D. SAMPERI, M.
SCHWEIZER, AND C. STRICKER (2002). Exponential Hedging and Entropic Penalties, Mathematical Finance, 12(1), 99 124.
[3] EL KAROUI, N., S. PENG, AND M.C. QUENEZ (1997). Backward Stochastic Diﬀerential Equations in Finance, Mathematical Finance, 7(1),1-71.
[4] KABANOV, Y. AND C. STRICKER (2002). On the Optimal Portfolio for
the Exponential Utility Maximization: Remarks to the Six-Author,
Mathematical Finance, 12(1), 125-134.
[5] KARATZAS, I. AND S. SHREVE (1998). “Methods of Mathematical Finance,” Springer.

An approximation for exponential hedging

299[6] KOBYLANSKI, M. (2000). Backward Stochastic Diﬀerential Equations andPartial Diﬀerential Equations with Quadratic Growth, Annals of Probability, 28(2), 558-602.
[7] ROUGE, R. AND N. EL KAROUI (2000). Pricing via Utility Maximizationand Entropy, Mathematical Finance. 10 (2), 259-276.
Graduate School of Engineering Science,
Osaka University,
Toyonaka, Osaka 560-8531,
Japan.

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 301-317

Orlicz Norm Equivalence
for the Ornstein-Uhlenbeck Operator
Ichiro Shigekawa

Dedicated to Professor Kiyosi It\^oon the occasion of his 88th birthday
Abstract.

The Meyer equivalence on an abstract Wiener space states thatthe-norm of square root of the Ornstein-Uhlenbeck operator isequivalent to-norm of the Malliavin derivative. We prove theequivalence in the framework of Orlicz space. We also discuss thesetting and higher order logalogarithmic Sobolev inequality inrithmic Sobolev inequality.
$L^{p}$$L^{p}$$L^{p}$\S 1.

Introduction

Let $(B, H, \mu)$ be an abstract Wiener space: $B$ is a separable real
Banach space, $H$ is a separable real Hilbert space which is embedded
densely and continuously in $B$ and is a Gaussian measure with$\mu$$\int_{B}\exp\{\sqrt{-1}B^{*}\langle l, x\rangle_{B}\}\mu(dx)=\exp\{-\frac{1}{2}|l|_{H^{*}}^{2}\}$,$l$$\in B^{*}\simeq\rightarrow H^{*}$.

On an abstract Wiener space, the Ornstein-Uhlenbeck semigroup is defined as(1)$T_{t}f(x)=\int_{B}f(e^{-t}x+\sqrt{1-e^{-2t}}y)\mu(dy)$.

is called the Ornstein-Uhlenbeck
The generator of the semigroup
operator and we denote it by . Then the following Meyer equivalenceis well-known: for any $ 1<p<\infty$ , there exists positive constants
andsuch that$\{T_{t}\}$$L$$C_{1}$$C_{2}$(2)$C_{1}\{||Df||_{p}+||f||_{p}\}\leq||\sqrt{1-L}f||_{p}\leq C_{2}\{||Df||_{p}+||f||_{p}\}$.

Received March 20, 2003.
Partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (A) 14204008, 2002.

302

I. Shigekawa

is theHere $D$ is the Malliavin $H$ -derivation and-norm. The
depend only on .
constants
andIn this paper we show that similar inequalities hold in the frameworkof Orlicz space, i.e., the above inequalities hold for the Orlicz norm inplace of -norm. Typical example we are in mind is $IP\log^{\beta}L$ . As anapplication, we discuss the logarithmic Sobolev inequality in IF setting
and higher order logarithmic Sobolev inequality.
$L^{p}$$||||_{p}$$C_{1}$$C_{2}$$p$$L^{p}$\S 2.

Orlicz space

In this section, we review the Orlicz space (see, e.g., [1] or [8] fordetails). First we need the notion of Young function. Young function isa function defined as$\Phi$(3)$\Phi(x)=\int_{0}^{x}\phi(t)dt$,$x\geq 0$

where is a non-negative, right continuous, non-decreasing function. If,
in addition, satisfies $\phi(0)=0$ , $\phi(t)>0$ for $t>0$ , $\phi(\infty)=\infty$ , thenis called a nice Young function or $N$ -function. Define by$\phi$$\Phi$$\phi$$\psi$$\psi(u)=\inf\{t;\phi(t)>u\}$ .
$\psi$

is right continuous and non-decreasing. The function$\Psi(y)=\int_{0}^{y}\psi(u)du$,$\Psi$

defined by$y\geq 0$

is called a complementary function. The following properties are fundamental.
(4)$xy\leq\Phi(x)+\Psi(y)$ ,(5)$x\phi(x)=\Phi(x)+\Psi(\phi(x))$ .

(4) is called the Young inequality.
The Orlicz space associated with is defined as follows. Let $(M, m)$be a measure space and be a nice Young function. Define a normby$\Phi$$\Phi$(6)$||||_{\Phi}$$||f||_{\Phi}:=\inf\{\lambda>0;\int_{M}\Phi(|f|/\lambda)dm\leq 1\}$.

is the set of all measurable functions which $satis5^{r}||f||_{\Phi}<\infty$ .
We call $L^{\Phi}(m)$ an Orlicz space. It is a Banach space with the norm
. If satisfies thecondition, i.e., there exists a constant $C$ such$L^{\Phi}(m)$$||||_{\Phi}$$f$$\Phi$$\Delta_{2}$Orlicz nom equivalence

303

that $\Phi(2x)\leq C\Phi(x)$ , then the dual space is identified with $L^{\Psi}(m)$ ,being the complementary function of .
We introduce some classes of functions.

$\Psi$$\Phi$

Definition 2.1. For non-negative constant,as follows:
functions$\alpha$, we define a set of$U(\alpha)$$L(\alpha)$(i)$\phi\in L(\alpha)\Leftrightarrow def\alpha\phi(t)\leq t\phi’(t)$,$\forall t>0$(ii)$\phi\in U(\alpha)\Leftrightarrow tdef\phi’(t)\leq\alpha\phi(t)$,$\forall t>0$.
.

The following inequality for semimartingales is important in ourlater argument.
Let $(Z_{t})(t\in[0, \infty])$ be a non-negative submartingale. We assume
is right continuous and has left hand limits. By the Doobthat
Meyer decomposition theorem,
can be decomposed as$(Z_{t})$$(Z_{t})$$Z_{t}=M_{t}+A_{t}$is an increasing process. We assume
whereis a martingale and$A_{0}=0$. If $\Phi\in U(\alpha)$ , then the following
thatis continuous andinequality holds (see [4, Theorem VI.99]):$(M_{t})$$(A_{t})$$(A_{t})$(7)$E[\Phi(A_{\infty})]\leq E[\Phi(\alpha Z_{\infty})]$.

Further, a generalization of the Doob’s inequality also holds. It
is stated as follows (see [4, Chapter $VI$ , Section 3]). We assume that, it holds thatfor an $\alpha>1$ . Then, setting$\Phi\in L(\alpha)$$Z_{t}^{*}:=\sup_{s\leq t}Z_{s}$(8)$E[\Phi(Z_{\infty}^{*})]\leq E[\Phi(\alpha Z_{\infty})]$.

From this inequality, we can have the following maximal ergodic inequality.
(9)

Here

$\int_{B}\Phi(\sup_{t\geq 0}|T_{t}f(x)|)\mu(dx)\leq\int_{B}\Phi(\alpha|f(x)|)\mu(dx)$$\{T_{t}\}$.

is the Ornstein-Uhlenbeck semigroup on an abstract Wiener

space $(B, H, \mu)$ .

\S 3.

Littlewood-Paley inequality

Let $(B, H, \mu)$ be an abstract Wiener space and $K$ be a separable
Hilbert space.
is the Ornstein-Uhlenbeck semigroup on $L^{p}(E, \mu;K)$
defined by (1). For $\alpha>0$ , set$\{T_{t}\}$$T_{t}^{(\alpha)}=e^{-\alpha t}T_{t}$.

I. Shigekawa

304

is $ L-\alpha$ . We further define a semigroup
Then the generator of
by subordination as follows:$\{T_{t}^{(\alpha)}\}$$\{Q_{t}^{(\alpha)}\}$$Q_{t}^{(\alpha)}=\int_{0}^{\infty}T_{s}^{(\alpha)}\lambda_{t}(ds)=\int_{0}^{\infty}e^{-\alpha s}T_{s}\lambda_{t}(ds)$is a probability measure on
Heregiven by$\lambda_{t}$$[0, \infty)$.

whose Laplace transform is$\int_{0}^{\infty}e^{-\gamma s}\lambda_{t}(ds)=e^{-\sqrt{\gamma}t}$.

and called the Cauchy
is simply denoted by
When $\alpha=0$ ,semigroup. For $F\in L^{\Phi}(B, \mu;K)$ , it holds that$Q_{t}^{(0)}$$Q_{t}$$||Q_{t}^{(\alpha)}F||_{\Phi}\leq e^{-\sqrt{\alpha}t}||F||_{\Phi}$. The generator
is a strongly continuous semigroup on
andwill be denoted by $-\sqrt{\alpha-L}$ .
We denote by $P(K)$ a set of all functions $f:B\rightarrow K$ which can be
expressed as,,$L^{\Phi}$$\{Q_{t}^{(\alpha)}\}$$\langle l_{n}, x\rangle)k_{i}$$\ldots$$f(x)=\sum_{i}p_{i}(\langle l_{1}, x\rangle)$

is a polynomial on
where
For $f\in P(K)$ , define$p_{i}$$\mathbb{R}^{n}$and$k_{1}$,$\ldots$, $k_{n}\in K$ ,$g^{\rightarrow}f(x, t)=|\partial_{t}Q_{t}^{(\alpha)}(x, f)|_{K}$$l_{1}$,$\ldots$,$l_{n}\in B^{*}$.

,$g^{\uparrow}f(x, t)=|DQ_{t}^{(\alpha)}(x, f)|_{HS}$,$gf(x, t)=\sqrt{g^{\rightarrow}f(x,t)^{2}+g\uparrow f(x,t)^{2}}$.

denotes the HilbertHere $Q_{a}^{(\alpha)}(x, f)=Q_{a}^{(\alpha)}f(x)$ and the norm
norm,
$gf$
,
,
all depend on but we fix it throughout
Schmidt
the argument and suppress it for simplicity. We further define$||_{HS}$$g^{\rightarrow}f$$g^{\uparrow}f$$\alpha$$G^{\rightarrow}f(x)=\{\int_{0}^{\infty}tg^{\rightarrow}f(x, t)^{2}dt\}^{1/2}$$G^{\uparrow}f(x)=\{\int_{0}^{\infty}tg^{\uparrow}f(x, t)^{2}dt\}^{1/2}$,,$Gf(x)=\{\int_{0}^{\infty}tgf(x, t)^{2}dt\}^{1/2}$

We call them Littlewood-Paley $G$-functions.
Our aim in this section is to prove the following theorem.

Orlicz norm equivalence

305

Theorem 3.1. Assume that $\Phi\in L(\alpha)\cap U(\beta)$ for constants $1<$
. Further assume that is either convex or concave. Then we
have$\alpha<\beta$$\phi$,(10)$||\Phi(Gf)||_{1}\sim<||\Phi(|f|)||_{1}$(11)$||\Phi(|f|)||_{1}\leq||\Phi(G^{\rightarrow}f)||_{1}$.

In the above theorem, $A\sim<B$ stands for $A\leq CB$ for a positive
constant $C$ that is independent of . We use this convention in thesequel without mentioning.
We give a probabilistic proof. To do this, we take the OrnsteinUhlenbeck process
on $B$ , i.e., the diﬀusion process generated by. We assume
generated by. We also take a processonthat the initial distribution ofis the stationary measure so thatthe process becomes stationary. We denote the starting point of theby N.
Brownian motion
stands for the integration with respect.
to this measure. Later we let$u(x,a)=Q_{a}^{(\alpha)}(x,f)$. Then $u(x, a)$ satisfies
Now, for $f\in P(K)$ , set$f$$(X_{t})$$L$$(B_{t})$$\mathbb{R}$$\frac{d^{2}}{da^{2}}$$(X_{t})$$\mu$$E_{N}$$(B_{t})$$ N\rightarrow\infty$$u(x, 0)=f(x)$

(12)

$\{$

$L_{x}u(x, a)+\partial_{a}^{2}u(x, a)-\alpha u(x, a)=0$

Define a stopping time$\tau$.

by$\tau=\inf\{t>0|B_{t}=0\}$ .

Then we can think of $u(X_{t}, B_{t})$ for$ t\leq\tau$. Set$M_{t}=Q_{B_{t\wedge\tau}}(X_{t\wedge\tau}, f)-\alpha\int_{0}^{t\wedge\tau}Q_{B_{3}}(X_{s}, f)ds$$=u(X_{t\wedge\tau}, B_{t\wedge\tau})-\alpha\int_{0}^{t\wedge\tau}Q_{B_{s}}(X_{s}, f)ds$.

Then
is a martingale with $M_{0}=Q_{B_{0}}f(X_{0})$ . The quadratic variation is given as$(M_{t})$(13)$\langle M\rangle_{t}=2\int_{0}^{t\wedge\tau}gf^{2}(X_{s}, B_{s})ds$.

Therefore we have(14)

$ d|u|^{2}=2(u, dM)+2\alpha|u|^{2}dt+\langle dM, dM\rangle$$=2(u, dM)+(2\alpha|u|^{2}+2gf^{2})dt$

306

I. Shigekawa

Now set$Z_{t}=|u(X_{t\wedge\tau}, B_{t\wedge\tau})|^{2}$.

is a non-negative submartingale. To compute, we approximate it as follows. Take $any\in>0$ and set $F(x)=\Phi(\sqrt{x+\in})$ . Recall
that is either convex or concave. We divide into tow cases.
$\Phi(\sqrt{Z_{t}})$$(Z_{t})$$\phi$$\frac{(i)\phi isconcave}{Weneedth}.e$following proposition.

Proposition 3.2. Assume $\Phi\in L(\alpha)(\alpha>1)$ . Then it holds that,for , $v\geq 0$ ,
$u$

(15)

$\Phi(v)\leq\frac{1}{\alpha-1}(\frac{1}{2}\phi’(u)v^{2}+\Phi(u))$

Proof

.

From the assumption, $\alpha\Phi(x)\leq x\phi(x)$ holds. Sinceis. Hence (15) clearly holds
which leads to$\phi$

concave,when $u\geq v$ .
If $v\geq u$ , we have$\Phi(x)\geq\frac{1}{2}x\phi(x)$$\alpha\leq 2$$|\{(x, y);0\leq x\leq u, \phi(x)\leq y\leq\phi(u)\}|\leq\frac{1}{2}u\phi(u)$$|\{(x, y);0\leq x\leq u, \phi(u)\leq y\leq\phi(v)\}|\leq u(\phi(v)-\phi(u))\leq u\phi’(u)(v-u)$$|\{(x, y);u\leq x\leq v, \phi(x)\leq y\leq\phi(v)\}|\leq\frac{1}{2}(v-u)^{2}\phi’(u)$These are easily obtained by observing the graph.
Summing up three terms of the left-hand side and$v\phi(v)$ . Therefore.

$\Phi(v)$, we have$\frac{1}{2}u\phi(u)+u\phi’(u)(v-u)+\frac{1}{2}(v-u)^{2}\phi’(u)+\Phi(v)\geq v\phi(v)\geq\alpha\Phi(v)$.

Hence we have$(\alpha-1)\Phi(v)\leq\frac{1}{2}u\phi(u)+\phi’(u)(v-u)(u+\frac{1}{2}v-\frac{1}{2}u)$$=\frac{1}{2}u\phi(u)+\frac{1}{2}\phi’(u)(v^{2}-u^{2})$$\leq\Phi(u)+\frac{1}{2}\phi’(u)v^{2}$

which is the desired result.

Q.E.D.

The derivatives of $F(x)=\Phi(\sqrt{x+\in})$ are$F’(x)=\Phi’(\sqrt{x+\epsilon})\frac{1}{2\sqrt{x+\in}}$,Orlicz norm equivalence

307$F’’(x)=\Phi’’(\sqrt{x+\in})\frac{1}{4(x+\in)}+\Phi’(\sqrt{x+\in})\frac{1}{2}(-\frac{1}{2})\frac{1}{\sqrt{x+\in}^{3}}$.

By the It\^o formula,$d\Phi(\sqrt{Z_{t}+\in})=\frac{\Phi’(\sqrt{Z_{t}+\in})}{2\sqrt{Z_{t}+\in}}dZ_{t}$$+\frac{1}{2}\{\frac{\Phi’’(\sqrt{Z_{t}+\in})}{4(Z_{t}+\in)}-\frac{1}{4}\frac{\Phi’(\sqrt{Z_{t}+\in})}{\sqrt{Z_{t}+\in}^{3}}\}\langle dZ, dZ\rangle$$=\frac{\phi(\sqrt{|u|^{2}+\in})}{2\sqrt{|u|^{2}+\in}}\{2(u, dM)+2(\alpha|u|^{2}+gf^{2})dt\}$$+\frac{1}{2}\{\frac{\phi’(\sqrt{|u|^{2}+\in})}{4(|u|^{2}+\in)}-\frac{1}{4}\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}^{3}}\}\langle dZ, dZ\rangle$$=\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}(u, dM)+\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}(\alpha|u|^{2}+gf^{2})dt$$+\frac{1}{8}\frac{1}{|u|^{2}+\in}\{\phi’(\sqrt{|u|^{2}+\in})-\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}\}\langle dZ, dZ\rangle$

Now we note $\langle dZ, dZ\rangle\leq 4|u|^{2}\langle dM, dM\rangle=8|u|^{2}gf^{2}dt$ .
$\phi’(t)\leq\phi(t)/t$ since
is concave. We therefore have.

Further$\phi$$\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}gf^{2}dt+\frac{1}{8}\frac{1}{|u|^{2}+\in}\{\phi’(\sqrt{|u|^{2}+\in})-\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}\}\langle dZ, dZ\rangle$$\geq\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}gf^{2}dt+\{\phi’(\sqrt{|u|^{2}+\in})-\frac{\phi(\sqrt{|u|^{2}+\in})}{\sqrt{|u|^{2}+\in}}\}gf^{2}dt$$=\phi’(\sqrt{|u|^{2}+\in})gf^{2}dt$

Integrating ﬀom 0 to
(16)$\tau$.

and taking expectation, we have$||\Phi(\sqrt{|f|^{2}+\in})||_{1}\geq E_{N}[\int_{0}^{\tau}\phi’(\sqrt{|u|^{2}+\in})gf^{2}dt]$.

We will give an estimate ﬀom below of the right-hand side. We notethat $f^{*}(x):=\sup_{t\geq 0}|T_{t}f(x)|\geq\sup_{a\geq 0}|u(x, a)|$ .
$E_{N}[\int_{0}^{\tau}\phi’(\sqrt{|u|^{2}+\in})gf^{2}dt]=||\int_{0}^{\infty}\phi’(\sqrt{|u|^{2}+\in})gf^{2}(\cdot, a)(a\wedge N)da||_{1}$$\geq||\int_{0}^{\infty}\phi’(\sqrt{f^{*2}+\in})gf^{2}(\cdot, a)(a\Lambda N)da||_{1}$I. Shigekawa

308($\cdot.\cdot\phi’$

Combining this with (16) and letting

is non-increasing)$ N\rightarrow\infty$$||\Phi(\sqrt{|f|^{2}+\in})||_{1}\geq||\int_{0}^{\infty}\phi’(\sqrt{f^{*2}+\in})gf^{2}(\cdot, a)ada||_{1}$$=||\phi’(\sqrt{f^{*2}+\in})Gf^{2}||_{1}$Now we use the inequality
tion 3.2 and getin Proposi-$\Phi(v)\leq\frac{1}{\alpha-1}(\frac{1}{2}\phi’(u)v^{2}+\Phi(u))$$||\Phi(Gf)||_{1}\sim<||\phi’(\sqrt{f^{*2}+\in})Gf^{2}||_{1}+||\Phi(\sqrt{f^{*2}+\in})||_{1}$$\leq||\Phi(\sqrt{|f|^{2}+\in})||_{1}+||\Phi(\sqrt{f^{*2}+\in})||_{1}$$Letting\in\rightarrow 0$.

and using the maximal ergodic inequality (9), we have$||\Phi(Gf)||_{1}\sim<||\Phi(|f|)||_{1}+||\Phi(f^{*})||_{1}\sim<||\Phi(|f|)||_{1}$This completes the proof in the case thatis convex.
Set
we have$\phi$.

is concave.

$\underline{(ii)\phi}$$\tilde{\Phi}(x)=\Phi(\sqrt{x})$. Then$\tilde{\Phi}$

is convex. In fact, by diﬀerentiating,$\frac{d}{dx}\Phi(\sqrt{x})=\Phi’(\sqrt{x})\frac{1}{2\sqrt{x}}=\frac{\phi(\sqrt{x})}{2\sqrt{x}}$.

The function is increasing since is convex and so the convexity offollows. Further
since$\overline{\Phi}$$\phi$$\tilde{\Phi}\in U(\alpha/2)$$\frac{x\overline{\Phi}’(x)}{\overline{\Phi}(x)}=\frac{x\Phi’(\sqrt{x})}{2\sqrt{x}\Phi(\sqrt{x})}=\frac{\sqrt{x}\Phi’(\sqrt{x})}{2\Phi(\sqrt{x})}$.

The submartingale
is decomposed as a sumof a martingale and an increasing process as in (14). By using (7), weget$Z_{t}=|u(X_{t\wedge\tau}, B_{t\wedge\tau})|^{2}$(17)$E_{N}[\overline{\Phi}(\int_{0}^{\tau}gf(X_{s}, B_{s})^{2}ds)]\leq E_{N}[\overline{\Phi}(Z_{\infty})]=E_{N}[\overline{\Phi}(|f(X_{\tau})|^{2})]$$=E_{N}[\Phi(|f(X_{\tau})|)]=||\Phi(|f|)||_{1}$Now we introduce $H$ -functions as follows.
$H^{\rightarrow}f(x)=\{\int_{0}^{\infty}tQ_{t}g^{\rightarrow}f(x, t)^{2}dt\}^{1/2}$

,

309

Orlicz norm equivalence$H^{\uparrow}f(x)=\{\int_{0}^{\infty}tQ_{t}g^{\uparrow}f(x, t)^{2}dt\}^{1/2}$,$Hf(x)=\{\int_{0}^{\infty}tQ_{t}gf(x, t)^{2}dt\}^{1/2}$

Then we have$||\Phi(Hf)||_{1}=||\tilde{\Phi}(Hf^{2})||_{1}$$=\lim_{N\rightarrow\infty}||\tilde{\Phi}(\int_{0}^{\infty}Q_{a}gf(\cdot, a)^{2}(a\wedge N)da)||_{1}$$=\lim_{N\rightarrow\infty}\int_{B}\tilde{\Phi}(E_{N}[\int_{0}^{\tau}gf^{2}(X_{t}, B_{t})dt|X_{\tau}=x])\mu(dx)$$\leq\lim_{N\rightarrow\infty}\int_{B}E_{N}[\overline{\Phi}(\int_{0}^{\tau}gf^{2}(X_{t}, B_{t})dt)|X_{\tau}=x])\mu(dx)$$\leq\lim_{N\rightarrow\infty}E_{N}[\overline{\Phi}(\int_{0}^{\tau}gf^{2}(X_{t}, B_{t})dt)]$$\leq||\Phi(|f|)||_{1}$.

$(\cdot.\cdot(17))$

It is well-known that $Gf$ is dominated by $Hf$ (see [7]) and so (10)follows. (11) can be shown by a standard duality argument. This completes the proof of Theorem 3.1.
Using this theorem, the Meyer equivalence in Orlicz space, which
is of our main interest, follows easily. In fact, the same proof as insetting works (see e.g., [9]).
$L^{p}$

Theorem 3.3. Assume that $\Phi\in L(\alpha)\cap U(\beta)$ for $ 1<\alpha<\beta$ andthat is either convex or concave. Then there exist positive constants
andsuch that$\phi$$C_{1}$$C_{2}$(18)$C_{1}\{||Df||_{\Phi}+||f||_{\Phi}\}\leq||\sqrt{1-L}f||_{\Phi}\leq C_{2}\{||Df||_{\Phi}+||f||_{\Phi}\}$

\S 4.

.

ExamplesWe give some example of nice Young functions that satisfy the con, $k\geq 1$ , we set
dition of Theorem 3.3. For indicies $p>1$ ,$\beta\in \mathbb{R}$(19)$\phi_{p,\beta,k}(x)=x^{p-1}\log^{p\beta}(k+x)$,$x\geq 0$

and define(20)$\Phi_{p,\beta,k}(x)=\int_{0}^{x}\phi_{p,\beta,k}(y)dy$.

I. Shigekawa

310

We regards this as a Young function. The function does not satisfy thecondition of Young function in general since might be negative. We
see when it is a Young function. To avoid complexity, we simply denoteand, respectively. Diﬀerentiating , weand in place of
have$\beta$$\Phi$$\phi$$\phi_{p,\beta,k}$$\phi$$\Phi_{p,\beta,k}$$\phi’(x)=(p-1)x^{p-2}\log^{p\beta}(k+x)+p\beta x^{p-1}\{\log^{p\beta-1}(k+x)\}\frac{1}{k+x}$$=x^{p-2}\log^{p\beta}(k+x)\{p-1+p\beta\frac{x}{(k+x)1og(k+x)}\}$ .

We look for the condition so that$\phi’$is positive. To do this, set$f(x)=\frac{x}{(k+x)1og(k+x)}$ .

(21)

If $k=1$ , takes its maximum 1 at $x=0$ . If $k>1$ , takes its maximum
at $ x=\alpha$ where is the solution of $k\log(k+x)-x=0$ . We can see. Therefore, in all cases of , it holds that
that
$f$

$f$

$\alpha$$k$$f(\alpha)\leq\frac{l}{1+1ogk}$(22)$0\leq\frac{x}{(k+x)1og(k+x)}\leq\frac{1}{1+\log k}$ .

Now it is easy to see that$1$$\Phi$

is a nice Young function if $ p(1+\frac{\beta}{1+\log k})\geq$. Further we easily have the following proposition.
Proposition 4.1.

$\phi$

satisfies following inequalities:(23)$(p-1)\phi(x)\leq x\phi’(x)\leq(p-1+\frac{p\beta}{1+1ogk})\phi(x)$

,

for

(24)

$(p-1+\frac{p\beta}{1+1ogk})\phi(x)\leq x\phi’(x)\leq(p-1)\phi(x)$,for $\beta<0$ .

Similar inequalities hold for
proposition.

$\Phi$$\beta\geq 0$,. To see this, we need the following

Proposition 4.2. For positive constant, then $\Phi\in L(\alpha+1)$ ,(i) if, then $\Phi\in U(\alpha+1)$ .
(ii) if$\alpha$, it holds that$\phi\in L(\alpha)$$\phi\in U(\alpha)$

Proof.

Supposehands, we have(25)$\phi\in L(\alpha)$, i.e., $\alpha\phi(t)\leq t\phi’(t)$ . By integrating both$\alpha\Phi(x)\leq\int_{0}^{x}t\phi’(t)dt$.

Orlicz norm equivalence311

On the other hand, since $(t\phi(t))’=\phi(t)+t\phi’(t)$ , we have$x\phi(x)=\int_{0}^{x}\{\phi(t)+t\phi’(t)\}dt$

and hence$x\phi(x)-\Phi(x)=\int_{0}^{x}t\phi’(t)dt$.

This combined with (25) leads$x\phi(x)\geq(\alpha+1)\Phi(x)$.

(ii) can be shown similarly.

Q.E.D.

Now the following proposition easily follows.
Proposition 4.3. The following inequalities hold:(26)$p\Phi(x)\leq x\Phi’(x)\leq p(1+\frac{\beta}{1+\log k})\Phi(x)$,$(\beta\geq 0)$

,

(27)

$p(1+\frac{\beta}{1+\log k})\Phi(x)\leq x\Phi’(x)\leq p\Phi(x)$,$(\beta<0)$.

Lastly, we will see the asymptotic behavior of the complementary
holds.
function . We use the notation $f\sim g$ when$\Psi$

of$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$

Proposition 4.4. Assume $p>1$ and let. The it hold: that$q$

be the conjugate exponent$p:\frac{1}{p}+\frac{1}{q}=1$(28)$p\Phi(x)\sim x^{p}\log^{p\beta}x$

(29)

,

$(q-1)^{q\beta}q\Psi(x)\sim x^{q}\log^{-q\beta}x$

Proof.

.

By the l’H\^opital theorem, we have$\lim_{x\rightarrow\infty}\frac{p\Phi(x)}{x^{p}1og^{p\beta}x}=\lim_{x\rightarrow\infty}\frac{p\phi(x)}{px^{p-1}\log^{p\beta}x+x^{p}p\beta(\log^{p\beta}x)/x}$$=\lim_{x\rightarrow\infty}\mathring{\frac{px^{p-1}1gp\beta(k+x)}{px^{p-1}1og^{p\beta}x+p\beta x^{p-1}1og^{p\beta-1}x}}$$=\lim_{x\rightarrow\infty}\frac{1og^{p\beta}(k+x)}{\log^{p\beta}x+\beta 1og^{p\beta-1}x}$$=1$which shows (28).

312

I. Shigekawa

As for$\Psi$, we have, by the l’H\^opital theorem,$\lim\underline{q\Psi(x)}$$x\rightarrow\infty x^{q}\log^{-q\beta}x$$=\lim_{x\rightarrow\infty}\frac{q\Psi(\phi(x))}{\phi(x)^{q}1og^{-q\beta}\phi(x)}$$=\lim\underline{q\Psi’(\phi(x))\phi’(x)}$$x\rightarrow\infty q\phi(x)^{q-1}\phi’(x)\log^{-q\beta}\phi(x)+\phi(x)^{q}(-q\beta)\{\log^{-q\beta-1}\phi(x)\}\phi’(x)/\phi(x)$$=\lim_{x\rightarrow\infty}\frac{x}{\phi(x)^{q-1}\log^{-q\beta}\phi(x)-\beta\phi(x)^{q-1}\log^{-q\beta-1}\phi(x)}$$=\lim_{x\rightarrow\infty}\frac{x}{\phi(x)^{q-1}\log^{-q\beta}\phi(x)(1-\beta/\log\phi(x))}$$=\lim_{x\rightarrow\infty}\frac{x}{\phi(x)^{q-1}1og^{-q\beta}\phi(x)}$$=\lim_{x\rightarrow\infty}\frac{x}{\{x^{p-1}\log^{p\beta}(k+x)\}^{q-1}1og^{-q\beta}(x^{p-1}\log^{p\beta}(k+x))}$$=\lim_{x\rightarrow\infty}\frac{x}{x^{(p-1)(q-1)}\log^{p\beta(q-1)}(k+x)\{(p-1)\log x+p\beta\log\log(k+x)\}^{-q\beta}}$$(\cdot.\cdot(p-1)(q-1)=1)$$=\lim_{x\rightarrow\infty}\frac{\{(p-1)1ogx+p\beta 1og\log(k+x)\}^{q\beta}}{\log^{p\beta(q-1)}(k+x)}$$=\lim_{x\rightarrow\infty}\{\frac{\{(p-1)\log x+p\beta 1og\log(k+x)}{1og(k+x)}\}^{q\beta}$$(\cdot.\cdot q=p(q-1))$$=(p-1)^{q\beta}$

which shows (29).

Q.E.D.

We denotes the Orlicz space.
associated withby
We do not specify since it does not aﬀect the asymptotic behavior at
infinity. Since the Wiener measure is finite,
is independentof . The above theorem means that the dual space ofis$L^{\Phi}$$\Phi$$L^{p}\log^{p\beta}L$$k$$L^{p}\log^{p\beta}L$$L^{p}\log^{p\beta}L$$k$$L^{q}\log^{-q\beta}L$

\S 5.

.

Logarithmic Sobolev inequality

The logarithmic Sobolev inequality insetting was discussed by
D. Bakry-P. A. Meyer [3] and higher order Logarithmic Sobolev inequality was discussed by G. F. Feissner [5] and R. A. Adams [2]. They all
used the interpolation theorem. Here we take a diﬀerent approach.
$L^{p}$

Orlicz norm equivalence313

The following logarithmic Sobolev inequality holds for the OrnsteinUhlenbeck process.
$E[f^{2}\log(f^{2}/||f||_{2}^{2})]\leq 2E[|Df|^{2}]$.

Here $E[]$ stands for the integration with respect to . Hereafter we
use this notation. Recall that $\int_{B}(Df, Dg)_{H^{*}}d\mu$ is the Dirichlet form
associated with the Ornstein-Uhlenbeck process. We remark that thefollowing argument works for the diﬀusion Dirichlet form satisfying thelogarithmic Sobolev inequality if we assume the Dirichlet form is of thegradient type.
We introduce a new Young function. Set
$\mu$

(30)

$\theta(x)=\{x^{2}\log(e+x^{2})\}^{(p-2)/4}\log^{p\beta/4}(k+x^{2}\log(e+x))$

and define(31)$\Theta(x)=\int_{0}^{x}\theta(y)dy$.

Then we have the following proposition.

Proposition 5.1. For suﬃcient large
positive constant $K$ such that
(32)

$k$

if necessary, there exists$x^{p}\log^{p/2}(e+x^{2})\log^{p\beta/2}(k+x^{2}\log(e+x^{2}))$$\leq K\Theta(x)^{2}\log(e+O-(x)^{2})$Proof.
(a)

We divide the proof into two cases,, $k=1$ .
Let us see the asymptotic behavior as$\beta\geq 0$$x\rightarrow 0$.

$LHS\sim x^{p}\cdot x^{(p\beta/2)2}=x^{p(1+\beta)}$.

On the other hand,$\theta(x)\sim x^{(p-2)/2+(p\beta)/2}=x^{p(\beta+1)/2-1}$

and hence$\Theta(x)\sim\frac{2}{p(\beta+1)}x^{p(\beta+1)/2}$$\Theta(x)^{2}\sim\frac{4}{p^{2}(\beta+1)^{2}}x^{p(\beta+1)}$.

Thus both hands have the same asymptotic behavior.

.

.

$a$

I. Shigekawa

314

As

$ x\rightarrow\infty$,$LHS\sim x^{p}2^{p/2}(\log^{p/2}x)2^{p\beta/2}\log^{p\beta/2}x=2^{p(\beta+1)/2}x^{p}\log^{p(\beta+1)/2}x$.

On the other hand,$\theta(x)\sim x^{(p-2)/2}2^{(p-2)/4}(\log^{(p-2)/4}x)2^{p\beta/4}\log^{p\beta/4}x$$\ominus(x)\sim(2/p)2^{(p+p\beta-2)/4}x^{p/2}\log^{(p+p\beta-2)/4}x$$\ominus(x)^{2}\log(e+\Theta(x)^{2})\sim p^{-2}2^{(p+p\beta+2)/2}x^{p}(\log^{(p+p\beta-2)/2}x)p\log x$$=p^{-1}2^{(p+p\beta+2)/2}x^{p}\log^{(p+p\beta)/2}x$.

Hence they have the same asymptotic behavior,(b) $\beta<0$ and large .
The asymptotic behavior at $ x=\infty$ can be obtained similarly as in
the case.
, $LHS\sim x^{p}$ is clear. Further we haveAs$k$$\beta\geq 0$$x\rightarrow 0$$\theta(x)\sim x^{(p-2)/2}$$\ominus(x)\sim\frac{2}{p}x^{p/2}$$\Theta(x)^{2}\log(e+\Theta(x)^{2})\sim\frac{4}{p^{2}}x^{p}$

Q.E.D.

Thus we have the desired result.

We recall the following fact. Let $U$ and $V$ be a non-negative functions on a measure space $(M, m)$ . Assume that$\int_{M}U\phi(U)dm<\infty$,$\int_{M}U\phi(U)dm\leq\int_{M}V\phi(U)dm+C$ .

Then it follows that(33)$\int_{M}\Phi(U)dm\leq\int_{M}\Phi(V)dm+C$ .

For the proof, see [4, Lemma VI.98]. Now we have the following theorem.
in place of
because the index
In the sequel, we denote by
is not essential.
$\Phi_{p,\beta}$$\Phi_{p,\beta,k}$

$k$

315

Orlicz norm equivalence

Proposition 5.2. For $p>2$ ,$C$ such that(34)$\beta\in \mathbb{R}$, there exists a positive constant$E[\Phi_{p,(\beta+1)/2}(|f|)]\leq CE[\Phi_{p,(1+\beta)/2-(1/p)}(|f|)]+CE[\Phi_{p,\beta/2}(|Df|)]$.

Proof. Set $g=\sqrt{O(|f|)^{2}+e}$ . Then
$Dg=\frac{2\Theta(|f|)\Theta’(|f|)D|f|}{2\sqrt{O(|f|)^{2}+e}}$

and hence $|Dg|\leq\theta(|f|)|Df|$ . Now, by using the logarithmic Sobolev
inequality$E[g^{2}\log(g^{2}/||g||_{2}^{2})]\leq 2E[|Dg|^{2}]$,we have$E[\{\Theta(|f|)^{2}+e\}\log(e+\ominus(|f|)^{2})]$$\leq E[\Theta(|f|)^{2}+e]\log E[e+\ominus(|f|)^{2}]$$+2E[|Df|^{2}\{|f|^{2}\log(e+|f|^{2})\}^{(p-2)/2}\log^{p\beta/2}(k+|f|^{2}\log(e+|f|^{2}))]$,

and$E[\Theta(|f|)^{2}\log(e+\ominus(|f|)^{2})]$$\leq E[\ominus(|f|)^{2}]\log E[e+\Theta(|f|)^{2}]$$+2E[|Df|^{2}\{|f|^{2}\log(e+|f|^{2})\}^{(p-2)/2}\log^{p\beta/2}(k+|f|^{2}\log(e+|f|^{2}))]$We set$\phi(x)=\phi_{p/2,\beta,k}(x)=x^{(p/2)-1}\log^{p\beta/2}(k+x)$,$U=|f|^{2}\log(e+|f|^{2})$ .

Then$U\phi(U)=|f|^{2}\log(e+|f|^{2})$$\times\{|f|^{2}\log(e+|f|^{2})\}^{(p/2)-1}\log^{p\beta/2}(k+|f|^{2}\log(e+|f|^{2}))$$=|f|^{p}\log^{p/2}(e+|f|^{2})\log^{p\beta/2}(k+|f|^{2}\log(e+|f|^{2}))$$\leq K\ominus(|f|)^{2}\log(e+\Theta(|f|)^{2})$.

$(\cdot.\cdot(32))$

Combining this with the previous result, we have$K^{-1}E[U\phi(U)]\leq E[e+\ominus(|f|)^{2}]\log E[e+\Theta(|f|)^{2}]+2E[|Df|^{2}\phi(U)]$ .

.

I. Shigekawa

316

Now, by (33), it follows that
$E[\Phi(U)]\leq KE[e+\ominus(|f|)^{2}]\log E[e+\Theta(|f|)^{2}]+2KE[\Phi(|Df|^{2})]$ .

Here$\Phi$

is the integral of . Since$\phi$$\Phi=\Phi_{p/2,\beta}$,$\Phi(x^{2})\leq c_{1}x^{2}\phi(x^{2})$$\leq c_{1}x^{2}(x^{2})^{(p/2)-1}\log^{p\beta/2}(k+x^{2})$$\leq c_{1}x^{p}\log^{p\beta/2}(k+x^{2})$$\leq c_{2}\Phi_{p,\beta/2}(x)$.

Further$\Phi(x^{2}\log(e+x^{2}))\geq c_{3}x^{2}\log(e+x^{2})\phi(x^{2}\log(e+x^{2}))$$=c_{3}x^{2}\log(e+x^{2})\{x^{2}\log(e+x^{2})\}^{(p/2)-1}$$\times\log^{p\beta/2}(k+x^{2}\log(e+x^{2}))$$=c_{3}x^{p}\log^{p/2}(e+x^{2})\log^{p\beta/2}(k+x^{2}\log(e+x^{2}))$$\geq c_{4}x^{p}\log^{p/2}(e+x)\log^{p\beta/2}(k+x)$$\geq c_{5}x^{p}\log^{p(1+\beta)/2}(k+x)$$\geq c_{6}\Phi_{p,(\beta+1)/2}(x)$and$\ominus(x)^{2}\leq x^{2}\theta(x)^{2}$$\leq x^{2}\{x^{2}\log(e+x^{2})\}^{(p-2)/2}\log^{p\beta/2}(k+x^{2}\log(e+x))$$\leq c_{7}x^{p}\log^{(p-2)/2}(e+x)\log^{p\beta/2}(k+x)$$\leq c_{7}x^{p}\log^{(p+p\beta-2)/2}(k+x)$$\leq c_{8}\Phi_{p,(1+\beta)/2-(1/p)}(x)$.

Thus we have eventually obtained$E[\Phi_{p,(\beta+1)/2}(|f|)]\leq CE[\Phi_{p,(1+\beta)/2-(1/p)}(|f|)]+CE[\Phi_{p,\beta/2}(|Df|)]$Q.E.D.

This completes the proof.
If $p=2$ andonly state the result.

$\beta\geq 0$$C$, the above proof works as well in this case. We

Proposition 5.3. For $p=2$ ,such that(35).

$\beta\geq 0$, there exists a positive constant$E[\Phi_{2,(\beta+1)/2}(|f|)]\leq CE[\Phi_{2,\beta/2}(|f|)]+CE[\Phi_{p,\beta/2}(|Df|)]$.

Orlicz nom equivalence

317

In Section 3, we showed that the right hand side of (34) is equivalent
to $E[\Phi_{p,\beta/2}(\sqrt{1-L}f)]$ . Therefore we easily get the following theorem.
Theorem 5.4. For $p>1$ ,(36)$\sqrt{1-L}^{-1}$$\beta\geq 0$, the following map is continuous:: $L^{p}\log^{p\beta}L\rightarrow L^{p}\log^{p(\beta+1/2)}L$ .

is $L^{q}\log^{-q\beta}L$ (see ProposiRecall that the dual space oftion 4.4). Hence, when $1<p<2$ , the above equation (36) is shown by
the duality. By iterating the map $\sqrt{1-L}^{-1}$ , we can have the continuityof $(1-L)^{-1}$ fromto $L^{p}\log^{p(\beta+1)}L$ .
$L^{p}\log^{p\beta}$$L^{p}\log^{p\beta}L$

References[1] R. A. Adams, “Sobolev spaces,” Academic press, New York, 1975.
[2] R. A. Adams, General logarithmic Sobolev inequalities and Orlicz imbeddings, J. Fund. Anal, 34 (1979), 292-303.
[3] D. Bakry and P. A. Meyer, Sur les in\’egalit\’es de Sobolev logarithmiques,
I, II, Lecture Notes in Math., 920 (1982), 138-145, 146-150, SpringerVerlag, Berlin.
[4] C. Dellacherie and P.-A. Meyer, “Probabilities and potential. B. Theoryof martingales” North-Holland Publishing Co., Amsterdam, 1982.
[5] G. F. Feissner, Hypercontractive semigroups and Sobolev’s inequality,
Trans. Amer. Math. Soc, 210 (1975), 51-62.
[6] P. A. Meyer, Notes sur les processus d’Ornstein-Uhlenbeck, S\’eminaire de
Prob. XVI, ed. par J. Azema et M. Yor, Lecture Notes in Math., vol.
920 (1982), 95-133, Springer-Verlag, Berlin-Heidelberg-New York.
[7] P. A. Meyer, Quelques results analytiques sur le semigroupe d’OrnsteinUhlenbeck en dimension infinie, Theory and application of random
fields, Proceedings of IFIP-WG 7/1 Working conf. at Bangalore, ed.
by G. Kallianpur, Lecture Notes in Cont. and Inform. Sci., vol. 49(1983), 201-214, Springer-Verlag, Berlin-Heidelberg-New York.
[8] M. M. Rao and Z. D. Ren, “ Theory of Orlicz spaces,” Marcel Dekker,
New York, 1991.
[9] I. Shigekawa, Littlewood-Paley inequality for a diﬀusion satisfying thelogarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary, Osaka J. Math., 39 (2002), 897-930.
[10] I. Shigekawa and N. Yoshida, Littlewood-Paley-Stein inequality for a symmetric diﬀusion, J. Math. Soc. Japan, 44 (1992), 251-280.
Department of Mathematics
Graduate School of Science
Kyoto University
Kyoto, 606-8502
Japan

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 319-331

Some Comments about It\^o’s Construction Procedure
Daniel W. Stroock
For K. It\^o on his 88th birthday
Abstract.
This article reviews It\^o’s procedure for constructing the Markov
process generated by variable coeﬃcient L\’evy-Khinchine operators.
In particular, it examines conditions under which It\^o’s procedure succeeds but more analytic procedures appear to fail.

Introduction\S 0

In his famous memoir [1], It\^o dealt with the construction of Markov
processes corresponding to variable coeﬃcient L\’evy-Khinchine operators. His method rests on the ability to represent of the action of L\’evyKhinchine operator with diﬀusion coeﬃcient $x\sim a(x)$ , driﬀ coeﬃcient$\sim b(x)$ , and L\’evy measure $x\sim M(x, \cdot)$ on ain theform$L$$\varphi\in C_{c}^{2}(\mathbb{R}^{7b};\mathbb{R})$$x$$L\varphi(x)=\frac{1}{2}\sum_{i,,,j=1}^{n}(\sigma(x)\sigma(x)^{T})_{ij}\partial_{i}\partial_{j}\varphi(x)+\sum_{i=1}^{n}c(x)_{i}\partial_{i}\varphi(x)$(0.1)$+\int_{\mathbb{R}^{n}\backslash \{0\}}(\varphi(x+F(x, y))-\varphi(x)$$-1_{[0,1]}(|y|)(F(x, y),$ $grad_{x}\varphi)_{\mathbb{R}^{n}})M(dy)$ ,,,::for appropriate functions
$F$ :
$M$
. In order for his method to, and L\’evy measure
a
of
must
be at least (Borel) meahave chance working, these functions
surable, and, in practice, they must be much better than that. Indeed,apart from refinements (cf. [6]), which are important but of restricted$\sigma$$\mathbb{R}^{n}\rightarrow Hom(\mathbb{R}^{n};\mathbb{R}^{n})$$c$$\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$\mathbb{R}^{7\iota}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$

Received January 7, 2003.
Research contained in this article was partially supported by NSF grant
DMS #6890047.

D. Stroock320

and be uniformly Lipschitz
applicability, what one needs is that$F$
satisfy conditions of the form:
continuous and that$\sigma$$c$$\lim_{r\searrow 0}\sup_{x\in \mathbb{R}^{n}}\frac{1}{1+|x|^{2}}\int_{0<|y|\leq r}|F(x, y)|^{2}\frac{dy}{|y|^{n+1}}=0$(0.2)$\sup_{x\in \mathbb{R}^{n}}\frac{1}{1+|x|^{2}}\int_{0<|y|\leq R}|F(x, y)|^{2}\frac{dy}{|y|^{n+1}}<\infty$$\sup_{x_{2}\neq x_{1}}\frac{1}{|x_{2}-x_{1}|^{2}}\int_{0<|y|\leq R}|F(x_{2}, y)-F(x_{1}, y)|^{2}\frac{dy}{|y|^{n+1}}<\infty$

for each $R\in(0, \infty)$ . Under these conditions it is possible to carry out(cf. \S 3.1 and \S 4.1 in [3]) It\^o’s procedure for constructing the Markov$(a(x), b(x)$ , $M(x. \cdot))$ by transforming theprocess corresponding topaths of the L\’evy process whose continuous part is standard Brownian
motion and whose L\’evy part is the symmetric Cauchy process whose.
L\’evy measure is
Assuming that $x\sim a(x)$ , $x\sim b(x)$ , and $x\sim M(x, \cdot)$ are measurable, it is always possible (cf. 3.2, and especially Theorem 3.2.5, in
and $(x, y)\sim F(x, y)$ so that[3] to make measurable choices of(0.1) holds. In addition, it is well-known (cf. 3.2.1 in [3]) that the nonnegative definite, symmetric square root of $x\sim a(x)$ will be uniformly
Lipschitz if either $x\sim a(x)$ is uniformly Lipschitz and uniformly positive definite or and its second derivatives are uniformly bounded.
On the other hand, it is much less clear what smoothness properties of$x\sim M(x, \cdot)$ will guarantee that $F$ can be chosen so that (0.2) holds.
Because it is the L\’evy term which poses the greatest challenge to traditional analytic techniques, it may be of interest to investigate how
successful It\^o’s theory is with it, and that is what we will be doing here.
$ x\wedge$$M_{0}(dy)=1_{\mathbb{R}^{n}\backslash \{0\}}(y)\frac{dy}{|y|^{n+1}}$$ x\sim\sigma$

$)$

$a$

$a$

Basic Result\S 1

In this section we will show how to construct an $F$ satisfying (0.2)
when $x\sim M(x, \cdot)$ can be expressed in the form

(1.1)

$M(x, \Gamma)=\omega_{n-1}\int_{S^{n-1}}(\int_{(0,\infty)}1_{\Gamma}(r\omega)\beta(x, \omega, r)dr)$$\mu(d\omega)$,, whereis the area of the unit sphere $S^{n-1}$ in$S^{n-1}$$\mu\in M_{1}(S^{n-1})$ (i.e.,), andis a Borel probability measure on:
is a measurable function with thefor$\mathbb{R}^{n}$$\omega_{n-1}$$\Gamma\in B_{\mathbb{R}^{n}\backslash \{0\}}$$\mu$$\beta$$\mathbb{R}^{n}\times S^{n-1}\times(0, \infty)\rightarrow(0, \infty)$

321

Some comments about It\^o ’s construction procedure

properties that
(1.2)$\int_{(0,\infty)}\beta(x, \omega, r)dr=\infty$

for all$(x, \omega)\in \mathbb{R}^{n}\times S^{n-1}$

,

(1.3)

$\inf_{(x,\omega)}\int_{[1,\infty)}\beta(x, \omega, s)ds>0\&\sup_{(x,(v)}\int_{(0,\infty)}\frac{r^{2}\beta(x,\omega,r)}{1+r^{2}}dr<\infty$(1.4)$\lim_{R\searrow 0}\sup_{(x,\omega)}\int_{(0,R]}r^{2}\beta(x, \omega, r)dr=0=\lim_{R\rightarrow\infty}\sup_{(x,\omega)}\int_{[R,\infty)}\beta(x, \omega, s)ds$

and, for each $(\omega, r)\in S^{n-1}\times(0, \infty)$ ,tive which satisfies$\beta(\cdot, \omega, r)$,,

has a continuous deriva-$\sup_{(x,\omega)}\int_{(0,R]}\frac{(\int_{[r,\infty)}|grad_{x}\beta(\cdot,\omega,s)|ds)^{2}}{\beta(x,\omega,r)}dr<\infty$(1.5)

for each $R\in(0, \infty)$ .
The construction of $F$ in this case can be carried out as follows.
First, one determines :so that$\rho$$\mathbb{R}^{n}\times S^{n-1}\times(0, \infty)\rightarrow(0, \infty)$$\int_{[\rho(x,\pm 1,r),\infty)}\beta(x, \pm 1, s)ds=\frac{2\mu(\{\pm 1\})}{r}$

when

$n$

$=1$

when

$n$

$\geq 2$

and$\int_{[\rho(x,\omega,r),\infty)}\beta(x, \omega, s)ds=\frac{1}{r}$.

Second, : $S^{n-1}\rightarrow S^{n-1}$ is chosen so thatwhen $=1$ and,
when $n\geq 2$ , is a measurable map with the property that
, wheredenotes the pushforward under of the standard
surface measure
on $S^{n-1}$ . (The existence of such an is assured by
Theorem 3.2.5 in [3].) Finally, one takes $F(x, r\omega)=\rho(x, f(\omega),$ $r)f(\omega)$ .
To see that this $F$ does the job, begin by observing that, by con$ f(\omega)=\omega$

$f$

$n$

$f_{*}\lambda_{S^{n-1}}=$$f$$\omega_{n-1}\mu$$f_{*}\lambda_{S^{n-1}}$$\lambda_{S^{n-1}}$

$f$

$f$

322

D. Stroock

struction,$\int_{\mathbb{R}^{n}\backslash \{0\}}\varphi\circ F(x, y)\frac{dy}{|y|^{n+1}}=\int_{S^{n-1}}(\int_{(0,\infty)}\varphi\circ F(x, r\omega)\frac{dr}{r^{2}})\lambda_{S^{n-1}}(d\omega)$$=\int_{S^{n-1}}$($\int_{(0,\infty)}\varphi(rf(\omega))\beta(x, f(\omega),$$r)dr$)$\lambda_{S^{n-1}}(d\omega)$$=\int_{\mathbb{R}^{n}\backslash \{0\}}\varphi(y)M(x, dy)$

for any(1.6). That is,$\varphi\in C(\mathbb{R}^{n}\backslash \{0\};[0, \infty))$$\int_{\mathbb{R}^{n}\backslash \{0\}}\varphi\circ F(x, y)M_{0}(dy)=\int_{\mathbb{R}^{n}\backslash \{0\}}\varphi(y)M(x, dy)$

. Thus, if$|\psi(y)-y|\leq C|y|^{2}$ for some $ C<\infty$ and we adopt

for$\varphi\in C(\mathbb{R}^{n}\backslash \{0\};[0, \infty))$$\psi\in C^{\infty}(B_{\mathbb{R}^{n}}(0,1);\mathbb{R}^{n})$satisfies(1.7) $K_{M}\varphi(x)=\int_{\mathbb{R}^{n}\backslash \{0\}}(\varphi(x+y)-\varphi(x)-(\psi(y), grad_{x}\varphi)_{\mathbb{R}^{n}})M(x, dy)$as the operator associated with

$x\leftrightarrow M(x, \cdot)$, then $K_{M}\varphi(x)$ is equal to$\sum_{i=1}^{n}c_{i}(x)\partial_{i}\varphi(x)+\int_{\mathbb{R}^{n}\backslash \{0\}}(\varphi(x+F(x, y))-\varphi(x)$

-1
where
(1.8)

$c:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$[0,1]$$(|y|)(F(x, y),$ $grad_{x}\varphi)_{\mathbb{R}^{n}})M_{0}(dy)$is given by$c(x)=\int_{\mathbb{R}^{n}\backslash \{0\}}(1_{[0,1]}(|y|)F(x, y)-\psi(F(x, y)))M_{0}(dy)$Next, we need to check that $F$ satisfies (0.2) and
Lipschitz. To this end, first observe that, by (1.6),$c$.

is uniformly$\int_{0<|y|\leq r}|F(x, y)|^{2}\frac{dy}{|y|^{n+1}}=\int_{\{y:0<|y|\leq R(r)\}}|y|^{2}M(x, dy)$$\leq\sup_{\omega}\int_{\{0<|y|\leq R(r)\}}r^{2}\beta(x, \omega, r)dr$,

where $R(r)\equiv\sup_{(x,\omega)}\rho(x, \omega, r)$ . Since, by the second part of (1.4),$ R(r)<\infty$ for all $r\in(0, \infty)$ , we know that the second line of (0.2) holds.

Some comments about It\^o ’s construction procedure323

At the same time, ﬀom second part of (1.3), we know that $R(r)\searrow 0$ as, and so the first line of (0.2) also holds.
Turning to the last line of (0.2), observe that$r\backslash 0$$\partial_{x}F(\cdot, r\omega)=\frac{\int_{[\rho(x,f(\omega),r),\infty)}\partial_{x}\beta(\cdot,f(\omega),s)ds}{\beta(x,f(\omega),\rho(x,f(\omega),r)}f(\omega)$,

and therefore, by (1.6), that$\int_{0<|y|\leq R}|F(x_{2}, y)-F(x_{1}, y)|^{2}\frac{dy}{|y|^{n+1}}$$\leq|x_{2}-x_{1}|^{2}\sup_{(x,\omega)}\int_{(0,R(r)]}\frac{(\int_{[r,\infty)}|grad_{x}\beta(\cdot,\omega,s)|ds)^{2}}{\beta(x,\omega,r)}$dr.

Hence the third line of (0.2) follows ﬀom (1.5).
Finally, we must check that the in (1.8) is uniformly Lipschitz
continuous. But, since$c$$c(x)=\int_{0<|y|\leq 1}(\psi(F(x, y))-F(x, y))M_{0}(dy)+\int_{|y|\geq 1}\psi(F(x, y))M_{0}(dy)$ ,we can use the first part of (1.3) and the same line of reasoning as above
to see that there is an $r\in(0, \infty)$ and a $ C<\infty$ for which$|\partial_{x}c|\leq C\sup_{\omega}\int_{(0,r]}\rho(x, \omega, s)|\partial_{x}\rho(\cdot, \omega, s)|\frac{ds}{s^{2}}$$\leq C\sup_{\omega}\int_{(0,R]}s(\int_{[s,\infty)}|\partial_{x}\beta(\cdot, \omega, \sigma)|d\sigma)ds$.

$\leq C\sup_{\omega}\sqrt{\int_{(0,R]}s^{2}\beta(x,\omega,s)ds}$

Hence, by the second part of (1.3) and (1.5), it is clear that is uniformly
Lipschitz.
By the results in \S 3.1 of [3], we can now say that when $M$ is givenby (1.1) with a satisfying (1.2)-(1.5), then It\^o’s construction leads toa Markov process which corresponds to the operator $K_{M}$ in (1.7) in the, the process solves the martingale
sense that, starting at each
$K_{M}$.
problem (cf. \S 3 below) foron$c$$\beta$$x\in \mathbb{R}^{n}$$C_{c}^{2}(\mathbb{R}^{n}; \mathbb{R})$D. Stroock

324

An Example: In order to demonstrate that It\^o’s theory can handle
more analytic methodology, consider the case whensituations which$de5^{r}$$\beta(x, \omega, r)=\alpha(x, \omega)r^{-1-\lambda(x,\omega)}$

,

:
:
and
are
where
measurable functions, $ 0<\alpha_{1}\leq\alpha_{2}<\infty$ , and 0 $<\lambda_{1}\leq\lambda_{2}<2$ .
Assuming thatandare continuously diﬀerentiable foreach and thatandarebounded, one can easily verify that ’s of this sort $satis\mathfrak{h}^{r}(1.2)-(1.5)$ .
The reason why traditional analytic approaches would have diﬃculties
with an operator $K_{M}$ of the form in (1.7) when $M$ is given by (1.1)
with these ’s is that, unless is independent of , $K_{M}$ will have no
principal part. For this reason, perturbative techniques, like those on
which standard pseudodiﬀerential arguments (cf. [2]) depend, do not
apply.
$\alpha$$\mathbb{R}^{n}\times S^{n-1}\rightarrow[\alpha_{1}, \alpha_{2}]$$\alpha(\cdot, \omega)$$\mathbb{R}^{n}\times S^{n-1}\rightarrow[\lambda_{1}, \lambda_{2}]$$\lambda$$\lambda(\cdot, \omega)$$(x, \omega)\sim grad_{x}\alpha(\cdot, \omega)$$\omega$$(x, \omega)\sim grad_{x}\lambda(\cdot, \omega)$$\beta$$\lambda$$\beta$$x$

Remark: It is reasonable to ask whether there is any advantage to begained by considering reference L\’evy measures other than $M_{0}(dy)=$. However, at least so far as the considerations in this and$1_{\mathbb{R}^{n}\backslash \{0\}}\frac{dy}{|y|^{n+1}}$

the next $sections^{1)}$ are concerned, the answer seems to be no. Indeed,without any change in the proof, one can show that It\^o’s procedure works
whenin (0.2) is replaced by any L\’evy measure $M$ and the conditions
there are replaced by$M_{0}$$\lim_{N\rightarrow\infty}\sup_{x\in \mathbb{R}^{n}}\frac{1}{1+|x|^{2}}\int_{\triangle_{N}}|F(x, y)|^{2}M(dy)=0$(1.9)$\sup_{x\in \mathbb{R}^{n}}\frac{1}{1+|x|^{2}}\int_{\Gamma_{\epsilon}}|F(x, y)|^{2}M(dy)<\infty$$\sup_{x_{2}\neq x_{1}}\frac{1}{|x_{2}-x_{1}|^{2}}\int_{\Gamma_{\epsilon}}|F(x_{2}, y)-F(x_{1}, y)|^{2}M(dy)<\infty$,

where, for each $N\geq 1,0\in\triangle_{N}\in B_{\mathbb{R}^{n}}$ satisfies,
and, for each,
. However,satisfies
because one can always find a measurable :such that$M$ $=f_{*}M_{0}$ , one can easily check that if $M$ and $F$ satisfies $F(x, \cdot)_{*}M=$$M(x, \cdot)$ and the conditions in (1.9), then $(x, y)\sim\tilde{F}(x, y)\equiv F(x, f(y))$
will satisfy $\tilde{F}(x, \cdot)_{*}M_{0}=M(x, \cdot)$ and (1.9) withreplacing
and$F$ and $M$ andreplacingandand .
$ M(\mathbb{R}^{n}\backslash \triangle_{N})<\infty$$\epsilon>0$$O\in\Gamma_{\epsilon}\in B_{\mathbb{R}^{n}}$$ M(\mathbb{R}^{n}\backslash \Gamma_{\epsilon})<\epsilon$$f$$\mathbb{R}^{n}\backslash \{0\}\rightarrow \mathbb{R}^{n}$$\overline{F}$$M_{0}$$f^{-1}(\Delta_{N})$1)$f^{-1}(\Gamma_{\epsilon})$$\triangle_{N}$$\Gamma_{\epsilon}$

See the concluding Remark in \S 3 for a consideration in which there is anadvantage to allowing more general reference L\’evy measures.

325

Some comments about It\^o ’s construction procedure\S 2

Some Extensions

It is important to note that there are situations in which it is impossible to construct an $F$ which satisfies (1.9) for any choice of L\’evymeasure $M$ , even though $ x\sim$ $M(x, \cdot)$ is as smooth as one could
hope. For example, consider the seemingly trivial case in which $=1$:
is smooth andand $M(x, dy)=\alpha(x)\delta_{1}(dy)$ , where$M$
is the unit point mass at 1. Clearly, if
is a L\’evy measure and$F$ :satisfies $F(x, \cdot)_{*}M=M(x, \cdot)$ , then, for each$F(x,\cdot)\in\{0,1\}$$M$,-almost everywhere and $M(\Gamma(x))=\alpha(x)$ when$\Gamma(x)\equiv\{y:F(x, y)=1\}$ . Thus, for each,$n$$\alpha$$\mathbb{R}\rightarrow[1, 2]$$\delta_{1}$$\mathbb{R}\times(\mathbb{R}\backslash \{0\})\rightarrow \mathbb{R}$$x$$\epsilon>0$$\int_{\Gamma_{\epsilon}}(F(x_{1}, y)-F(x_{0}, y))^{2}M(dy)\geq\alpha_{\epsilon}(x_{1})+\alpha_{\epsilon}(x_{0})-2\alpha_{\epsilon}(x_{1})\wedge\alpha_{\epsilon}(x_{0})$$=|\alpha_{\epsilon}(x_{1})-\alpha_{\epsilon}(x_{0})|$,. In particular,
uniformly as
where
be constantthe only way that the third line of (1.9) could hold is thatfor each, which means that itself would have to be constant. Of
course, one can object that this example is a little ridiculous since it iseasy to carry out It\^o’s construction whenever $x\sim M(x, \mathbb{R}^{n})$ is bounded,even if the third line of (1.9) fails. On the other hand, one can overwherecome this objection by considering. Proceeding aseachsatisfies
before, we know that $M(\Gamma_{m}(x))=\alpha_{m}(x)$ and, $M$ -almost everywhere,$F(x, \cdot)=\sum_{m=0}^{\infty}3^{-m}1_{\Gamma_{m}(x)}$, where $\Gamma_{m}(x)\equiv\{y : ^{F}(x, y)=3^{-m}\}$ .
Hence, if, then$\alpha_{\epsilon}(x)\equiv M(\Gamma(x)\cap\Gamma_{\epsilon})\nearrow\alpha(x)$$\epsilon\searrow 0$$\alpha_{\epsilon}$$\epsilon>0$$\alpha$$M(x, \cdot)=\sum_{m=0}^{\infty}\alpha_{m}(x)\delta_{3^{-m}}$$\alpha_{m}\in C_{b}^{\infty}(\mathbb{R};(0, \infty))$$||\alpha_{m}||_{C_{b}^{1}(\mathbb{R};\mathbb{R})}\leq C8^{m}$$M_{\epsilon}(dy)=1_{\Gamma_{\epsilon}}(y)M(dy)$$\int_{\Gamma_{\epsilon}}|F(x_{1}, y)-F(x_{0}, y)|^{2}M(dy)$$=\sum 9^{-m}\infty(M_{\epsilon}(\Gamma_{m}(x_{1}))+M_{\epsilon}(\Gamma_{m}(x_{1}))-2M_{\epsilon}(\Gamma_{m}(x_{0})\cap\Gamma_{m}(x_{1})))$

$m=0$

-2

$\sum 3^{-m}\sum\infty 3^{-n}(M_{\epsilon}(\Gamma_{m}(x_{0})\cap\Gamma_{n}(x_{1}))+M_{\epsilon}(\Gamma_{n}(x_{0})\cap\Gamma_{m}(x_{1})))$

$m=0$

$n>m$

$\geq\sum_{m=0}^{\infty}9^{-m}(M_{\epsilon}(\Gamma_{m}(x_{1}))+M_{\epsilon}(\Gamma_{m}(x_{1}))-2M_{\epsilon}(\Gamma_{m}(x_{0})\cap\Gamma_{m}(x_{1})))$$-\frac{2}{3}\sum_{m=0}^{\infty}9^{-m}(M_{\epsilon}(\Gamma_{m}(x_{0})\cap\Gamma_{m}(x_{1})G)+M_{\epsilon}(\Gamma_{m}(x_{1})\cap\Gamma_{m}(x_{0})G))$326

D. Stroock$=\frac{1}{3}\sum_{m=0}^{\infty}9^{-m}(M_{\epsilon}(\Gamma_{m}(x_{0})\cup\Gamma_{m}(x_{1}))-M_{\epsilon}(\Gamma_{m}(x_{0})\cap\Gamma_{m}(x_{1})))$$\geq\frac{1}{3}\sum_{m=0}^{\infty}9^{-m}|M_{\epsilon}(\Gamma_{m}(x_{1}))-M_{\epsilon}(\Gamma_{m}(x_{1}))|$.

Hence, by the same argument as the one just used, the third line of (1.9)’s is constant.
can hold only if each of theIn view of the preceding example, it is interesting to note that the
problems encountered there disappear if the measure has a suﬃciently:
strong absolutely continuous part. To be more precise, let$S^{n-1}\rightarrow(0, \infty)$ be a function which satisfies the conditions in (1.2)(1.4), and suppose that $(x, \omega)\sim\mu(x, \omega, \cdot)$ is a measurable map ﬀom
into measures onsuch that$\alpha_{m}$$\mathbb{R}^{n}\times$$\beta$$\mathbb{R}^{n}\times S^{n-1}$$(0, \infty)$$\mu(x, \omega, dr)=\beta(x, \omega, r)dr+\iota/(x, \omega, dr)$

where$\sup_{(x,\omega)}\int_{(0,\infty)}\frac{r^{2}}{1+r^{2}}\nu(x, \omega, dr)<\infty$(2.1)$\lim_{R\searrow 0}\sup_{(x,\omega)}\int_{(0,R]}r^{2}\iota/(x, \omega, dr)=0=\lim_{R\rightarrow\infty}\sup_{(x,\omega)}\iota/(x, \omega, [R, \infty))$.

Further, assume that $x\sim\beta(x, \omega, r)$ and $x\sim\iota/(x, \omega, [r, \infty))$ are continuously diﬀerentiate for each $(x, r)\in \mathbb{R}^{n}\times(0, \infty)$ . Finally, choosewith total integral 1, set$\eta\in C_{c}^{\infty}((0, \infty);[0, \infty))$$\beta_{\epsilon}(x, \omega, r)=\beta(x, \omega, r)+\int_{(0,\infty)}\eta_{\epsilon}(s-r)\nu(x, \omega, ds)$

where

(2.2)

$\eta_{\epsilon}(s)\equiv\epsilon^{-1}\eta(\frac{s}{\epsilon})$for$\epsilon\in(0,1]$,, and assume that, for each $R\in(0, \infty)$ ,$\sup_{(x,\omega)}\int_{(0,R]}\frac{(\int_{[r,\infty)}|grad_{x}\beta_{\epsilon}(\cdot,\omega,s)|ds)^{2}}{\beta(x,\omega,r)}dr<\infty$.

$\epsilon\in(0,1]$

Next, given a probability measure on $S^{n-1}$ , define::andrelative to by the prescription used in\S 1 (cf. the discussion preceding (1.6)). By the arguments used in \S 1, weknow that, when $F$ and are replaced throughout by
and , theneverything in (0.2) as well as the Lipschitz continuity of the associatedin (1.8) can be controlled in terms of quantity in (1.5). But clearly the$\mathbb{R}^{n}\times S^{n-1}\rightarrow$$\mu$$(0, \infty)$$F_{\epsilon}$$\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$\beta$$c_{\epsilon}$$\rho_{\epsilon}$$\beta_{\epsilon}$$F_{\epsilon}$$\beta_{\epsilon}$

327

Some comments about It\^o ’s construction procedure

quantity in (1.5) is dominated uniformly for(2.2). At the same time, if(2.3)$\epsilon\in(0,1]$

by the quantity in$M(x, \Gamma)=\int_{S^{n-1}}(\int_{(0,\infty)}1_{\Gamma}(r\omega)\mu(x, \omega, dr))\mu(d\omega)$,thenwhere $F(x, \cdot)_{*}M=M(x, \cdot)$ . Hence, when $x\vee M(x, \cdot)$
is given by (2.3) for any $\mu\in M_{1}(S^{n-1})$ and a$\mu(x, \omega, \cdot)\in M_{1}((0, \infty))$ which satisfies (2.1) and (2.2), then a choice of$F$ satisfying (0.2) is available.
$F_{\epsilon}\rightarrow F$$(x, \omega)\in \mathbb{R}^{n}\times S^{n-1}\mapsto$

Uniqueness for the Martingale Problem\S 3

Suppose that (cf. (1.7))(3.1)$L\varphi(x)=\frac{1}{2}\sum_{i,,,j=1}^{n}a(x)_{ij}\partial_{i}\partial_{j}\varphi(x)+\sum_{i=1}^{n}b(x)_{i}+K_{M}\varphi(x)$for, where $x\vee*a(x)$ and $x\sim b(x)$ are continuous map-matrices
pings into, respectively, non-negative definite, symmetricandand $x\sim M(x, \cdot)$ takes its values in L\’evy measures and satisfies$\varphi\in C_{c}^{2}(\mathbb{R}^{n};\mathbb{R})$$n\times n$$\mathbb{R}^{n}$$\sup_{|x|\leq R}\int_{\mathbb{R}^{n}\backslash \{0\}}\frac{|y|^{2}}{1+|y|^{2}}M(x, dy)<\infty$

for all $R\in(0, \infty)$ .

be the space of right continuous paths :which possess a left limit $p(t-)$ at each $t\in(0, \infty)$ , and use
to denote the -algebra overgenerated by $p\sim p(\tau)$ for$\tau\in[0, t]$ . We will say thatsolves the martingale
problem for ifLet$D([0, \infty);\mathbb{R}^{n})$$p$$[0, \infty)\rightarrow$$\mathbb{R}^{n}$$B_{t}$$D([0, \infty);\mathbb{R}^{n})$$\sigma$$\mathbb{P}\in M_{1}(D([0, \infty);\mathbb{R}^{n}))$$L$($\varphi(p(t))-\int_{0}^{t}L\varphi(p(\tau))d\tau$,$B_{t}$,$\mathbb{P}$) is a martingale. If, in addition, $\mathbb{P}(p(0)=x)=1$ , then we will say
for allthat solves the martingale problem for starting from .
In 3.1.5 of [3], it is shown that when admits a representation of
the form in (0.1) with uniformly Lipschitz continuous and and an$x\sim F(x, \cdot)$ satisfying (0.2), It\^o’s construction leads to a solution to themartingale problem for starting from . On the other hand, there arelots of other ways in which one might go about constructing solutions
to this martingale problem. (In fact, even if one restricts ones attention$\varphi\in C_{c}^{2}(\mathbb{R}^{n};\mathbb{R})$$L$$\mathbb{P}$

$x$

$L$

$\sigma$

$L$

$x$

$b$

D. Stroock328

to It\^o’s method, there is are lots of choices of and $F$ , and each one
gives rise to a diﬀerent construction.) Thus, it is of some importance todetermine conditions which guarantee that there is only one solution to
the martingale problem for a given starting $bom$ a given .
Under the condition that $M$ $=0$ , the problem of determining when
uniqueness holds for the martingale problem was studied systematicallyin Chapter 6 of [4]. The methods used there are of two types. Methodsof the first type work by duality and yield (cf. Theorem 6.3.2 in $opcit$)
uniqueness for solutions to the martingale problem as a consequence ofexistence of solutions to the evolution equation$\sigma$

$L$

(3.2)

$\partial_{t}u=Lu$$x$

with $ u(0, \cdot)=\varphi$

for suﬃciently many ’s. This duality method is quite powerful andleads to the most refined results obtained in [4]. For example, when
$M=0$ and and have two bounded, continuous derivatives, it is shownin \S 3.2 of [4] that (3.2) admits classical solutions for,enough
is
more
than
to
and thischeck uniqueness for the associatedmartingale problem. In \S 4.2 of [3], this sort of reasoning is extended tosituations with $M\neq 0$ , when the quantities and entering (0.1) have
two bounded, continuous derivatives and $x\sim F(x, \cdot)$ has continuous
second derivatives which satisfies appropriate (cf. (H2)2 in $opcit$) meansquare bounds. For example, these conditions are often met by $F$ ’s of
the sort constructed in \S 1. Unfortunately, they seem unlikely to hold forsituations requiring the extension introduced in \S 2.
The second method introduced in [4] is more directly dependent on
It\^o’s theory. Namely, when $M$ $=0$ , it is shown there (cf. Theorem 5.3.2
in $opcit$ ) that any solution to the martingale problem can be realized as
the solution of an It\^o stochastic integral equation. Thus, when $M$ $=0$
and
and are Lipschitz continuous, uniqueness for the martingale
problem comes quite easily as a consequence of It\^o’s theory. (This is
the result which was refined in [6].) In this concluding section, we will
examine possible extensions of this line of reasoning to the case when
$M$.
Suppose that solves the martingale problem for starting from .
Using the techniques developed in \S 1 of [5], one can make an It\^o decompostion of the paths into their “continuous” and “discontinuous parts”
parts. More precisely, given $p\in D([0, \infty);\mathbb{R}^{n})$ , awith, define
to be the number of $\tau\in(0, \infty)$
such that $(\tau,p(\tau)-p(\tau-))\in\Gamma$ . One can then show that there exists
a measurable map $p\in D([0, \infty);\mathbb{R}^{n})\mapsto p_{1}\in D([0, \infty);\mathbb{R}^{n})$ such that$\varphi$

$a$

$b$

$\varphi\in C_{c}^{2}(\mathbb{R}^{n}; \mathbb{R})$$\sigma$$\sigma$$b$$b$$\neq 0$$L$$\mathbb{P}$$p$$\Gamma\in B_{[0,\infty)}\times B_{\mathbb{R}^{n}}$$([0, \infty)\times\{0\})\cap\overline{\Gamma}=\emptyset$$\nu(\Gamma;p)$

$x$

329

Some comments about It\^o ’s construction procedure(cf. (1.7))(3.4)$p_{1}(t)=\lim_{r\searrow 0}$$\int\int$$(y\nu(d\tau\times dy;p)-\psi(y)d\tau\times M(p(\tau), dy))$,$[0,t]\times B_{R^{n}}(0,r)G$

uniformly for ’s in compacts, in -probability. Moreover, if $p_{0}=p-p_{1}$ ,then(a)-almost surely,(b) for each,$t$$\mathbb{P}$$p_{0}\in C([0, \infty);\mathbb{R}^{n})\mathbb{P}$$\varphi\in C_{c}^{2}(\mathbb{R}^{n}; \mathbb{R})$$(\varphi(p_{0}(t))-\int_{0}^{t}L_{0}\varphi(p(\tau))d\tau$,$B_{t}$,$\mathbb{P})$is a martingale, where$L_{0}\varphi(x)=\frac{1}{2}\sum_{i,,,j=1}^{n}a(x)_{ij}\partial_{i}\partial_{j}\varphi(x)+\sum_{i=1}^{n}b(x)_{i}\partial_{i}\varphi(x)$(c)

for each$\varphi\in C_{c}^{2}(\mathbb{R}^{n}; \mathbb{R})$,,$(\varphi(p_{1}(t))-\int_{0}^{t}K_{M}\varphi(p(\tau))d\tau$,$B_{t}$,$\mathbb{P})$is a martingale.

In spite of the obvious ambiguity in this decomposition, we will callthe continuous part and the discontinuous part of .
andGiven a measurable $x\vee\sigma(x)$ satisfying $a(x)=\sigma(x)\sigma(x)^{T}$ , one canstart from (b) above and, by mimicking the procedure in Theorem 5.3.2
of [4], produce a Brownian motion such that
$p_{0}$$p$$p_{1}$$\beta$(3.5)$ p0(t)=x+\int_{0}^{t}\sigma(p(\tau))d\beta(\tau)+\int_{0}^{t}b(p(\tau))d\tau$,$t\in[0, \infty)$.

There are technical diﬃculties which arise when becomes degenerate,
and these necessitate the introduction of a larger probability space, one
which is big enough to support a full blown Brownian motion. However,as is explained in the theorem just cited, the resolution of such diﬃculties
is well understood. On the other hand, it is not so clear how to treat. Specifically,
the analogous diﬃculties for the discontinuous part$measure^{2)}M$
and a L\’evy
forgiven a measurable $F$ :$\sigma$$p_{1}$$\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$2)

For reasons which will be explained below in the concluding Remark, it isbest to allow general reference L\’evy measures here rather than always taking$M_{0}$.

330

D. Stroock

which $F(x, \cdot)_{*}M=M(x, \cdot)$ , it is not clear in general how one canuse $F$ to produce a L\’evy process from which
can be -constructed$F$
via It\^o’s procedure. Nonetheless, when is non-degenerate in the sense
that, for each
, $F(x, \cdot)$ is one-to-one
onto itself,$F^{-1}(x,\cdot)$ to
then one can construct such a L\’evy process. Namely, takebe the $inverse^{3)}$ of $F(x, \cdot)$ , and set$re$$p_{1}$$x\in \mathbb{R}^{n}$$q_{r}(t,p)=$$\int\int$$bom\mathbb{R}^{n}\backslash \{0\}$$\Phi^{-1}(\tau, y;p)(\nu(d\tau\times dy;p)-d\tau\times M(p(\tau), dy))$,$[0,t]\times B_{R^{n}}(0,r)G$

where $\Phi^{-1}(\tau, y;p)\equiv F^{-1}(p(\tau-), y)$ . Then one can show that there
exists a $\{B_{t} : t\geq 0\}$ -progressively measurable$q(\cdot,p)\in D([0, \infty);\mathbb{R}^{n})$ such that, as, $q_{r}(\cdot,p)\rightarrow q(\cdot,p)$ uniformly on compacts in -probability. Moreover, the -distribution of$p\sim q(\cdot,p)$ is that of the L\’evy process corresponding to $M$ in the sense,
that, for each$ p\in D([0, \infty);\mathbb{R}^{n})\rightarrow$$r\searrow 0$$\mathbb{P}$$\mathbb{P}$$\xi\in \mathbb{R}^{n}$$E\mathbb{P}[e^{\sqrt{-1}(\xi,q(1,p))_{R^{n}}}]$.

$=\exp[\int_{\mathbb{R}^{n}\backslash \{0\}}(e^{\sqrt{-1}(\xi,y)_{\mathbb{R}^{n}}}-1-1_{[0,1]}(|y|)(\xi, y)_{\mathbb{R}^{n}})M(dy)]$

In addition, it should be clear that, -almost surely, $(\cdot ; q(\cdot,p))=$. In particular, if $\Phi(\tau, y;q)\equiv F(q(\tau-), y)$ , then$\mathbb{P}$$\nu$$\Phi^{-1}$$(\cdot ; p)_{*}\nu(.

; p)$$\nu(\cdot ; p)=\Phi(.

and so (cf. (3.4))$p_{1}(t)$; q(\cdot,p))_{*}\nu(.

; q(p))$-almost surely,$\mathbb{P}$is equal to$r\searrow 0\lim_{[0,t]\times}\int\int_{(B_{\mathbb{R}^{n}}0,r)}(F(p(\tau-), y)\nu(d\tau\times dy;q(\cdot ^{;} p))$$-\psi(y)(F(p(\tau), y))_{\mathbb{R}^{n}}d\tau\times M(dy))$

Hence, after putting this together with (3.5), the path
via It\^o’s procedure ﬀom a L\’evy process for which$p$.

can be recovered$\xi\sim\exp[-\frac{|\xi|^{2}}{2}+\int_{\mathbb{R}^{n}\backslash \{0\}}(e^{\sqrt{-1}(\xi,y)_{\mathbb{R}^{n}}}-1-1_{[0,1]}(|y|)(\xi, y)_{\mathbb{R}^{n}})M(dy)]$3)

By a famous theorem due to C. Kuratowski, this inverse will be Borel
measurable with respect to $(x, y)$ .

331

Some comments about It\^o ’s construction procedureis the characteristic function of the distribution at time 1.
These considerations yield the following uniqueness theorem.
are
and :
Theorem. Suppose that :uniformly Lipschitz continuous functions, $M$ is a L\’evy measure, and the$F$ :is a measurable function which satis,
fies the conditions in (1.9). Further, assume that, for each$F(x, \cdot)$ is one-to-one
onto itself Then, for each startingfrompoint, It\^o ’s construction yields the one and only solution to the martingale problem for operator
described in (3.1) when $a(x)=\sigma(x)\sigma(x)^{T}$.
and $M(x, \cdot)=F(x, \cdot)_{*}M$ for all$\sigma$$\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$b$$\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$\mathbb{R}^{n}\times \mathbb{R}^{n}\backslash \{0\}\rightarrow \mathbb{R}^{n}\backslash \{0\}$$x\in \mathbb{R}^{n}$$\mathbb{R}^{n}\backslash \{0\}$$L$$x\in \mathbb{R}^{n}$Remark: As distinguished $bom$ our earlier results, there is an advanwhen applying
tage to allowing reference L\’evy measures other than$(x,\omega,r)\sim\mu(x, \omega, r)$
the preceding theorem. For instance, suppose thatsatisfies the conditions in (2.1) and (2.2), and let $x\vee’ M(x, \cdot)$ be givenby (2.3) for some $\mu\in M_{1}(S^{n-1})$ . Then the function $F$ which was constructed so that $M(x, \cdot)=F(x, \cdot)_{*}M_{0}$ need not be one-to-one and onto
because the map : $S^{n-1}\rightarrow S^{n-1}$ may fail to be. On the other hand,
if we take $M(dy)=\frac{1}{|y|^{n+1}}\mu(dy)$ , then the construction given in \S 1 does
not require the use of and leads to an $F$ which is one-to-one and ontoand satisfies $M(x, \cdot)=F(x, \cdot)_{*}M$ as well as the conditions in (1.9).
$M_{0}$$f$$f$

References[1]

It\^o, K., “On stochastic diﬀerential equations”, Mem. Amer. Math. Soc.
#4, A.M.S., Providence, RI, 1951.

[2]

Jacob, N. &Schilling, R., L\’evy-Type processes and pseudodiﬀerential
operators, in “L\’evy Processes, Theory and Applications” , (Barndorﬀ-Nielsen, O., Mikosch, T., and Resnik, S., eds.), Birkh\"auser, 2001.

[3]

Stroock, D., “Markov Procsses from It\^o’s Perspective”, Annals of Math.
Studies #155, Princeton Univ. Press, 2003.

[4]

Stroock, D. &Varadhan, S.R.S., “Multidimensional Diﬀusion Processes”,
Grundlehren #233, Springer-Verlag, Heidelberg&NYC, 1979.

[5]

Stroock, D., Diﬀusion processes associated with L\’evy processes, Z. Wahr.
verw. Geb. 32 (1975), 209-244.

[6]

Watanabe, S. &Yamada, T., On the uniqueness of solutions of stochastic
diﬀerential equations, J. Math. Kyoto Univ. #11 (1971), 155-167.

M. I. T., 2-272
Cambridge, MA, 02139-4307 USA
dws@math.mit.edu

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 333-350

Criticality of Generalized Schr\"odinger Operatorsand Diﬀerentiability of Spectral Functions
Masayoshi Takeda and Kaneharu Tsuchida
Abstract.
Let be a positive Radon measure in the Kato class. We considerof a generalized
the spectral boundSchr\"odinger operator, and show that theonspectral bound is diﬀerentiable if $d\leq 4$ and is Green-tight.
$\mu$$C(\lambda)=-\inf\sigma(H^{\lambda\mu})(\lambda\in \mathbb{R}^{1})$$\mathbb{R}^{d}$$ H^{\lambda\mu}=-\frac{1}{2}\triangle-\lambda\mu$$\mu$Introduction

\S 1.

Let
be the classical Dirichlet integral and a positive
Radon measure in the Kato class. For a Schr\"odinger operatorby,, define the spectral function$(D, H^{1}(\mathbb{R}^{d}))$$\mu$$H^{\lambda\mu}=$$-\frac{1}{2}\Delta-\lambda\mu$$C(\lambda)$$\lambda\in \mathbb{R}^{1}$$C(\lambda)$$=$$-\inf\{\theta :\theta\in\sigma(tt^{\lambda\mu})\}$$=$$-\inf\{\frac{1}{2}D(u, u)-\lambda\int_{\mathbb{R}^{d}}\tilde{u}^{2}d\mu$:$u$$\in H^{1}(\mathbb{R}^{d})$,$\int_{\mathbb{R}^{d}}u^{2}dx=1\}$

andis a quasiwhereis the set of the spectrum ofcontinuous version of . In this paper, we study the diﬀerentiability.
of the function
When the potential is a function in a certain Kato class, Arendt
and Batty [3] proved that the spectral function is diﬀerentiable atand its derivative equals to zero ([3, Corollary 2.10]). Using a large deviation principle for additive functionals of the Brownian motion, Wu[27] obtained a necessary and suﬃcient condition for the spectral function being diﬀerentiable at 0. In [24] one of the authors extended Wu’s
result to measures which may be singular with respect to the Lebesgue
measure. Furthermore, one of the authors showed that if $d\leq 2$ and the), the spectral function ismeasure is Green-tight (in notation,. Here the class
was introduced in Zhao [29](seediﬀerentiable on$H^{\lambda\mu}$$\sigma(H^{\lambda\mu})$$\tilde{u}$$u$$C(\lambda)$$\mu$$\lambda=0$$\mu\in \mathcal{K}_{d}^{\infty}$$\mu$$\mathbb{R}^{1}$$\mathcal{K}_{d}^{\infty}$

Received February 12, 2003.

,

334

M. Takeda and K. Tsuchida

Definition 2.1 (II) below). A main objective of this paper is to extend
the results in [24] as follows:

Theorem 1.1. If $d\leq 4$ andis diﬀerentia for all.

$\mu\in \mathcal{K}_{d\prime}^{\infty}$

then the spectral function$\lambda\in \mathbb{R}^{1}$$te$$C(\lambda)$

Define $\lambda^{+}=\inf\{\lambda>0 : C(\lambda)>0\}$ . We then see that $\lambda^{+}=0$
for $d\leq 2$ and $\lambda^{+}>0$ for $d\geq 3$ and the proof of Theorem 1.1 isat. In
reduced to the proof of the diﬀerentiability of[24], the diﬀerentiablity at
is derived from the fact that for $d\leq 2$
the Brownian motion is a Harris recurrent process with infinite invariant
measure, the Lebesgue measure. We will extend this method for $d=3,4$
by applying the criticality theory of Schr\"odinger operators.
We first extend the criticality theory to the generalized Schr\"odinger
operator; we show in Corollary 3.5 below that if $d\geq 3$ , then the operator
does not admit the minimal positiveis critical, that is,Green function but admits a positive continuous-harmonic function. This harmonic function is called a ground state, which is uniquelydetermined up to constant multiplication. Moreover, if $d=3,4$ ,. In
is null critical, that is, the ground state does not belong tofact, denoting by the ground state, we prove in section 5 that $h(x)$ is
equivalent to the Green function $G(0, x)$ of the Brownian motion on aneighbourhood of the infinity; there exist positive constants , $C$ suchthat
$C(\lambda)$$\lambda=\lambda^{+}$$\lambda=0$$\mathcal{H}^{\mu}$$H^{\lambda^{+}\mu}$$H^{\lambda^{+}\mu}$$H^{\lambda^{+}\mu}$$H^{\lambda^{+}\mu}$$L^{2}$

$h$

$c$

(1)

$\frac{c}{|x|^{d-2}}\leq h(x)\leq\frac{C}{|x|^{d-2}}$,$|x|>1$ .

The criticality and the null criticality are regarded as extended notionsof recurrence and null recurrence respectively. Using these facts, we seethat if $d=3,4$ , the -transformed process generated by the Markov
semigroup$h$$P_{t}^{\lambda^{+}\mu,h}f(x)=\frac{1}{h(x)}$$\exp(-tH^{\lambda^{+}\mu})(hf)(x)$becomes a Harris recurrent Markov process with infinite invariant measure $h^{2}dx$ . Furthermore, through the -transformation a functional inequality for the critical Schr\"odinger form is derived (Theorem 4.4) ; theinequality is an extenstion of Oshima’s inequality ([11]) which holds forthe Dirichlet forms generated by symmetric Harris recurrent Markov
processes. We now obtain Theorem 1.1 by applying the argument in[24] to the transformed process. This is a key idea of the proof of Theobecomes positive
rem 1.1. The equation (1) tells us that if $d\geq 5$ ,. Thus we can not use
critical, that is, the ground state belongs to$h$$H^{\lambda^{+}\mu}$$L^{2}$

Diﬀerentiability of spectral functions

335

our method and have not known yet whether
is difFerentable ornot.
The criticality of Schr\"odinger operators is studied by many people(M. Murata, Y. Pinchover, R. Pinsky,...). In particular, the equation(1) was shown by Murata [10] for classical Schr\"odinger operators on
and extended by Pinchover [12] to second order elliptic operators in a.
domain of
Our motivation lies in the proof of the large deviation principle forin the Revuz correspondence with
continuous additive functional. The function
is regarded as a logarithmic moment generating(see [21]), and the diﬀerentiability
function of the additive functionalof logarithmic moment generating functions play a crucial role in the
G\"artner-Ellis Theorem (see [7]). In fact, using Theorem 1.1, we canassociatedshow the large deviation principle for additive functional.
with$C(\lambda)$$\mathbb{R}^{d}$$\mathbb{R}^{d}$$A_{t}^{\mu}$$C(\lambda)$$\mu$$A^{\mu}$$A_{t}^{\mu}$$\mu\in \mathcal{K}_{d}^{\infty}$\S 2.

Preliminaries. Let $p(t, x, y)$
Let $W=(P_{x}, B_{t})$ be a Brownian motion on
be the transition density function of $W$ and $G(x, y)$ its Green function,$G(x, y)=C(d)|x-y|^{2-d}$ , where $C(d)=(2\pi)^{-1}\Gamma(\frac{d}{2}-1)$ . For a measure, the 0-potential of is defined by $G\mu(x)=\int_{\mathbb{R}^{d}}G(x, y)\mu(dy)$ . Let
be the semigroup of $W$ , $P_{t}f(x)=\int_{\mathbb{R}^{d}}p(t, x, y)f(y)dy=E_{x}[f(B_{t})]$ . The
Dirichlet form of $W$ is given by $(1/2D, H^{1}(\mathbb{R}^{d}))$ where $D$ denotes theis the Sobolev space of order 1classical Dirichlet integral and([8, Example 4.4.1]). Let
denote the extended Dirichlet$(1/2D,H^{1}(\mathbb{R}^{d}))$is a Hilbert space( , p.36]). Note thatform of
$D$
$W$
([8,
with inner productbecause
Theorem 1.5.3]). Let
is transient
be the -resolvent kernel of W.
Throughout this paper, the Lebesgue measure is denoted by $m$ and$m(dx)$ is abbriviated to $dx$ . For $r>0$ , we denote by $B(r)$ an open
ball with radius $R$ centered at the origin. We use , $C$ , , etc as positive
constants which may be diﬀerent at diﬀerent occurrences. We now define
classes of measures which play an important role in this paper.
$\mathbb{R}^{d}(d\geq 3)$$\mu$$P_{t}$$\mu$$H^{1}(\mathbb{R}^{d})$$(1/2D, H_{e}^{1}(\mathbb{R}^{d}))$$H_{e}^{1}(\mathbb{R}^{d})$$[8$$G_{\alpha}(x, y)$$\alpha$$c$Definition 2.1. (I) A positive Radon measure
be in the Kato class (in notation), if$\mu\in \mathcal{K}_{d} $(2)$\ldots$

$\mu$

$ $

$\lim_{a\rightarrow 0}\sup_{x\in \mathbb{R}^{d}}\int_{|x-y|\leq a}G(x, y)\mu(dy)=0$.

on$\mathbb{R}^{d}$is said to336

M. Takeda and K. Tsuchida(II) A measure

$\mu$

(3)

is in

$\mathcal{K}_{d}^{\infty}$

if

is in

$\mu$

$\mathcal{K}_{d}$and satisfies$\lim_{R\rightarrow\infty}\sup_{x\in \mathbb{R}^{d}}\int_{|y|>R}G(x, y)\mu(dy)=0$For$\mu\in \mathcal{K}_{d}$(4), define a symmetric bilinear form$\mathcal{E}^{\mu}(u, u)=\frac{1}{2}D(u, u)-\int_{\mathbb{R}^{d}}\overline{u}^{2}d\mu$,$\mathcal{E}^{\mu}$.

by$u\in H^{1}(\mathbb{R}^{d})$,where is a quasi continuous version of ([8, Theorem 2.1.3]). In the seis represented byquel, we always assume that every function
charges no set of zero capacity
its quasi continuous version. Sinceis well defined. We see from [2, Theoremby [2, Theorem 3.3], the form
4.1] that
becomes a lower semi-bounded closed symmetric
the selfform. We calla Schr\"odinger form. Denote by. Let
adjoint operator generated by: $P_{t}^{\mu}=\exp(-tH^{\mu})$ . We see-semigroup generated by
be the$bom$ [ , Theorem 6.3(iv)] that
admits a symmetric integral kernel$p^{\mu}(t, x, y)$ which is jointly continuous on.
,continuousFordenotes a positive
additive functional
which is in the Revuz correspondence with : for any positive Borel
function and -excessive function ,$\overline{u}$$u$$u\in H_{e}^{1}(\mathbb{R}^{d})$$\mu\in \mathcal{K}_{d}$$\mathcal{E}^{\mu}$$(\mathcal{E}^{\mu}, H^{1}(\mathbb{R}^{d}))$$H^{\mu}$$(\mathcal{E}^{\mu}, H^{1}(\mathbb{R}^{d}))$$(\mathcal{E}^{\mu}, H^{1}(\mathbb{R}^{d})):\mathcal{E}^{\mu}(u, v)=(H^{\mu}u, v)$$L^{2}$$P_{t}^{\mu}$$H^{\mu}$$2$$P_{t}^{\mu}$$(0, \infty)$$\mu\in \mathcal{K}_{d}$$\times \mathbb{R}^{d}\times \mathbb{R}^{d}$$A_{t}^{\mu}$

$\mu$

$f$

(5)

$h$

$\gamma$$<h\mu$,,$f>=\lim_{t\rightarrow 0}\frac{1}{t}E_{hm}[\int_{0}^{t}f(B_{s})dA_{s}^{\mu}]$( , p.188]). By the Feynman-Kac formula, the semigroup$[8$$P_{t}^{\mu}$is written

$as$

(6)

\S 3.

$P_{t}^{\mu}f(x)=E_{x}[\exp(A_{t}^{\mu})f(B_{t})]$.

Criticality and ground state

Definition 3.1. A real-valued function is said to be harmonic ona domain $D$ with respect toif for any relatively compact open set$h$$H^{\mu}$$G\subset\overline{G}\subset D_{f}$

(7)

where

$h(x)=E_{x}[\exp(A_{\tau_{G}}^{\mu})h(B_{\tau_{G}})]$$\tau_{G}$,$x\in G$,is the first exit time from $G$ , $\tau_{G}=\inf\{t>0:B_{t}\not\in G\}$ .

We formally write aharmonic functionas $7\{^{\mu}h=0$ . An
possesses the minimal positive
operatoris said to be subcritical if$h$$H^{\mu_{-}}$$H^{\mu}$$H^{\mu}$

337

Diﬀerentiability of spectral functions

Green function $G^{\mu}(x, y)$ , that is,$ G^{\mu}(x, y)=\int_{0}^{\infty}p^{\mu}(t, x, y)dt<\infty$,$x\neq y$.

The operator
is said to be critical if $ G^{\mu}(x, y)=\infty$ and a positiveis neither
continuous-harmonic function exists. If the operator
subcritical nor critical, it is said to be supercritical (see [13, p.145]).
The spectral function
is defined by the bottom of the spectrum
: for
,
of
$\mathcal{H}^{\mu}$$H^{\mu}$$H^{\mu}$$C(\lambda)$$H^{\lambda\mu}$$\mu\in \mathcal{K}_{d}^{\infty}$(8)$C(\lambda)=-\inf\{\mathcal{E}^{\lambda\mu}(u, u)$;$u\in H^{1}(\mathbb{R}^{d})$,$\int_{\mathbb{R}^{d}}u^{2}dx=1\}$.

Define$\lambda^{+}$$=$$\inf\{\lambda>0:C(\lambda)>0\}$We then see that $C(\lambda)=0$ for
Lemma 3.1. Fortion such that$\mu\in \mathcal{K}_{d}^{\infty}$$\mathcal{H}^{\lambda^{+}\mu}h=0$$\lambda\leq\lambda^{+}([23])$.

.

, $t/iere$ exists a positive continuous func-.

for the Dirichlet
be the bottom of spectrum of$0=-C(\lambda^{+})<\lambda_{r\iota+1}<\lambda_{7\iota}$
is subcrit,problem on $B(n)$ . Since
on $B(n)$ .
ical on $B(n)$ . Let
denotes the Green operator ofis theby $h_{n}(x)=c_{n}G^{n+1}I_{A_{n}}(x)$ , where
We define a functionis the normalized
indicator function of $A_{n}(=B(n+1)\backslash B(n))$ and$c_{n}=(G^{7l+1}I_{A_{n}}(0))^{-1}$
constant,
is a harmonic function on. Then$B(m)$ , $m<n$ . Indeed, for $x\in B(m)$Proof.

Let$\mathcal{H}^{\lambda^{+}\mu}$$\lambda_{r\iota}$$H^{\lambda^{+}\mu}$$H^{\lambda^{+}\mu}$$G^{n}$$h_{n}$$I_{A_{n}}$$c_{n}$$h_{n}$$E_{x}[\exp(\lambda^{+}A_{\tau_{m}}^{\mu})h_{n}(B_{\tau_{m}})]=c_{n}E_{x}[\exp(\lambda^{+}A_{\tau_{m}}^{\mu})G^{\tau\iota+1}I_{A_{n}}(B_{\tau_{m}})]$$=c_{n}E_{x}[\exp(\lambda^{+}A_{\tau_{m}}^{\mu})E_{B_{\tau_{m}}}[\int_{0}^{\tau_{n+1}}\exp(\lambda^{+}A_{t}^{\mu})I_{A_{n}}(B_{t})dt]]$,

where $\tau_{m}=\inf\{t>0 : B_{t}\not\in B(m)\}$ . By the strong Markov property,
the right hand side is equal to$c_{n}E_{x}[\int_{0}^{\tau_{n+1}o\theta_{\tau_{m}}}\exp(\lambda^{+}(A_{\tau_{m}}^{\mu}+A_{t}^{\mu}\circ\theta_{\tau_{m}})I_{A_{n}}(B_{t+\tau_{m}})dt]$$=$$c_{n}E_{x}[\int_{\tau_{m}}^{\tau_{n+1}o\theta_{\tau_{m}}+\tau_{m}}\exp(\lambda^{+}A_{t}^{\mu})I_{A_{n}}(B_{t})dt]$

andNoting that
we see that the last term is equal to $h_{n}(x)$ . Therefore
for $G=B(m)$ .
$\tau_{n+1}\circ\theta_{\tau_{m}}+\tau_{m}=\tau_{n+1}$.

,
satisfies (7)$\int_{0}^{\tau_{m}}\exp(\lambda^{+}A_{t}^{\mu})I_{A_{n}}(B_{t})dt=0$$h_{n}$

338

M. Takeda and K. TsuchidaNow by [4, Corollary 7.8],
is uniformly bounded and equicontinuous on $B(1)$ , so we can choose a subsequence of
which converges uniformly on $B(1)$ . We denote the subsequence by. Nexttake a subsequenceofso that it converges uniformly on$B(2)$ .
By the same procedure, we take a subsequenceofso that it converges uniformly on $B(m+1)$ . Then the function,$h(x)=\lim_{n\rightarrow\infty}h_{n}^{(n)}(x)$, is a desired one.
Q.E.D.
$\{h_{n}\}$$\{h_{n}\}$$\{h_{n}^{(1)}\}$$\{h_{n}^{(2)}\}$$\{h_{n}^{(1)}\}$$\{h_{n}^{(m+1)}\}$$\{h_{n}^{(m)}\}$

Lemma 3.2. The following statements are equivalent:(i) $\inf\{\frac{1}{2}D(u, u)$ :$u\in H^{1}(\mathbb{R}^{d})$:$u\in H^{1}(\mathbb{R}^{d})$(ii)$\inf\{\mathcal{E}^{\mu}(u, u)$

Proof.

, $\int_{\mathbb{R}^{d}}u^{2}d\mu=1\}<1$ ;, $\int_{\mathbb{R}^{d}}u^{2}dx=1\}<0$ .

Assume (i). Then there exists aand $1/2D(\varphi_{0}, \varphi_{0})<1$ . Letting
have.
$(ii)\Rightarrow(i)$ follows similarly.
$\int_{\mathbb{R}^{d}}\varphi_{0}^{2}d\mu=1$$\varphi_{0}\in C_{0}^{\infty}(\mathbb{R}^{d})$such that$u_{0}=\varphi_{0}/\sqrt{\int_{\mathbb{R}^{d}}\varphi_{0}^{2}dx}$, we$\mathcal{E}^{\mu}(u_{0}, u_{0})<0$

Q.E.D.

Remark 3.3. We see ﬀom [25, Lemma 3.5] that if:$\inf\{\frac{1}{2}D(u, u)$$\int_{\mathbb{R}^{d}}u^{2}d\mu=1\}\leq 1$

,

then
:

$\inf\{\mathcal{E}^{\mu}(u, u)$$\int_{\mathbb{R}^{d}}u^{2}dx=1\}\leq 0$.

However, the converse does not hold in general. Indeed, let $\mu=\sigma_{R}$ ,the surface measure of the sphere. Then if $R<\frac{d-2}{2}$ , the firstinfimum is greater than 1, while the second infimum is equal to 0 ([25]).
$\partial B(R)$. Then the number
Lemma 3.4. Leta unique positive number such that$\mu\in \mathcal{K}_{d}^{\infty}$(9)$\inf\{\frac{1}{2}D(u, u)$

Proof.

:$\lambda^{+}$

is characterized as$\lambda^{+}\int_{\mathbb{R}^{d}}u^{2}d\mu=1\}=1$.

Define$F(\lambda)=\inf\{\frac{1}{2}D(u, u)$:$\lambda\int_{\mathbb{R}^{d}}u^{2}(x)\mu(dx)=1\}$,

Note that $ F(\lambda)=F(1)/\lambda$ . Then $F(1)$ is nothing but the bottom ofspectrum of the time changed process by the additive functional,$A_{t}^{\mu}([22$339Diﬀerentiability of spectral functionsof theLemma 3.1]). We see by [23, Lemma 3.1] that 1-resolvent. Hence it follows $bom$time changed process satisfies[ $17$ , Corollary 3.2] and [23, Corollary 2.2] that $F(1)>0$ . Consequently
we see that $\lambda^{0}=F(1)$ is a unique positive constant such that.
.
Lemma 3.2 leads us that
$R_{1}^{\mu}$$R_{1}^{\mu}1\in C_{\infty}(\mathbb{R}^{d})$$F(\lambda^{0})=1Q.E.D.$$\lambda^{0}=\lambda^{+}$

Corollary 3.5. For$\mu\in \mathcal{K}_{d}^{\infty}$, the operator$H^{\lambda^{+}\mu}$is critical.

Letbe the function in the proof of Lemma 3.4. Thenis subcritical if
it is known in [25, Theorem 3.9] that the operatorisand only if $F(\lambda)>1$ . Hence by Lemma 3.1 and Lemma 3.4,
Q.E.D.
critical.

Proof.

$F(\lambda)$$\mathcal{H}^{\lambda\mu}$$H^{\lambda^{+}\mu}$Lemma 3.6. A positive$P_{t}^{\lambda^{+}\mu}h(x)\leq h(x)$Proof.

$H^{\lambda^{+}\mu}$-harmonic function$h$satisfies.

Let $x\in B(m)$ . By Definition 3.1,

satisfies$h$$h(x)=E_{x}[\exp(\lambda^{+}A_{\tau_{n}}^{\mu})h(B_{\tau_{n}})]$

for any $>m$ . Hereis the first exit time from $B(n)$ . It follows $bom$
the Markov property that$n$$\tau_{n}$$E_{x}[\exp(\lambda^{+}A_{t}^{\mu})h(B_{t});t<\tau_{m}]$$=$$E_{x}[\exp(\lambda^{+}A_{t}^{\mu})\exp(\lambda^{+}A_{\tau_{n}}^{\mu}\circ\theta_{t})h(B_{\tau_{n}}\circ\theta_{t});t<\tau_{m}]$$=$$E_{x}[\exp(\lambda^{+}A_{\tau_{n}}^{\mu})h(B_{\tau_{n}});t<\tau_{m}]\leq h(x)$.

Hence we have$P_{t}^{\lambda^{+}\mu}h(x)=\lim_{m\rightarrow\infty}E_{x}[\exp(\lambda^{+}A_{t}^{\mu})h(B_{t});t<\tau_{m}]\leq h(x)$.
Q.E.D.

Let
be a positive semigroup with integral kernel $p(t, x, y)$ . A
positive function
is called-excessive if satisfies $P_{t}h(x)\leq h(x)$ .
is
For a -excessive function $h(x)$ , the -transformed semigroupdefined by$P_{t}$$h$$h$$P_{t}$$h$$P_{t}$(10)$P_{t}^{h}f(x)=\int_{\mathbb{R}^{d}}\frac{1}{h(x)}p(t, x, y)h(y)f(y)dy$$P_{t}^{h}$, $t>0$ ,$x$,$y\in \mathbb{R}^{d}$.

Thenbecomes a Markovian semigroup.
Let be the function defined in Lemma 3.1. We see from Lemma 3.6
generates a $h^{2}m$ -symmetricthat the -transformed semigroup. Note thatis recurrent
Markov process
because of the criticality of.
$P_{t}^{h}$$h$$P_{t}^{\lambda^{+}\mu,h}$$h$$W^{\lambda^{+}\mu,h}=(P_{x}^{\lambda^{+}\mu,h}, X_{t})$$H^{\lambda^{+}\mu}$$W^{\lambda^{+}\mu,h}$340

M. Takeda and K. Tsuchida

Lemma 3.7. Finely continuous
up to constant multiplication.

$P_{t}^{\lambda^{+}\mu}$-excessive functions are unique

We follow the argument in [13, Theorem 4.3.4]. Let ,finely continuous-excessive functions. Since

Proof.

$h$$h^{/}$

be$P_{t}^{\lambda^{+}\mu}$$E_{x}[\exp(\lambda^{+}A_{t}^{\mu})h(B_{t})(\frac{h^{J}}{h})(B_{t})]\leq h\cdot\frac{h^{/}}{h}(x)$,we have$E_{x}^{\lambda^{+}\mu,h}[\frac{h^{/}}{h}(X_{t})]\leq\frac{h^{/}}{h}(x)$where$E_{x}^{\lambda^{+}\mu,h}$andfinely open,, we have,is the expectation of -transformed process. For$U_{\epsilon}(y)=\{z:|h(z)-h(y)|<\epsilon\}$, we put. Sinceis,[ , Problem 4.6.3]. Replacing by$W^{\lambda^{+}\mu,h}$$h$$\epsilon>0$$y\in \mathbb{R}^{d}$$U_{\epsilon}(y)$$P_{x}^{\lambda^{+}\mu,h}- a.s$$\sigma_{U_{\epsilon}(y)}<\infty$

$8$

$t$

$\sigma_{\epsilon}$(11)$E_{x}^{\lambda^{+}\mu,h}[\frac{h^{/}}{h}(X_{\sigma_{\epsilon}})]\leq\frac{h^{J}}{h}(x)$.

Note that the left hand side of (11) converges tothen have$\frac{h^{/}}{h}(y)$$=$$\leq$

Since

$x$

and

$y$

$\frac{h’}{h}(y)$as$\epsilon\rightarrow 0$. We$E_{x}^{\lambda^{+}\mu,h}[\lim_{\epsilon\rightarrow}\inf_{0}\frac{h’}{h}(X_{\sigma_{\epsilon}})]\leq\lim_{\epsilon\rightarrow}\inf_{0}E_{x}^{\lambda^{+}\mu,h}[\frac{h^{J}}{h}(X_{\sigma_{\epsilon}})]$$\frac{h^{/}}{h}(x)$.

are arbitrary,$h^{J}/h$is a constant function.

Q.E.D.

Now we give known facts on the Kato class.
Theorem 3.8 ([20]). Let(12)$\mu\in \mathcal{K}_{d}$. Then for any$u\in H^{1}(\mathbb{R}^{d})$$\int_{\mathbb{R}^{d}}u^{2}(x)\mu(dx)\leq||G_{\alpha}\mu||_{\infty}(D(u, u)+\alpha\int_{\mathbb{R}^{d}}u^{2}(x)dx)$

It is known from [1] (also see [28]) that(13)$\mu\in \mathcal{K}_{d}$$\lim_{\alpha\rightarrow\infty}||G_{\alpha}\mu||_{\infty}=0$

Therefore we see that for any
for any$\epsilon$

if and only if.

there exists a constant$M(\epsilon)$$u\in H^{1}(\mathbb{R}^{d})$

(14)

.

$\int_{\mathbb{R}^{d}}u^{2}(x)\mu(dx)\leq\epsilon D(u, u)+M(\epsilon)\int_{\mathbb{R}^{d}}u^{2}(x)dx$.

such that

341

Diﬀerentiability of spectral functions

For a measure$\mu$, let $\mu_{R}(\cdot)=\mu(\cdot\cap B(R))$ and $\mu_{R^{c}}=\mu(\cdot\cap B(R)^{c})$ .

Lemma 3.9. If
is compact.

Proof.

Let$\mu\in \mathcal{K}_{d}^{\infty}$$\{u_{n}\}$, then the embedding of

be a sequence in$u_{n}\rightarrow u_{0}\in H_{e}^{1}(\mathbb{R}^{d})$$H_{e}^{1}(\mathbb{R}^{d})$$u_{n}I_{K}\rightarrow u_{0}I_{K}$Now, for$\varphi\in C_{0}^{\infty}(\mathbb{R}^{d})$to$L^{2}(\mu)$such that, $D$ -weakly.

Rellich’s theorem says that for any compact set(15)$H_{e}^{1}(\mathbb{R}^{d})$$K\subset \mathbb{R}^{d}$$L^{2}(m)- stroIlgly$.

with $\varphi=1$ on $B(R)$$\int_{\mathbb{R}^{d}}|u_{n}-u_{0}|^{2}\mu_{R}(dx)=\int_{\mathbb{R}^{d}}|u_{n}\varphi-u_{0}\varphi|^{2}\mu_{R}(dx)$$\leq\epsilon D(u_{n}\varphi-u_{0}\varphi, u_{n}\varphi-u_{0}\varphi)+M(\epsilon)\int_{\mathbb{R}^{d}}|u_{n}\varphi-u_{0}\varphi|^{2}dx$by (14), and the second term converges to 0 as

$ n\rightarrow\infty$by (15). Since$\sup_{n}D(u_{n}\varphi-u_{0}\varphi, u_{n}\varphi-u_{0}\varphi)<\infty$by the principle of uniform boundedness and is arbitrary,toin. Moreover, since by Theorem 3.8,$\epsilon$$u_{0}$$u_{n}$

converges$L^{2}(\mu_{R})$$\int_{\mathbb{R}^{d}}|u_{n}-u_{0}|^{2}\mu(dx)=\int_{\mathbb{R}^{d}}|u_{n}-u_{0}|^{2}\mu_{R}(dx)+\int_{\mathbb{R}^{d}}|u_{n}-u_{0}|^{2}\mu_{R^{c}}(dx)$$\leq\int_{\mathbb{R}^{d}}|u_{n}-u0|^{2}\mu_{R}(dx)+||G\mu_{R^{c}}||_{\infty}D(u_{n}-u_{0}, u_{n}-u_{0})$,$\lim_{n\rightarrow}\sup_{\infty}\int_{\mathbb{R}^{d}}|u_{7\iota}-u_{0}|^{2}\mu(dx)\leq||G\mu_{R^{c}}||_{\infty}\sup_{n}D(u_{n}-u_{0}, u_{n}-u_{0})$Hence according to the definition of
0 by letting $R$ to . Therefore

the right hand side converges tois an-convergent sequence.
Q.E.D.

$\mathcal{K}_{d}^{\infty}$$\{u_{n}\}$$\infty$.

$L^{2}(\mu)$

is subcritical or critical. Let be a positive
Assume that
harmonic function. We denote by
the family of $m$ -measurable$<\infty$
m-a.e. and there exists anfunction onsuch thatsuch that
Cauchy sequenceof functions in.
m-a.e. We callas above an approximating sequence for$h$$\mathcal{H}^{\mu}$$H^{\mu_{-}}$$D_{e}(\mathcal{E}^{\mu})$$\mathbb{R}^{d}$$\mathcal{E}^{\mu_{-}}$

$|u|$

$u$

$\{u_{n}\}$$\{u_{n}\}$$H^{1}(\mathbb{R}^{d})$$\lim_{n\rightarrow\infty}u_{n}=u$$u\in D_{e}(\mathcal{E}^{\mu})$342

M. Takeda and K. Tsuchida

Note that the Dirichlet form
Markov semigroupis given by

associated with the$(\mathcal{E}^{\mu,h}, D(\mathcal{E}^{\mu,h}))$$P_{t}^{\mu,h}$$\mathcal{E}^{\mu,h}(u, v)$$=$$\mathcal{E}^{\mu}(hu, hv)$

$D(\mathcal{E}^{\mu,h})$$=$$\{u\in L^{2}(\mathbb{R}^{d};h^{2}dx) :hu\in D(\mathcal{E}^{\mu})\}$.

if and only if, whereThen we see thatis the entended Dirichlet space of. Consequently, the Schr\"odinger formas
can be well extended toa symmetric form: forand its approximating sequence$u/h\in D_{e}(\mathcal{E}^{\mu,h})$$u\in D_{e}(\mathcal{E}^{\mu})$$D_{e}(\mathcal{E}^{\mu,h})$$(\mathcal{E}^{\mu,h}, D(\mathcal{E}^{\mu,h}))$$\mathcal{E}^{\mu}$$D_{e}(\mathcal{E}^{\mu})$$\{u_{n}\}$$u\in D_{e}(\mathcal{E}^{\mu})$$(16)$$\mathcal{E}^{\mu}(u, u)=\lim_{n\rightarrow\infty}\mathcal{E}^{\mu}(u_{n}, u_{n})$,$u\in D_{e}(\mathcal{E}^{\mu})$(see [8, p.35]). We callthe extended Schr\"odinger form. We
see from [18, Definition 1.6] that a function belongs to
if thereof functions inexists a sequencesuch thatra-a.e. and$(\mathcal{E}^{\mu}, D_{e}(\mathcal{E}^{\mu}))$$u$$D_{e}(\mathcal{E}^{\mu})$$\lim_{n\rightarrow\infty}u_{n}=u$$H^{1}(\mathbb{R}^{d})$$\{u_{n}\}$$\sup_{n}\mathcal{E}^{\mu}(u_{n}, u_{n})<\infty$.

Ifis a subcritical Schr\"odinger form, that is, the associated operator
be subcritical, thenbecomes a Hilbert
space by [8, Lemma 1.5.5]. In particular, a positive-harmonic func. Iftion does not belong to
is a critical Schr\"odinger
form, that is, the associated operator
be critical, its ground state
belongs toon account of [8, Theotem 1.6.3]. Noting that for$(\mathcal{E}^{\mu}, H^{1}(\mathbb{R}^{d}))$$H^{\mu}$$(\mathcal{E}^{\mu}, D_{e}(\mathcal{E}^{\mu}))$$\mathcal{H}^{\mu}$$h$$D_{e}(\mathcal{E}^{\mu})$$(\mathcal{E}^{\mu}, H^{1}(\mathbb{R}^{d}))$$h$$H^{\mu}$$D_{e}(\mathcal{E}^{\mu})$$\mu\in \mathcal{K}_{d}^{\infty}$$\mathcal{E}^{\mu}(u, u)\leq(1/2+||G\mu||_{\infty})D(u, u)$

by Theorem 3.8, we see that.
includesFor
define $=\lambda^{+}\mu-w\cdot m$ . We then see thatissubcritical. Let $G^{\nu}(x, y)$ be the Green function of
andthe Greenoperator,$H_{e}^{1}(\mathbb{R}^{d})$$D_{e}(\mathcal{E}^{\mu})$$w\geq 0\in C_{0}(\mathbb{R}^{d})$$H^{\nu}$$iJ$$G^{\nu}$$H^{\nu}$(17)$G^{\nu}f(x)=\int_{\mathbb{R}^{d}}G^{\nu}(x, y)f(y)dy$.

By [26, Theorem 3.1], the Green function $G^{\nu}(x, y)$ is equivalent to$G(x, y)$ : there exist positive constants , $C$ such that
$c$

(18)

$cG(x, y)\leq G^{\nu}(x, y)\leq CG(x, y)$Lemma 3.10. For a positive function$D_{e}(\mathcal{E}^{\iota/})$for $x\neq y$ .

$\varphi\in C_{0}(\mathbb{R}^{d})$,$ G^{\nu}\varphi$

belongs toDiﬀerentiability of spectral functions

Proof

Let

belongs to. Thenbe the -resolvent associated with
. Moreover,asand$H^{\nu}$$\beta$$G_{\beta}^{\nu}$$H^{1}(\mathbb{R}^{d})$343$ G_{\beta}^{\nu}\varphi$$\beta\rightarrow 0$$ G_{\beta}^{\nu}\varphi\rightarrow G^{\nu}\varphi$$\mathcal{E}^{\nu}(G_{\beta}^{\nu}\varphi, G_{\beta}^{\nu}\varphi)\leq \mathcal{E}_{\beta}^{\nu}(G_{\beta}^{\nu}\varphi, G_{\beta}^{\nu}\varphi)=(\varphi, G_{\beta}^{\nu}\varphi)\leq(\varphi, G^{\nu}\varphi)$and the right hand side is not greater than $ C(\varphi, G\varphi)<\infty$ by (18).
Q.E.D.
The next theorem is first obtained by Murata [10, Theorem 2.2]
is absolutely continuous with respect to thewhen the potential
Lebesgue measure.
$\mu$

Theorem 3.11. For. The positive continuous$w\in C_{0}(\mathbb{R}^{d})$$\lambda^{+}\mu-w\cdot m$

(19)

$=$
, let
with $w\geq 0$ ,-ha7monic function satisfies$w\not\equiv 0$$7\{^{\lambda^{+}\mu}$$h$$h(x)=\int_{\mathbb{R}^{d}}G^{\nu}(x, y)h(y)w(y)dy$

Proofsuch that$u_{0}$.

Note that by Lemma 3.9 there exists a function
attains the infimum:$\inf\{\frac{1}{2}D(u, u)$

The function$u_{0}$:$u\in H_{e}^{1}(\mathbb{R}^{d})$

,

$iJ$

$\lambda^{+}\int_{\mathbb{R}^{d}}u^{2}d\mu=1\}=1$$u_{0}\in H_{e}^{1}(\mathbb{R}^{d})$.

then satisfies the following equation:$\frac{1}{2}D(u_{0}, f)=\lambda^{+}\int_{\mathbb{R}^{d}}u_{0}fd\mu$

and thus by the definition of$\mathcal{E}^{\nu}(u_{0}, f)=\int_{\mathbb{R}^{d}}u_{0}$for all$f\in H_{e}^{1}(\mathbb{R}^{d})$

,

$iJ$

fwdx

for all$f\in H_{e}^{1}(\mathbb{R}^{d})$.

On account of the definition of the extended Schr\"odinger form, we see. Sincethat the equation above is extended to any
by Lemma 3.10, we obtain, by substituting
for anyfor$f\in D_{e}(\mathcal{E}^{\nu})$$\varphi\in C_{0}(\mathbb{R}^{d})$$D_{e}(\mathcal{E}^{\nu})$$ G^{\nu}\varphi$$ G^{\nu}\varphi\in$$f$$\int_{\mathbb{R}^{d}}u_{0}(x)\varphi(x)dx=\int_{\mathbb{R}^{d}}u_{0}(x)w(x)G^{\nu}\varphi(x)dx=\int_{\mathbb{R}^{d}}G^{\nu}(u_{0}w)(x)\varphi(x)dx$

thus$u_{0}(x)=\int_{\mathbb{R}^{d}}G^{\nu}(x, y)u_{0}(y)w(y)dy$, m-a. .
$e$

,

344

M. Takeda and K. TsuchidaLet$v(x)=E_{x}[\int_{0}^{\infty}\exp(A_{t}^{\nu})u_{0}(B_{t})w(B_{t})dt]$,

Then the function $v(x)$ equak to $u_{0}(x)$ m-a.e. and satisfies$v(x)=\int_{\mathbb{R}^{d}}G^{\nu}(x, y)v(y)w(y)dy$

Moreover, $v(x)$ is a finely continuous$v(B_{s})$(20)$P_{t}^{\lambda^{+}\mu}$, m-a.e.

-excessive function. Indeed,$=E_{B_{s}}[\int_{0}^{\infty}\exp(A_{t}^{\nu})u_{0}(B_{t})w(B_{t})dt]$$=E_{x}[\int_{0}^{\infty}\exp(A_{t}^{\nu}\circ\theta_{s})u_{0}(B_{t+s})w(B_{t+s})dt|F_{s]}$$=\exp(-A_{s}^{\nu})E_{x}[\int_{0}^{\infty}\exp(A_{t}^{\nu})u_{0}(B_{t})w(B_{t})dt|F_{s}]$$-\exp(-A_{s}^{\nu})\int_{0}^{s}\exp(A_{t}^{\nu})u_{0}(B_{t})w(B_{t})dt$.

and the first term of the last equality is right continuous because of. Hencethe right continuity ofis finely continuous $([10,Theorem$
A.2.7]), and thus $v(x)=u_{0}(x)q.e$ . Consequently$\mathcal{F}_{s}$

(21)

$v$

for any .

$v(x)=E_{x}[\int_{0}^{\infty}\exp(A_{t}^{\nu})v(B_{t})w(B_{t})dt]$$x$

Let $M_{t}=E_{x}[\int_{0}^{\infty}\exp(A_{t}^{\nu})v(B_{t})w(B_{t})dt|F_{s}]$ . Then according to (20) and(21)$\exp(A_{t}^{\lambda^{+}\mu})v(B_{t})=\exp(\int_{0}^{t}w(B_{u})du)(\exp(A_{t}^{\nu})v(B_{t}))$$=$$v(B_{0})+\int_{0}^{t}\exp(\int_{0}^{s}w(B_{u})du)dM_{s}-\int_{0}^{t}\exp(A_{s}^{\lambda^{+}\mu})v(B_{s})w(B_{s})ds$$+\int_{0}^{t}\exp(A_{s}^{\nu})v(B_{s})\exp(\int_{0}^{s}w(B_{u})du)w(B_{s})ds$$=$$v(B_{0})+\int_{0}^{t}\exp(\int_{0}^{s}w(B_{u})du)dM_{s}$,which implies that
$E_{x}[\exp(A_{t}^{\lambda^{+}\mu})v(B_{t})]\leq v(x)$.

Diﬀerentiability of spectral functions

Hence $h(x)=cv(x)$ by Lemma 3.7, and thus for aU(22)$h(x)=\int_{\mathbb{R}^{d}}G^{\nu}(x, y)h(y)w(y)dy$

345

$x$

.
Q.E.D.

\S 4.

An extension of Oshima’s inequality

In this section, we extend Oshima’s inequality in [11] to critical
Schr\"odinger forms. The inequality plays a crucial role for the proof of.
the diﬀerentiability of$C(\lambda)$

Lemma 4.1. Let be a positive continuoustion. Then the -transformed semigroupofFeller property.

$H^{\lambda^{+}\mu}$$h$$P_{t}^{\lambda^{+}\mu,h}$$h$-harmonic funchas the strong$P_{t}^{\lambda^{+}\mu}$Following the argument in [6, Corollary 5.2.7], we can prove
Q.E.D.
this lemma.

Proof.

Proposition 4.2. For the ground state , the -trasformed process
is Harris recurrent, that is, for a non-negative
$h$

$h$

$W^{\lambda^{+}\mu,h}=(P_{x}^{\lambda^{+}\mu,h}, X_{t})$function

$f$

,

(23)

$\int_{0}^{\infty}f(X_{t})dt=\infty$,$P_{x}^{\lambda^{+}\mu,h}$

-a. .
$s$

whenever $m(\{x:f(x)>0\})>0$ .

generates the $h^{2}m$ -symmetric recurrent Markov
Since
process, we see ﬀom [8, Theorem 4.6.6] that

Proof.

(24)$P_{t}^{\lambda^{+}\mu,h}$$P_{x}[\sigma_{A}\circ\theta_{n}<\infty, \forall n\geq 0]=1$

for $q.e$ .

$x\in \mathbb{R}^{d}$,where $A=\{x : f(x)>0\}$ . Moreover, since the Markov process
has transition density with respect to $h^{2}m$ , (24) holds for all
by [8, Problem 4.6.3]. Hence according to [16, Chapter $X$ , Proposition
Q.E.D.
(3.11) , we have the equation (23).
$W^{\lambda^{+}\mu,h}$$x\in \mathbb{R}^{d}$$]$and its ground state
Theorem 4.3. For the form$g\inL^{1}(h^{2}m)$and a function
exist a positive function
such that for$\mathcal{E}^{\lambda^{+}\mu}$$h$, therewith$\psi\in C_{0}(\mathbb{R}^{d})$$\int_{\mathbb{R}^{d}}\psi h^{2}dx=1$(25)$u\in D(\mathcal{E}^{\lambda^{+}\mu,h})$$\int_{\mathbb{R}^{d}}|u(x)-h(x)L(\frac{u}{h})|g(x)h(x)dx\leq C\mathcal{E}^{\lambda^{+}\mu}(u, u)^{1/2}$

,

346

M. Takeda and K. Tsuchida

where $C$ is a positive constant and$L(u)=\int_{]R^{d}}u\psi h^{2}dx$.

We can apply Oshima’s inequality to the Dirichlet form
the Harris recurrence condition: there ex$g\in
L^{1}(h^{2}m)$
ist a positive function
and a function
withsuch that for any

Proof.

$(\mathcal{E}^{\lambda^{+}\mu,h}, D(\mathcal{E}^{\lambda^{+}\mu,h}))satis\mathfrak{h}^{\gamma}ing$$\psi\in C_{0}(\mathbb{R}^{d})$$\int_{\mathbb{R}^{d}}\psi h^{2}dx=1$

(26)

$u$

$\in D(\mathcal{E}^{\lambda^{+}\mu,h})$$\int_{\mathbb{R}^{d}}|u(x)-L(u)|g(x)h^{2}(x)dx\leq C\mathcal{E}^{\lambda^{+}\mu,h}(u, u)^{1/2}$

,

where
$L(u)=\int_{\mathbb{R}^{d}}u\psi h^{2}dx$

Therefore substituting $v/h$ for$u$.

in (26) and noting the relation$\mathcal{E}^{\lambda^{+}\mu,h}(v, v)=\mathcal{E}^{\lambda^{+}\mu}(hv, hv)$,

we obtain the equality (25).

Q.E.D.

Diﬀerentiability of spectral function\S 5.

Lemma 5.1 ([24, Lemma 4.3]). Let. Then for any, the negative spectrum ofconsists of isolated eigenvalues withfinite multiplicities.
$\lambda>$$\mu\in \mathcal{K}_{d}^{\infty}$$\lambda^{+}$$\sigma(\mathcal{E}^{\lambda\mu})$Let
be critical and its groung state. Then we callcritical if the function does not belong to $L^{2}(m)$ ,$h$$H^{\mu}$$H^{\mu}$

null

$h$

Theorem 5.2. Let. If
is diﬀentiable.
spectral function$H^{\lambda^{+}\mu}$$\mu\in \mathcal{K}_{d}^{\infty}$

is null $critical_{f}$ then its$C(\lambda)$

Note that by Lemma 5.1, for, $-C(\lambda)$ is the principal eigenvalue of Schr\"odinger operator. By analytic
perturbation theory [9, Chapter VII], we can see thatis diﬀeren. Hence we only need to prove the diﬀerentiability of
tiable onat. Sinceis convex, it is enough to prove that there
exists a sequencesuch that. By theandperturbation theory [9, p.405, Chapter VII (4.44)], we see

Proof.

$\lambda>\lambda^{+}$$ H^{\lambda\mu}=-\frac{1}{2}\triangle-\lambda\mu$$C(\lambda)$$\lambda>\lambda^{+}$$C(\lambda)$$\lambda=\lambda^{+}$$C(\lambda)$$\{\lambda_{n}\}$(27)$\lambda_{n}\downarrow\lambda^{+}$$\frac{dC(\lambda)}{d\lambda}=\int_{\mathbb{R}^{d}}u_{\lambda}^{2}d\mu$$dC(\lambda_{n})/d\lambda\downarrow 0$

,

347

Diﬀerentiability of spectral functions

where
that is,$u_{\lambda}$

is the(28)$L^{2}$-normalized eigenfunction corresponding to$C(\lambda)=\lambda\int_{\mathbb{R}^{d}}u_{\lambda}^{2}d\mu-\frac{1}{2}D(u_{\lambda}, u_{\lambda})$Using (14) and taking$\epsilon>0$so small that

Noting that

$C(\lambda_{n})\rightarrow 0$

(29)

as

$n$

$\rightarrow\infty$,.

$\lambda_{n}\epsilon<1/2$$D(u_{\lambda_{n}}, u_{\lambda_{n}})\leq\frac{-C(\lambda_{n})+\lambda_{n}M(\epsilon)}{1/2-\lambda_{n}\epsilon}$$-C(\lambda)$, we have.

, we see$\sup_{n}D(u_{\lambda_{n}}, u_{\lambda_{n}})<\infty$

.

Since
$\mathcal{E}^{\lambda^{+}\mu}(u_{\lambda_{n}}, u_{\lambda_{n}})-\mathcal{E}^{\lambda_{n}\mu}(u_{\lambda_{n}}, u_{\lambda_{n}})\leq(\lambda_{n}-\lambda^{+})||G\mu||_{\infty}D(u_{\lambda_{n}}, u_{\lambda_{n}})$the right hand side converges to 0 as
(30)

$n$

$\rightarrow\infty$by (29). Thus we obtain$\lim_{n\rightarrow\infty}\mathcal{E}^{\lambda^{+}\mu}(u_{\lambda_{n}}, u_{\lambda_{n}})=0$

For the ground state ofwithtor. For
Then we have$h$$\psi\in C_{0}(\mathbb{R}^{d})$$H^{\lambda^{+}\mu}$let$H^{\lambda^{+}\mu,h}$$\int_{\mathbb{R}^{d}}\psi h^{2}dx=1$.

be the -transformed opera$h$, let $L(u)=\int_{\mathbb{R}^{d}}u(x)\psi(x)h^{2}(x)dx$ .

$|L(\frac{u_{\lambda_{n}}}{h})|\leq\sqrt{\int_{\mathbb{R}^{d}}u_{\lambda_{n}}^{2}dx}\sqrt{\int_{\mathbb{R}^{d}}\psi^{2}(x)h^{2}(x)dx}<\infty$.

tending toHence we can choose a sequencesuch that$C$.
Noting by Thorem 4.3,
converges to a certain constant$\lambda^{+}$$\{\lambda_{n}\}$$L(u_{\lambda_{n}}/h)$$\int_{\mathbb{R}^{d}}|u_{\lambda_{n}}-Ch|ghdx$

we see$\leq$$\int_{\mathbb{R}^{d}}|u_{\lambda_{n}}-hL(\frac{u_{\lambda_{n}}}{h})|ghdx+\int_{\mathbb{R}^{d}}|hL(\frac{u_{\lambda_{n}}}{h})-Ch|ghdx$$\leq$$C\mathcal{E}^{\lambda^{+}\mu}(u_{\lambda_{n}}, u_{\lambda_{n}})^{1/2}+\int_{\mathbb{R}^{d}}|L(\frac{u_{\lambda_{n}}}{h})-C|gh^{2}dx\rightarrow 0$$u_{\lambda_{n}}\rightarrow Cha.e$,. by choosing a subsequence if necessary. Since$1=\lim_{n\rightarrow}\inf_{\infty}\int_{\mathbb{R}^{d}}u_{\lambda_{n}}^{2}dx\geq\int_{\mathbb{R}^{d}}\lim_{n\rightarrow}\inf_{\infty}u_{\lambda_{n}}^{2}dx=C^{2}\int_{\mathbb{R}^{d}}h^{2}dx$

,

348

M. Takeda and K. Tsuchida

the constant $C$ must be equal to 0 on account of the null criticality.
Since
is an eigenvalue for.
,Thus we have by [2, Theorem 6.1 (iii)]$-H^{\lambda_{n}\mu}$$C(\lambda_{n})$$u_{\lambda_{n}}=e^{-C(\lambda_{n})t}P_{t}^{\lambda_{n}\mu}u_{\lambda_{n}}$$||u_{\lambda_{n}}||_{\infty}\leq e^{-C(\lambda_{n})t}||P_{t}^{\lambda_{n}\mu}||_{2,\infty}\leq||P_{t}^{\lambda_{1}\mu}||_{2,\infty}<\infty$

Also we can assume thatquence. Therefore we have$u_{\lambda_{n}}\rightarrow 0q.e$. as$ k\rightarrow\infty$.

by choosing a subse-$\lim_{n\rightarrow}\sup_{\infty}\int_{\mathbb{R}^{d}}u_{\lambda_{n}}^{2}d\mu$$\leq$$\lim_{n\rightarrow}\sup_{\infty}\int_{\mathbb{R}^{d}}u_{\lambda_{n}}^{2}d\mu_{R}+\lim_{n\rightarrow}\sup_{\infty}||G\mu_{R^{c}}||_{\infty}D(u_{\lambda_{n}}, u_{\lambda_{n}})$$\leq$$||G\mu_{R^{c}}||_{\infty}M$

By letting $R$ to$\infty$.

, we complete the proof.

Q.E.D.

Finally we consider the situation in Theorem 5.2. By Theorem 3.11
we have$c\int_{K}G^{\nu}(x, y)w(y)dy\leq h(x)\leq C\int_{K}G^{\nu}(x, y)w(y)dy$,

where $K$ is the support of $w$ . Let $B(R)\supset K$ . Applying the Harnack, $x\in B(R)^{c}$ , we see that
inequality to$G^{\nu}(x, \cdot)$

Gv $(x, O)\leq h(x)\leq CG^{\nu}(x, 0)$ on $x\in B(R)^{c}$ .
We see from the equation (18) that the ground state(31)$cG(x, O)\leq h(x)\leq CG(x, 0)$ ,$h$satisfies

on $x\in B(R)^{c}$ .

, that is,Hence we see that if $d\leq 4$ , is not in
is null critical.
Therefore combining [24, Theorem 4.3] and Theorem 5.2, we obtain
Theorem 1. 1.
$h$$L^{2}$$H^{\lambda^{+}\mu}$References[1] Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for
Schr\"odinger operators, Comm. Pure Appl. Math. 35, 209-273, (1982).
[2] Albeverio, S., Blanchard, P., Ma, Z.M.: Feynman-Kac semigroups interms of signed smooth measures, in “Random Partial Diﬀerential
Equations” ed. U. Hornung et al., Birkh\"auser, 1-31, (1991).
[3] Arendt, W., Batty, C.J.K.: The spectral function and principal eigenvalues for Schr\"odinger operators, Potential Anal. 7, 415-436, (1997).

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349[4] Boukricha, A., Hansen, W., Hueber, H.: Continuous solutions of thegeneralized Schr\"odinger equation and perturbation of harmonic spaces,
Expo. Math. 5, 97-135, (1987).
[5] Brasche, J.F., Exner, P., Kuperin, Y., Seba, P.: Schr\"odinger operatorswith singular interactions, J. Math. Anal. Appl. 184, 112-139, (1994).
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Cambridge, U.K., (1989).
[7] Dembo, A., Zeitouni, O.: Large deviation techniques and applications,
Second edition, Applications of Mathematics 38, Springer-Verlag, New
York, (1998).
[8] Fukushima, M., Oshima, Y., Takeda, M. : Dirichlet Forms and SymmetricMarkov Processes, Walter de Gruyter, Berlin, (1994).
[9] Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag,
Berlin-Heidelberg-New York, (1984).
,[10] Murata, M.: Structure of positive solutions to $(-\triangle+V)u=0$ in
Duke Math. J. 53, 869-943, (1986).
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[12] Pinchover, Y.: Criticality and ground states for scond-order elliptic equations, J. Diﬀerential Equations, 80, 237-250, (1989).
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I, Functional Analysis, Academic Press, (1972).
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IV, Analysis of Operators, Academic Press, (1979).
[16] Revuz, D., Yor, M. : Continuous Martingales and Brownian Motion, 3rdedition, Springer, New York, (1998).
[17] Sato, S.: An inequality for the spectral radius of Markov processes, Kodai
Math. J. 8, 5-13, (1985).
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[19] Simon, B.: Schr\"odinger semigroups, Bull. Amer. Math. Soc. 7, 447-536,(1982).
[20] Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures,
Potential Anal. 5, 109-138, (1996).
[21] Takeda, M.: Asymptotic properties of generalized Feynman-Kac functionals, Potential Analysis 9, 261-291, (1998).
[22] Takeda, M. : Exponential decay of lifetimes and a theorem of Kac on total
occupation times, Potential Analysis 11, 235-247, (1999).
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M. Takeda and K. Tsuchida[23] Takeda, M.:-independence of the spectral radius of symmetric Markov
semigroups, Stochastic Processes, Physics and Geometry: New Interplays. II: A Volume in Honor of Sergio Albeverio, Edited by Fritz
Gesztesy, et.al, (2000).
[24] Takeda, M.: Large deviation principle for additive functionals of Brownian motion corresponding to Kato measures, To appear in Potential
Analysis.
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Schr\"odinger operators, J. Funct. Anal. 191, 343-376, (2002).
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$L^{p}$$\alpha$Masayoshi Takeda
Mathematical InstituteTohoku University
Aoba, Sendai, 980-8578
Japan
Kaneharu Tsuchida
Mathematical InstituteTohoku University
Aoba, Sendai,980-8578
Japan

Advanced Studies in Pure Mathematics 41, 2004
Stochastic Analysis and Related Topics
pp. 351-373

On Spectra of Noises Associated with Harris Flows
Jon Warren and Shinzo Watanabe
Dedicated to Professor Kiyosi It\^o on his 88th birthday
Abstract.

A Harris flow is a stochastic flow on the real line given by SDE(2.1) below. We study the noise generated by Harris flows, particularly spectra of the noise. Our aim is to understand what lies beyond
the finite order terms in the chaos expansion (the Wiener-It\^o expansion) for nonstrong solutions of SDE (2.1).

Definitions and main results\S 1.

The notion of noises in continuous time (i.e., the case of time $t\in R$ )has been introduced by Tsirelson (cf. $[T1]$ , $[T2]$ , $[T5]$ ):is a two paramDefinition 1.1. $A$ noise$s\leqt$,eter family of sub,of events defifined on a probability
space $(\Omega, F, P)$ which is stationary in time and possesses the following
property:$N=[\{\mathcal{F}_{s,t}\}_{s\leq t}, \{T_{h}\}_{h\in R}]$$\sigma- fifields$(1.1)$\mathcal{F}_{s,t}$$F_{s,u}=F_{s,t}\otimes F_{t,u}$,$s\leq t\leq u$,, for every
andare independent and generate. By the stationarity in time, we mean the existence of a measurable, a measurable one-parameter group of automorphisms, on,
flflow.
is sent toby, in whichthat is,$t$$F_{s,t}$$F_{t,\tau\iota}$$ s\leq$$\mathcal{F}_{s,\tau\nu}$$\leq u$$\{T_{h}\}$$i.e.$$\mathcal{F}_{s,t}$$(\Omega, \mathcal{F}_{-\infty,\infty}.--\ovalbox{\tt\small REJECT}_{s\leq t}\mathcal{F}_{s,t})$$\mathcal{F}_{s+h,t+h}$$T_{h}$In this article, it is always assumed that the probability space is complete
and separable and that a sub -field contains all $P$-null sets.
In the discrete time case (i.e., the case of time $n\in Z$ ), a noise can be
defined similarly but it is essentially equivalent to giving an i.i.d. random
sequence. In the continuous time case, noises generated by incrementsof a Wiener process (of finite or countably infinite dimension), a stationary Poisson point process, or an independent pair of them, are typical$\sigma$

Received March 31, 2003.

J. Warren and S. Watanabe

352

examples which we call white, linearizable or classical noises. There aremany non-classical noises, however. Every noise $N=\{F_{s,t}\}$ contains aunique maximal (i.e., the largest) classical subnoise which is denoted by$N^{lin}=\{\mathcal{F}_{s,t}^{lin}\}$.

A Harris flflow (as will be defined precisely in Def.1.3 below) is astochastic flow on the real line $R$ determined uniquely by giving a realpositive definite function $b(x)$ such that $b(0)=1$ , (cf. [H]). Note that$b(x)=b(-x)$ . We assume that either $b(x)=1_{\{0\}}(x)$ or $b(x)$ is continuous,onand strictly positive-defifinite in the sense that the$\{b(x_{i}-x_{j})\}$
matrixis strictly positive-definite for any choice of finite
diﬀerent points
in R. The Harris flow in the discontinuous case of$b(x)=1_{\{0\}}(x)$ is known as the Arratia flflow ([A]).
Here is a formal definition of stochastic flows on the real line: Letbe the set of all non-decreasing right-continuous functions : $ x\in R\mapsto$$\varphi(x)\in R$ with the metric defined by
where$C^{2}$$R\backslash \{0\}$$\{x_{i}\}$$\mathcal{T}$$\varphi$$\rho(\varphi, \psi)=\sum_{n=1}^{\infty}2^{-n}(\rho_{r\iota}(\varphi, \psi)\wedge 1)$$\rho_{n}(\varphi, \psi)=\inf\{\in>0|\varphi(x-\epsilon)-\in\leq\psi(x)\leq\varphi(x+\in)+\in$

for all $x\in[-n, n]\}$ .
Thenis a Polish space: The composite,
defined by, and the evaluation map$\varphi(x)\in R$ are all Borel measurable even though they are generally notcontinuous.
$\mathcal{T}$$(\varphi, \psi)\in \mathcal{T}\times \mathcal{T}\mapsto\psi\circ\varphi\in \mathcal{T}$$\psi\circ\varphi(x)=\psi(\varphi(x))$$\mathcal{T}\times R\ni(\varphi, x)\mapsto$

Definition 1.2. By stochastic flow on $R$ , we mean a family $X=$$\{X_{s,t}; s\leq t\}$having the followingof -valued random variables
properties:(1) (Flow property), $X_{s,u}=X_{t,u}\circ X_{s,t}$ and $X_{t,t}=id$ , $a.s$ . forevery $s\leq t\leq u$ ,(2) (Independence property), for any sequence,-valued random variables, $k=1$ ,, , are independent,(3) (Stationarity), for any $h>0$ , $X_{s,t}=dX_{s+h,t+h}$ ,(4) (Stochastic continuity),.
in probability as$a$$\mathcal{T}$$X_{s,t}$$t_{0}\leq t_{1}\leq\cdots\leq t_{n}$$\mathcal{T}$$\cdots$$X_{t_{k-1},t_{k}}$$X_{0,h}\rightarrow id$$n$$h\downarrow 0$

Given a stochastic flow $X=\{X_{s,t}\}$ , it generates a noise $N^{X}=[\{F_{s,t}^{X}\}$ ,by letting
to be the -field generated by -valued random, $s\leq u\leq v\leq t$ , andvariables
to be a unique one-parameterfamily of automorphisms onsuch that $(T_{h})_{*}(X_{u,v}(x))=$$X_{u+h,v+h}(x)$ , $u\leq v$ , $x\in R$ .
Now we give a formal definition of Harris flows. Generally, for agiven filtration, we denote by
the space of all$\{T_{h}\}]$$\mathcal{F}_{s,t}^{X}$$\mathcal{T}$$\sigma$$\{T_{h}\}$$X_{u,v}$$(\Omega, \mathcal{F}_{-\infty,\infty}^{X})$$F=\{\mathcal{F}_{t}\}_{t\geq 0}$$\mathcal{M}_{2}(F)$353

On spectra of noises associated with Hams flflows

locally square-integrable $F$ -martingales $M$ $=(M_{t})_{t\geq 0}$ with $M_{0}=0$ and.
by
the subspace formed of all continuous elements in$\mathcal{M}_{2}(F)$$\mathcal{M}_{2}^{c}(F)$Definition 1.3. The Harris flow $X=\{X_{s,t}\}$ associated with thecorrelation function $b(x)$ is a stochastic flflow on $R$ such that, for every$x\in R$ ,if we defifine the process $M(x)=(M_{t}(x))_{t\geq 0}$ by setting $M_{t}(x)=$$X_{0,t}(x)-x$ and the fifiltration $F^{X}=\{F_{t}^{X}\}$ by setting, thenand, for every , $y\in R$ , we have$F_{t}^{X}=F_{0,t}^{X}$$M(x)\in \mathcal{M}_{2}^{c}(F^{X})$

(1.2)

$x$

$\langle M(x), M(y)\rangle_{t}=\int_{0}^{t}b(X_{0,s}(x)-X_{0,s}(y))ds$.

The law of a Harris flow is uniquely determined under our assumption
on functions $b(x)$ . The existence of Harris flows has been established in[H] (cf. also $[LR1]$ ). A Harris flow is equivalently given by a stochastic
diﬀerential equation (SDE) (2.1) in Section 2.
Let $X=\{X_{s,t}\}$ be a Harris flow associated with the function $b(x)$and $N^{X}$ be the noise generated by it. Suppose that $b(x)$ is continuous.
Then we can construct a centered Gaussian system $W=\{W(t, x);t\in$$R$ , $x\in R\}$ contained insuch that $(T_{h})_{*}[W(t, x)-W(s, x)]=$$x\in
R$ and, if we set $w_{t}(x)=W(t, x)-$$s\leqt$,$W(t+h, x)-W(s+h, x)$ ,$W(0, x)$ , then $w(x)=(w_{t}(x))_{t\geq 0}\in \mathcal{M}_{2}^{c}(F^{X})$ and, for every , $y\in R$ , wehave $\langle w(x), w(y)\rangle_{t}=tb(x-y)$ . Indeed, $W(t, x)-W(s, x)$ is the -limit. Here, for a sequence of times: $s=t_{0}<$ofas$M_{\triangle}^{x}(s,t)=\sum_{k=1}^{n}(X_{t_{k-1},t_{k}}(x)-x)$$t_{1}<\cdots<t_{n-1}<t_{n}=t$ and $x\in R$ ,and $|\Delta|=\max_{k}|t_{k}-t_{k-1}|$ . $W$ defines a Gaussian white noise $N^{W}=$where $\mathcal{F}_{s,t}^{W}=\sigma[W(v, x)-W(u, x);s\leq u\leq v\leq t, x\in R]$ .
It is obvious that $N^{W}$ is a subnoise of $N^{X}$ .
$L_{2}(\mathcal{F}_{-\infty,\infty}^{X})$$x$$L_{2}$$M_{\triangle}^{x}(s, t)$$\triangle$$|\triangle|\rightarrow 0$$[\{F_{s,t}^{W}\}, \{T_{h}\}]$

Theorem 1.1. Suppose that the function $b(x)$ is continuous. Then,it holds that $[N^{X}]^{lin}=N^{W}$ . Furthermore, $N^{X}=N^{W}$ holds, that is,
the noise $N^{X}$ generated by the Harris flflow $X$ is classical, if and only if(1.3)$\int_{0+}^{1}(1-b(x))^{-1}dx=\infty$.

Hence, the noise $N^{X}$ is nonclassical if and only if(1.4)$\int_{0+}^{1}(1-b(x))^{-1}dx<\infty$.

In the case of the Arratia flow, it generates a nonclassical noise:
Tsirelson $[T3]$ (cf. also $[LR2]$ ) showed that this noise is black in thesense thatfor every $s\leq t$ .
$(\mathcal{F}_{s,t}^{X})^{lin}=\{\emptyset, \Omega\}$

J. Warren and S. Watanabe

354

Tsirelson $([T2], [T5])$ introduced the notion of spectral measures fornoises which is an invariant under the isomorphism of noises and which
can measure the degree of non-linearity (or sensitivity in the discretetime approximation) of noises. Let be the space formed of all compact
sets in $R$ endowed with the Hausdorﬀ distance andbe its subclass$C^{f}=$$\{ S\in C||S|<\infty\}$ . Here,formed of all finite sets:denotesthe number of elements in .
$C$$C^{f}$

$|S|$

$S$

Definition 1.4. Let $N=[\{F_{s,t}\}, \{T_{h}\}]$ be a noise. To every, there corresponds a unique fifinite Borel measure
on
that
such

$\Phi\in$$L_{2}(F_{-\infty,\infty})$(1.5)$\mu_{\Phi}$$C$$\mu_{\Phi}(\{S\in C|S\subset J\})=E[E(\Phi|F(J))^{2}]$. Here, by an elementary set , we mean
$J=\bigcup_{k}[t_{k}, t_{k+1}]$
of non-overlapping intervals and we set.
is called the spectral measure of the noise $N$.
associated withfor every elementary set
fifinite union$J\subset R$

$J$

$a$

$F(J)=\ovalbox{\tt\small REJECT}_{k}\mathcal{F}_{t_{k},t_{k+1}}$$\mu_{\Phi}$$\Phi\in L_{2}(F_{-\infty,\infty})$

When $\Phi\in L_{2}(F_{s,t})$ , we havewhere $C_{[s,t]}=\{S\in$$|S\subset[s, t]\}$ , so thatis a measure on. The following is an
important characterization of classical noises due to Tsirelson: a noiseis classical if and only if.
for every$L_{2}^{us}(F_{s,t})=$$\{ \Phi\in L_{2}(F_{s,t})|||\Phi||_{2}=1\}$ ; the unit sphere inSet. If, thenis a Borel probability on sothat we can speak of a -value random variable with the distribution.
We denote it by
and call it the spectral set of the noise associatedwith .
We wish to describe the spectral set
for the noise $N^{X}$ generated
by a Harris flow $X$ when
. The random set
by.
in this case is denoted
We would also obtain some informationonfor general . We consider naturally the case when the noise isnonclassical so that we assume (1.4). Furthermore, we assume that$\mu_{\Phi}(C\backslash C_{[s,t]})=0$$C$$C_{[s,t]}$$\mu_{\Phi}$$\mu_{\Phi}(C\backslash C^{f})=0$$C$$\Phi\in L_{2}^{us}(\mathcal{F}_{-\infty,\infty})$$L_{2}(\mathcal{F}_{s,t})$$\Phi\in L_{2}(F_{-\infty,\infty})$$\mu_{\Phi}$$C$$\mu_{\Phi}$$S_{\Phi}$$\Phi$$S_{\Phi}$$\Phi=X_{0,1}(0)\in L_{2}^{us}(\mathcal{F}_{0,1}^{X})$$S_{\Phi}$$S_{X}$$\Phi$$S_{\Phi}$(1.6)$b(x)$

is non-increasing in$(0, \infty)$

and satisfifies$\lim_{x\rightarrow\infty}b(x)=0$.

Functions $b(x)=\exp(-c|x|^{\alpha})$ for $c>0$ and $0<\alpha<1$ are typicalexamples. Also, $b(x)=1_{\{0\}}(x)$ (the case of the Arratia flow) is anothertypical example.
be the the set of all accumulation points of ,For $S\in C$ , let
so that
if and only if.
$S^{acc}$$ S^{acc}\neq\emptyset$$S$$S\not\in C^{f}$

Theorem 1.2. Let $X$ be the Harris flflow associated with the function$b(x)$ which
be the spectral set
satisfifies (1.4) and (1.6) and letof$S_{X}$$S_{\Phi}$

On spectra of noises associated with Harris flflows

the noise $N^{X}$ for $\Phi=X_{0,1}(0)$ . Then the random set
law as the random set in $[0, 1]$ defifined by$S_{X}^{acc}$

355

has the same

$S$

(1.7)

$\overline{S}=$wherethe generator$\{ t|0\leq t\leq\tau,\hat{\xi}^{+}(\tau-t)=0\}$$\hat{\xi}^{+}=\{\hat{\xi}^{+}(t)\}_{t\geq 0}$(1.7)

is the reflflecting diﬀusion process on$[0, \infty)$

with$\hat{L}=\frac{d}{dx}(1-b(x))\frac{d}{dx}$and the initial distribution $\mu(dx):=-db(x)$ . Here,is $a[0,1]$ -valued
and uniformly distributed random variable independent of .
$\tau$$\hat{\xi}^{+}$

In particular, we have$P(S_{X}^{acc}\neq\emptyset)=P(|S_{X}|=\infty)=P(\overline{S}\neq\emptyset)=P\{\exists t\in[0, \tau];\hat{\xi}^{+}(t)=0\}$

and this probability is also equal to $E[\int_{0}^{1}(1-b(\xi^{+}(t)))dt]$ wherewith the generator
is the reflecting diﬀusion process on$\xi^{+}=$$\{\xi^{+}(t)\}_{t\geq 0}$(1.9)$[0, \infty)$$L=(1-b(x))\frac{d^{2}}{dx^{2}}$which starts at 0. Still another expression of this probability is given, where $A(t)$ is an additive functional ofby the expectation
the one-dimensional Wiener processwith $\beta(0)=0$ , defined by$\frac{1}{2}E[A^{-1}(1)]$$\beta(t)$(1.10)$A(t)=\frac{1}{2}\int_{0}^{t}(1-b(\beta(s)))^{-1}ds$,.
is the inverse function of
In the case of the Arratia flow, $S_{X}^{acc}=S_{X}$ and it is a perfect set,$a.s.$ .
It is described as a zero points set of a (double speed) reflecting
Brownian motion starting at 0 as in the theorem. This recovers a resultof Tsirelson $([T4])$ who obtained it by an approximation by coalescing
random walks.
In the following, we consider the class of Harris flows associatedwith the correlation functions $b(x)$ which satisfy (1.4), (1.6) and, forsome $0\leq\alpha<1$ ,and$t\rightarrow A^{-1}(t)$(1.11)$t\rightarrow A(t)$$1-b(x)\vee\wedge|x|^{\alpha}$as$x\rightarrow 0$.

Again, functions $b(x)=\exp(-c|x|^{\alpha})$ for $c>0$ and $0<\alpha<1$ aretypical examples. Note also that the function $b(x)=1_{\{0\}}(x)$ (the case

356

J. Warren and S. Watanabeof the Arratia flow) is a typical example of the case when $\alpha=0$ . From
Theorem 1.2, we can obtain the following: Denoting by $dim(S)$ theHausdorﬀ dimension of a subset in $R$ ,$S$

Corollary 1.1.
is not empty.

$dim(S_{X}^{acc})=\frac{1-\alpha}{2-\alpha}$$a.s.$, under the condition that it

Theorem 1.3. Let $\gamma=\inf\{\beta|dim(S_{\Phi})\leq\beta$ , $a.s$ . for any. Then$\Phi\in$$L_{2}^{us}(F_{-\infty,\infty}^{X})\}$$\gamma=\frac{1-\alpha}{2-\alpha}$.

The proof of these theorems will be given in the subsequent sections
by appealing to two main tools: joinings of Harris flflows and certain
duality relations between the reflecting (absorbing) -diﬀusion and theabsorbing (resp. reflecting) -diﬀusion.
$L$$\hat{L}$\S 2.

The joining of Harris flows: The proof of Th. 1.1.

Suppose that the correlation function $b(x)$ of a Harris flow $X$ iscontinuous. Let $H(\subset C_{b}(R\rightarrow R))$ be the (real) reproducing kernel
Hilbert space associated with $b(x)$ so that, defining $f_{x}\in H$ by $f_{x}(y)=$$b(y-x)$ , linear combinations
are dense in $H$ and $(f_{x}, f_{y})_{H}=$$b(x-y)$ . The Gaussian system $W$ introduced in Section 1 can be given
equivalently by a Gaussian system $\{W(t, f);t\in R, f\in H\}$ contained insuch that $(T_{h})_{*}[W(t, f)-W(s, f)]=W(t+h, f)-W(s+$$s\leq
t$
$f)$
$f\in
H$ and, if we set $w_{t}(f)=W(t, f)-W(0, f)$ , then $w(f)=$, ,,and, for every , $g\in H$ , we have$t(f, g)_{H}$ . Indeed, we set $W(t, f)=\sum_{i}c_{i}W(t, x_{i})$ when $f=\sum c_{i}f_{x_{i}}$ andextend this to general $f\in H$ by routine arguments.
We define an It\^o-type stochastic integral
for $F^{X}-$predictable processes and satisfying that
, $a.s.$ , by$\sum c_{i}f_{x_{i}}$$L^{2}(F_{-\infty,\infty}^{X})$$h$$(w_{t}(f))_{t\geq 0}\in \mathcal{M}_{2}^{c}(F^{X})$$f$$\langle w(f), w(g)\rangle_{t}=$$\int_{0}^{t}\psi_{s}\cdot W(ds, \varphi_{s})$$\varphi$$\psi$$\int_{0}^{t}|\psi_{s}|^{2}ds<\infty$$\int_{0}^{t}\psi_{s}\cdot W(ds, \varphi_{s})=\sum_{k}\int_{0}^{t}\psi_{s}\cdot e_{k}(\varphi_{s})db_{k}(s)$,is an orthonormal basis (ONB) in $H$ and $b_{k}(t)=W(t, e_{k})$ ,is an independent family of one-dimensional Wiener processes. As is easily seen, the definition is independent of a particular
choice of ONB. Note that, so that, inparticular,. Now, (1.2) is equivalently given in theform of SDE for $X_{t}:=X_{0,t}(x)$ :where
so that$\{e_{k}\}$$\{b_{k}(t)\}$$\sum_{k}e_{k}(\varphi_{s})e_{k}(\varphi_{s}’)=b(\varphi_{s}-\varphi_{s}’)$$\sum_{k}|e_{k}(\varphi_{s})|^{2}\equiv 1$(2.1)$X_{t}=x+\int_{0}^{t}W(ds, X_{s})=x+\sum_{k}\int_{0}^{t}e_{k}(X_{s})db_{k}(s)$ .

On spectra of noises associated with Harris flflows

357

Since $\sum_{k}|e_{k}(x)-e_{k}(y)|^{2}=2(1-b(x-y))$ , the condition (1.3) implies
the pathwise uniqueness of solutions for SDE (2.1) (cf. [IW], p.182).
Hence, if the function satisfies the condition (1.3), thenis a unique-measurable for every
strong solution to SDE (2.1) so that $X_{0,t}(x)$ is-measurable for every. By the stationarity, we see that $X_{s,t}(x)$ is
and $s\leq t$ . Therefore, $N^{X}=N^{W}$ holds. Thus, the if part of Th. 1.1
is proved.
To prove the only if part, we first remark the following martingale
representation theorem for Harris flows.
$b$$X_{t}$$F_{0,t}^{W}$$F_{s,t}^{W}$

$x$

$x$

Proposition 2.1. Suppose the correlation function $b(x)$ of the Harif and only if there exists
flflow is continuous. Then,-predictable processes satissequence, $k=1,2$ ,, of, $a.s.$ , for each $t>0$ , andfying that$ris$

$M\in \mathcal{M}_{2}(F^{X})$$a$$F^{X}$$\varphi_{k}=(\varphi_{k}(t))$$\ldots$$\sum_{k}\int_{0}^{t}\varphi_{k}^{2}(s)ds<\infty$$M(t)=\sum_{k}\int_{0}^{t}\varphi_{k}(s)db_{k}(s)$In particular, it holds that$\mathcal{M}_{2}(F^{X})=\mathcal{M}_{2}^{c}(F^{X})$.

.

-valued process, , $\ldots x_{n}\in R$ , any
Given distinct
SDE(2.1) startcomponentof which eachsolves theing from
and these components satisfy the coalescing property, has
the same law as the -point motion of the Harris flow $(X_{0,t}(x_{1}), X_{0,t}(x_{2})$, $X_{0,t}(x_{n}))$ . Rom this uniqueness in law, it follows by the usualthat is measurable with respect tomethods that any $M$
this -point motion is continuous and has the desired representation asa stochastic integral. The result can then be extended to an arbitrary$M$using the fact that the set of representable martingales
is closed in this space.

Proof.

$x_{1}$$R^{n}$$x_{2}$$(X_{t}^{1}, X_{t}^{2}, \ldots X_{t}^{n})$$X_{t}^{k}$$x_{k}$$n$$\ldots$$\in \mathcal{M}_{2}(F^{X})$$n$$\in \mathcal{M}_{2}(F^{X})$$\blacksquare$

From this proposition, we can easily deduce that $[N^{X}]^{lin}=N^{W}$ ,,see also Lemma $6a5$ of $[T5]$ . Indeed, if $N^{W}$ is smaller thanwhich cannot bethen there should exist some martingale ingiven by a sum of stochastic integrals by . Hence, in order to prove
the only if part, it is suﬃcient to show that (1.4) implies that $N^{W}$ isstrictly smaller than $N^{X}$ . For this, we introduce the following notion.
$[N^{X}]^{lin}$$\mathcal{M}_{2}(F^{X})$$b_{k}$

joining of a Harris flow, we mean a pair $(X=$, $X’=\{X_{s,t}’\})$ of copies of the Harris flflow defifined on a same probability space such that the joint process $---=\{_{-s,t}^{-}-=(X_{s,t}, X_{s,t}’);s\leq t\}$has the independence property (2) in Def. 1.2. Given $0\leq\rho\leq 1$ , it isDefinition 2.1. By$\{X_{s,t}\}$

$a$

358

J. Warren and S. Watanabe-joining if it satisfifies further the following: $X$ and $X’$ are stacalled
tionarily correlated in the sense that the joint process has the stationar,
ity property (3) of Def. 1.2 and, if fili ations
and martingales $M(x)=(M_{t}(x))$ , $M’(x)=(M_{t}’(x))$ are defifined sim, respectively, thenandareilarly as in Def. 1.3 for $X$ and,, and, forjointly immersed,
$y\in
R$
,
every ,$a$$\rho-$—$F^{X}=\{\mathcal{F}_{t}^{X}\}$$F^{X’}$$F^{X}$$X^{J}$$i.e.$$F^{X’}=\{\mathcal{F}_{t}^{X’}\}$$\mathcal{M}_{2}(F^{X})\cup \mathcal{M}_{2}(F^{X’})\subset \mathcal{M}_{2}(F^{X}\ovalbox{\tt\small REJECT} F^{X’})$

$x$

(2.2)
$b(x)$$\langle M(x), M’(y)\rangle_{t}=\int_{0}^{t}\rho\cdot b(X_{0,s}(x)-X_{0,s}’(y))ds$,

being the correlation function of the Harris flflow.

It is obvious that, for a -joining, the corresponding Gaussian noises $W$
and $W’$ are jointly Gaussian and correlated$\rho-$$\rho$

Lemma 2.1. For $0\leq\rho<1$ , a -joining exists and is unique inlaw. If, in particular, $\rho=0$ , then it is a pair of independent copies.
$\rho$

This lemma can be deduced from the fact that the following diﬀerential operator
with variables $x=(x_{1}, \cdots, x_{n})\in R^{n}$ and $x’=$$(x_{1}’, \cdots, x_{m}’)\in R^{m}$ is non degenerate at all such points $(x, x’)\in R^{n}\times$$R^{m}$as all coordinates in are diﬀerent and also all coordinates inare diﬀerent:$\Lambda$

$x’$

$x$

$\Lambda$,$=$$\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}b(x_{i}-x_{j})\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}+\frac{1}{2}\sum_{k=1}^{m}\sum_{l=1}^{m}b(x_{k}’-x_{l}’)\frac{\partial^{2}}{\partial x_{k}\partial x_{l}’}$$+$$\rho\sum_{i=1}^{n}\sum_{k=1}^{m}b(x_{i}-x_{k}’)\frac{\partial^{2}}{\partial x_{i}\partial x_{k}’}$.

Note that, for a -joining $(X, X’)$ , the process$\rho$$[o, \infty)\ni t\mapsto(X_{0,t}(x_{1}), \cdots, X_{0,t}(x_{n}), X_{0,t}’(x_{1}’), \cdots, X_{0,t}’(x_{m}’))$

is a solution to the$\Lambda$-martingale problem.

We now assume (1.4) and prove that $N^{W}$ is strictly smaller than$N^{X}$. Take -joinings
for $\rho\in[0,1)$ . By (2.2), the process
is a Feller diﬀusion on $R$ with the canonical scale $s(x)=x$ and the speed measure $m(dx)=(1-\rho\cdot b(x))^{-1}dx$
which starts from the origin at time 0, (cf. [IM] for a general theory of, the processes
Feller diﬀusions). As
converge to the Feller
diﬀusionwith the canonical scale $s(x)=x$ and the speed measure$m(dx)=(1-b(x))^{-1}dx$ which starts from the origin 0 at time 0. As is$(X^{(\rho)}, X^{\prime(\rho)})$$p$$\xi^{(\rho)}(t)=X_{0,t}^{(\rho)}(0)-X_{0,t}^{\prime(\rho)}(0)$$\rho\nearrow 1$$\xi(t)$$\xi^{(\rho)}(t)$359

On spectra of noises associated with Hams flflows

well-known, $\xi(t)=\beta(A^{-1}(t))$ for a one-dimensional Wiener process
and $A(t)$ is defined by (1.10). Then we have$\beta(t)$$\lim_{\rho\nearrow 1}E[|\xi^{(\rho)}(t)|^{2}]=E[|\xi(t)|^{2}]=\frac{1}{2}E[A^{-1}(t)]>0$

for $t>0$ .
$L_{2}(\mathcal{F}_{0,t}^{W})$Suppose $N^{X}\subset N^{W}$ be true.

and $ E[X_{0,t}^{\prime(\rho)}(0)|W]=P_{-\log\rho}\Phi$ where

Then $ X_{0,t}^{(\rho)}(0):=\Phi\in$$(P_{s})_{s\geq 0}$

is the Ornstein-. Hence $E[|\xi^{(\rho)}(t)|^{2}]=2(||\Phi||_{2}^{2}$
Uhlenbeck semigroup acing on$-(\Phi, P_{-\log\rho}\Phi)_{2})$ . By the-continuity of the Ornstein-Uhlenbeck semigroup, we have$L_{2}(F_{0,t}^{W})$$L^{2}$$\lim_{\rho\nearrow 1}E[|\xi^{(\rho)}(t)|^{2}]=\lim_{\rho\nearrow 1}2(||\Phi||_{2}^{2}-(\Phi, P_{-\log\rho}\Phi)_{2})=0$.

Thus we have a contradiction and hence we cannot have $N^{X}\subset N^{W}$ .
This proves the only if part of Th. 1.1 so that its proof now is completed.
In the following, we assume that (1.4) holds so that the noise generated by the Harris flow is nonclassical. In this case, 1-joinings are not-joining.
unique. We specify two of them as the-joining and the$1^{+}$$1^{-}$,
Definition 2.2. The -joining $(X, X’)$ is the identity joining:$X=X’$ . The -joining is the limit in law of the -joiningsas
It is such that $[0, \infty)\ni t\mapsto X_{0,t}(x)-X_{0,t}’(y)$ , for fifixed, $y\in R$ , is the Feller diﬀusion on $R$ with the canonical scale $s(x)=x$
and the speed measure $m(dx)=(1-b(x))^{-1}dx$ which starts at $x-y$ attime 0.
$1^{+}$$i.e.$$(X^{(\rho)}, X^{\prime(\rho)})$$1^{-}$$\rho$$\rho\nearrow 1$$x$

For $\rho\in[0,1)$ , let $(X, X’)$ be a -joining with corresponding -correlated
Gaussian processes $W$ and $W’$ . It is easy to see that the joint lawof $(X, X’, W, W’)$ is given by$p$$\rho$$\square (d\mathcal{X}d\mathcal{X}’d\mathcal{W}d\mathcal{W}’)$$P(X\in d\mathcal{X}|W=\mathcal{W})P(X’\in d\mathcal{X}’|W’=\mathcal{W}’)P(W\in d\mathcal{W}, W’\in d\mathcal{W}’)$.

From this, we deduce that
$E[\Phi\cdot\pi_{*}(\Psi)]$

whenever:$\pi_{*}$$\Phi$,$\Psi$$=$$E[E[\Phi|W]\cdot E[\pi_{*}(\Psi)|W’]]$$=$$E[E[\Phi|W]\cdot E(E[\pi_{*}(\Psi)|W’]|W)]$$=$$E[E[\Phi|W]\cdot E[\pi_{*}(E(\Psi|W))|W]]$$=$$E[E[\Phi|W]\cdot P_{-\log\rho}(E(\Psi|W))]$$\in L_{2}(F_{-\infty,\infty}^{X})$.

$L_{0}(\mathcal{F}_{-\infty,\infty}^{X})\rightarrow L_{0}(F_{-\infty,\infty}^{X’})$Here,is the unique isomorphism
such that $\pi_{*}(X_{s,t}(x))=X_{s,t}’(x)$ for$\pi_{*}$

360

J. Warren and S. Watanabe

every , and , andis the Ornstein-Uhlenbeck semigroup act-continuity of the Ornstein-Uhlenbeck. By theing onsemigroup, the above expectation converges to $E[E[\Phi|W]\cdot E[\Psi|W]]$ as. This proves existence of the -joining as the limit of -joinings.
-joining $(X, X’)$ the corresponding Gaussian systems
Moreover for a$W$ and $W’$ are equal and $X$ and $X’$ are conditionally independent given
this common Gaussian process.
$s$

$t$

$x$

$(P_{s})$$L^{2}$$L_{2}(F_{-\infty,\infty}^{W})$$1^{-}$$\rho\nearrow 1$$\rho$$1^{-}$

Remark 2.1. For the Arratia flflow, its -joining for $\rho\in[0,1)$ isindependent of and coincides with 0-joining, that is, a pair of independent copies of the Arratia flflow. Hence, its-joining is also a pairof independent copies of the Arratia flflow.
$\rho$$\rho$$1^{-}$

Let $F=\bigcup_{k=1}^{n}[t_{2k-2}, t_{2k-1}]$ be an elementary set in $R$ defined for asequence $t_{0}<t_{1}<\cdots<t_{2n-2}<t_{2n-1}$ of times. We would introduce-joining $(X, X’)$ of the Harris flow when $\rho\in[0,1)$ ,the notion ofwhich is roughly the -joining on $F$ and the identity joining outside$F$ .
To be more precise, set $ t_{-1}=-\infty$ and $ t_{2n}=\infty$ by convention.
Take a -joining $(Y, Y’)$ and a-joining $(Z, Z’)$ which are mutually$X=[\{X_{s,t}\}_{s\leqt}]$independent. Define
as follows: First, set $X_{s,t}=Y_{s,t}$ if$t_{2k-2}\leq s\leq t\leq t_{2k-1}$ , $k=1$ ,, and $X_{s,t}=Z_{s,t}$ if $ t_{2k-1}\leq s\leq t\leq$, $k=0$ ,, . Then, definefor general $s\leq t$ , by$(\rho, F)$$\rho$$1^{+}$$\rho$$\cdots$$\cdots$$t_{2k}$$n$$n$$X_{s,t}$$X_{S},t=Xt_{l},toXt_{1-1},t_{t}o\cdots oXt_{k},t_{k+1}oX_{s,t_{k}}$when $t_{k-1}<s\leq t_{k}\leq t_{l}\leq t<t_{l+1},0\leq k\leq l\leq 2n-1$ . Define $X’=$similarly ﬀom $Y’$ and $Z’$ . Then $(X, X’)$ defines a joiningof the Harris flow in which, however, $X$ and $X’$ are not stationarily
correlated.
$[\{X_{s,t}’\}_{s\leq t}]$Definition 2.3. The pair $(X, X’)$ defifined above is called the $(\rho, F)-$joining of the Harris flflow.
Next, take mutually independent -joining $(Y,Y’)$ andand construct the pair $(X, X’)$ in the same way.
$1^{-}$$1^{+}$-joining $(Z,Z’)$Definition 2.4. The pair $(X, X’)$ defifined above is called the $(1^{-}, F)-$joining of the Harris flflow

We turn now to the notion of the spectral measureassociatedwith someas defined in Def.1.4. This notion is intimately related to chaos expansions. The spectral measure of a random, measurable with respect to $W$ , can be exvariable
pressed by expanding as a sum of multiple Wiener-It\^o integrals withrespect to the Brownian motions . To be more precise,$\mu_{\Phi}$$\Phi\in L_{2}(F_{-\infty,\infty}^{X})$$\Phi\in L_{2}(F_{-\infty,\infty}^{W})$$\Phi$$b_{k}$$\Phi=\sum_{m=1}^{\infty}I_{m}$361

On spectra of noises associated with Hams flflows

where $L$ is a constant and
It\^o stochastic integral$I_{m}$, for $m=1,2$ .

, is given by an iterated$\cdots$$\cdots$$I_{m}=\sum_{(k_{1},,k_{m})}\cdots\int\cdots\int_{-\infty<t_{m}<<t_{1}<\infty}\cdots f_{\Phi}^{(k_{1},,k_{m})}\cdots(t_{1},$,$\ldots$$\mu_{\Phi}$$t_{m})db_{k_{m}}(t_{m})\cdots db_{k_{1}}(t_{1})$.

is supported on $C^{f}=\{S\in C:|S|<\infty\}$ and$\mu_{\Phi}(C^{f})=E(\Phi^{2})=\sum_{m=0}^{\infty}E(|I_{m}|^{2})$$=$$\cdots$$\sum_{m=0(k_{1}}^{\infty},\cdot\cdot\sum_{k_{m})}$$,\int\cdots\int_{-\infty<t_{m}<<t_{1}<\infty}\cdots|f_{\Phi}^{(k_{1},,k_{m})}\cdots(t_{1},$.

, $ t_{m})|^{2}dt_{m}\cdots dt_{1}<\infty$ .

$\ldots$

The restriction of
to $\{S\in C : |S|=m\}$ is determined (denoting$S=\{t_{m}, \cdots, t_{1}\}$ , $-\infty<t_{m}<\cdots<t_{1}<\infty)$ by$\mu_{\Phi}$$\mu_{\Phi}(dS;|S|=m)=|f_{\Phi}^{(k_{1},,k_{m})}\cdots(t_{1}, \cdots, t_{m})|^{2}dt_{m}\cdots dt_{1}$.

In particular, $\mu_{\Phi}(|S|=m)=E(|I_{m}|^{2})$ .
, the chaos expansion of $E[\Phi|W]$
For a generalgiven by $E[\Phi|W]=\sum_{m=0}^{\infty}I_{m}$ , yields in the same fashion the restrictionoftoand in particular$\Phi\in L_{2}(F_{-\infty,\infty}^{X})$$C^{f}$$\mu_{\Phi}$$E[E[\Phi|W]^{2}]=\mu_{\Phi}(C^{f})$.

If $(X, X’)$ is a -joining for $\rho\in[0,1)$ and $\Phi’=\pi_{*}(\Phi)$ as above, we
have$\rho$$E(E[\Phi’|W’]|W)=P_{-log\rho}(E[\Phi|W])=\sum_{m=0}^{\infty}\rho^{m}I_{m}$.

As was remarked above, the relation $E(\Phi\Phi’)=E(E[\Phi|W]E[\Phi’|W’])$holds. Hence,(2.3)$E(\Phi\Phi’)=\sum_{m=0}^{\infty}\rho^{m}E(|I_{m}|^{2})=\int_{C}\rho^{|S|}\mu_{\Phi}(dS)$In the same way, we deduce for a(2.4)$1^{-}$-joining $(X, X’)$ ,$E(\Phi\Phi’)=\mu_{\Phi}(C^{f})$

.

.

362

J. Warren and S. Watanabe

Example 2.1.
continuous function
whereConsider the case $\Phi=g(X_{0,1}(x))$ for a boundedon R. Note that $E(\Phi^{2})=\int_{R}p(1, x-y)g(y)^{2}dy$$g$$p(t, x)=\frac{1}{\sqrt{2\pi t}}\exp\{-\frac{x^{2}}{2t}\}$,$t>0$ ,

$x\in R$ .

The chaos expansion of $E[\Phi|W]$ was obtained explicitly by Veretennikov
and Krylov (cf. $[VKJ$): By setting$T_{t}f(x)=\int_{R}p(t, x-y)f(y)dy$

and$Q_{t}^{k}f(x)=e_{k}(x)\frac{\partial}{\partial x}T_{t}f(x)$,we have$g(X_{0,1}(x))=\sum_{m=0}^{n}I_{m}+R_{n}$, $m=1$ ,where, , andstochastic integrals:$I_{m}$$I_{m}$$=$$n$$R_{n}$$I_{0}=T_{1}g(x)=E[\Phi]$ ,are given by the following iterated It\^o$\sum_{(k_{1},k_{2},,k_{m})}\cdots\int\cdots\int_{0<t_{m}<t_{m-1}<<t_{2}<t_{1}<1}\cdots[T_{t_{m}}Q_{t_{m-1}-t_{m}}^{k_{m}}\cdots$$Q_{t_{1}-t_{2}}^{k_{2}}Q_{1-t_{1}}^{k_{1}}g(x)]db_{k_{m}}(t_{m})db_{k_{m-1}}(t_{m-1})\cdots db_{k_{2}}(t_{2})db_{k_{1}}(t_{1})$$\ldots$$R_{n}$$\ldots$

,

$=$

,

$\cdots\sum_{(k_{1},k_{2},,k_{n},k_{n+1})}\int\cdots$$\ldots\int_{0<t_{n+1}<t_{n}<<t_{2}<t_{1}<1}\cdots[Q_{t_{n}-t_{n+1}}^{k_{n+1}}Q_{t_{n-1}-t_{n}}^{k_{n}}\cdots$$\ldots Q_{t_{1}-t_{2}}^{k_{2}}Q_{1-t_{1}}^{k_{1}}g(X_{0,t_{n+1}}(x))]db_{k_{n+1}}(t_{n+1})db_{k_{n}}(t_{n})\cdots$$\ldots$$db_{k_{2}}(t_{2})db_{k_{1}}(t_{1})$.

From this, we obtain that
$E[\Phi|W]=\sum_{m=0}^{\infty}I_{m}$.

The following is a key lemma for the proof of Theorem 1.2 which
records various generalizations of the identities (2.3) and (2.4). As above,
we denote by
the spectral set
when $\Phi=X_{0,1}(0)$ which is avalued random variable.
$S_{X}$$S_{\Phi}$$C_{[0,1]^{-}}$

363

On spectra of noises associated with Hams flflows

If $(X, X’)$ is

Lemma 2.2. (i)for $\rho\in[0,1)$ , then,(2.5)$a(\rho, F)$-joining of the Hanis flflow$E[\rho^{|S_{X}\cap F|]}=E[X_{0,1}(0)X_{0,1}’(0)]$,equivalently,(2.6)$E[1-\rho^{|S_{X}\cap F|]}=\frac{1}{2}E[|X_{0,1}(0)-X_{0,1}’(0)|^{2}]$.

If $(X, X’)$ is $a(1^{-}, F)$ -joining of the Harris flflow, then,

(i)
(2.7)

$P$ $(|S_{X}\cap F|<\infty)=E[X_{0,1}(0)X_{0,1}’(0)]$ ,equivalently,(2.8)$P$ $( |S_{X}\cap F|=\infty)=\frac{1}{2}E[|X_{0,1}(0)-X_{0,1}’(0)|^{2}]$.

More generally, let $(X, X’)$ be $a(\rho, F)$ -joining for $0\leq\rho<1(a$$(1^{-}, F)$ -joining) and. There is a unique isomorphism:
such that $\pi_{*}(X_{s,t}(x))=X_{s,t}’(x)$ for
every , and . Set $\Phi’=\pi_{*}(\Phi)$ . Then we have(iii)$\Phi\in L_{2}^{us}(F_{-\infty,\infty}^{X})$$L_{0}(\mathcal{F}_{-\infty,\infty}^{X})\rightarrow L_{0}(F_{-\infty,\infty}^{X’})$$\pi_{*}$

$s$

$t$

(2.8)

$x$

$E[\rho^{|S_{\Phi}\cap F|}]$( resp.

$P$( $|S_{\Phi}\cap F|<\infty)$ ) $=E[\Phi\Phi’]$ ,equivalently,(2.10) $E[1-\rho^{|S_{\Phi}\cap F|}]$ ( resp. $P$ ( $|S_{\Phi}\cap F|=\infty)$ ) $=\frac{1}{2}E[|\Phi-\Phi’|^{2}]$and $F=[s, t]$ , (2.8) is
In the case whennothing but (2.3) and (2.4). Rom this, we can deduce (2.9) in thegeneral case of an elementary set $F=\bigcup_{k=1}^{n}[t_{2k-2}, t_{2k-1}]$ , $t_{-1}=-\infty<$$ t_{0}<\cdots<t_{2n-1}<t_{2n}=\infty$ , by considering the following-spacefactorization:

Proof.

$\Phi\in L_{2}^{us}(F_{s,t}^{X})$$L^{2}$$L_{2}(F_{-\infty,\infty}^{X})=\otimes L_{2}(F_{t_{k-1},t_{k}}^{X}k=02n)$.

We omit the details.

\S 3.

$\blacksquare$

Duality relations forand -diﬀusions in the time
reversal: The proof of Th. 1.2.
$L$ -

Let
processes on$\{\xi^{+}(t), P_{x}\}$$[0, \infty)$$\hat{L}$be the reflecting L- and -diﬀusion
andintroduced in Section 1. The associated Markovian$\{\hat{\xi}^{+}(t),\hat{P}_{x}\}$$\hat{L}$J. Warren and S. Watanabe

364

semigroups of operators acting on the space $B([0, \infty))$ of real bounded
Borel functions are defined by
(3.1)

and

$T_{t}^{+}f(x)=E_{x}[f(\xi^{+}(t))]$$\hat{T}_{t}^{+}f(x)=\hat{E}_{x}[f(\xi^{\hat{+}}(t))]$.

Define also the semigroups for absorbing processes by
and(3.2) $T_{t}^{-}f(x)=E_{x}[f(\xi^{+}(t\wedge\sigma_{0}))]$$\hat{T}_{t}^{-}f(x)=\hat{E}_{x}[f(\hat{\xi}^{+}(t\wedge\hat{\sigma}_{0}))]$

,

,
and
and
where
are the first hitting time to 0 of
respectively. Introduce, further, the semigroups for processes with theextinction at hitting to 0 by
(3.3)
$T_{t}^{0}f(x)=E_{x}[f(\xi^{+}(t))\cdot 1_{[t<\sigma_{0}]}]$.
and$\hat{\xi}^{+}(t)$$\xi^{+}(t)$$\hat{\sigma}_{0}$$\sigma_{0}$$\hat{T}_{t}^{0}f(x)=\hat{E}_{x}[f(\hat{\xi}^{+}(t))\cdot 1_{[t<\hat{\sigma}_{0}]]}$

are sub-Markovian
are Markovian semigroups and
and
and
,
and
have the strong Feller
semigroups. Note $ako$ thatproperty but
andhave the Feller property only. It holds that$\hat{T}_{t}^{-}$$T_{t}^{-}$$\hat{T_{t}}^{0}$$T_{t}^{0}$$T_{t}^{+},\hat{T_{t}}^{+}$$T_{t}^{0}$$\hat{T}_{t}^{0}$$\hat{T}_{t}^{-}$$T_{t}^{-}$(3.4)$T_{t}^{0}f=T_{t}^{-}(1_{(0,\infty)}\cdot f)$and$\hat{T_{t}}^{0}f=\hat{T_{t}}^{-}(1_{(0,\infty)}\cdot f)$.

We have the following duality relations which form another key
lemma in the proof of Th. 1.2:

Lemma 3.1.
(3.5)

For , $y\in[0, \infty)$ and $t>0$ ,$x$$T_{t}^{+}1_{[0,y]}(x)=\hat{T_{t}}^{0}1_{[x,\infty)}(y)$and$T_{t}^{-}1_{[0,y]}(x)=\hat{T_{t}}^{+}1_{[x,\infty)}(y)$.

More generally, for , $y\in[0, \infty)$ and $0\leq t_{0}<t_{1}<\ldots<t_{2n-1}<t_{2n}<$$t_{2n+1}$ ,
$x$

(3.6)

$T_{t_{1}-t_{O}}^{+}T_{t_{2}-t_{1}}^{-}T_{t_{3}-t_{2}}^{+}\cdots T_{t_{2n-1}-t_{2n-2}}^{+}T_{t_{2n}-t_{2n-1}}^{-}1_{[0,y]}(x)$$=$$\hat{T}_{t_{2n}-t_{2n-1}}^{+}\hat{T}_{t_{2n-1}-t_{2n-2}}^{0}\cdots\hat{T}_{t_{3}-t_{2}}^{0}\hat{T_{t_{2}-t_{1}}}^{+}\hat{T}_{t_{1}-t_{0}}^{0}1_{[x,\infty)}(y)$

,

and
(3.7)
$=$

$T_{t_{1}-t_{0}}^{+}T_{t_{2}-t_{1}}^{-}T_{t_{3}-t_{2}}^{+}\cdots T_{t_{2n-1}-t_{2n-2}}^{+}T_{t_{2n}-t_{2n-1}}^{-}T_{t_{2n+1}-t_{2n}}^{+}1_{[0,y]}(x)$$\hat{T_{t_{2n+1}-t_{2n}}}^{0}\hat{T}_{t_{2n}-t_{2n-1}}^{+}\hat{T}_{t_{2n-1}-t_{2n-2}}^{0}\cdots\hat{T}_{t_{3}-t_{2}}^{0}\hat{T_{t_{2}-t_{1}}}^{+}\hat{T_{t_{1}-t_{0}}}^{0}1_{[x,\infty)}(y)$.

Admitting this lemma for a moment, we now proceed to proveTh. 1.2.

Proof of Th. 1.2. Letelementary set in$[0, 1]$

and

be an
-coupling of the Harris$F=[t_{0}, t_{1}]\cup[t_{2}, t_{3}]\ldots\cup[t_{2n-2}, t_{2n-1}]$$(X, X’)$be a$(1^{-}, F)$

365

On spectra of noises associated with Hams flflows

is a time-inhomogeneous
flow. Set $\xi(t)=X_{0,t}(0)-X_{0,t}’(0)$ . Thendiﬀusion process which behaves as a reflecting diﬀusion when $t\in F$
and as an absorbing -diﬀusion (i.e., -diﬀusion with 0 as a trap) when$t\in[0,1]\backslash F$ . It is known that $P(S_{X}\ni t)=0$ for every $t\in[0,1]$ (cf.
. Then (2.8), combined with this remark, yields that$|\xi(t)|$

$L$

$L$

$L$

$[T$

$2])$

$P(|S_{X}\cap F|=\infty)=P(S_{X}^{acc}\cap F\neq\emptyset)=\frac{1}{2}E[|\xi(1)|^{2}]$By applying the It\^o formula for$\xi(t)$on each interval.

$[t_{k}, t_{k+1}]$, we have$\frac{1}{2}E[|\xi(1)|^{2}]=\int_{0}^{1}E[(1-b)(\xi(t))]dt=1-\int_{0}^{1}E[b(\xi(t))]dt$,

and hence,(3.8)$P(S_{X}^{acc}\cap F=\emptyset)=\int_{0}^{1}E[b(\xi(t))]dt$.

On the other hand,$E[b(\xi(t))]$$T_{t_{1}-t_{0}}^{+}T_{t_{2}-t_{1}}^{-}\cdots T_{t_{2k}-t_{2k-1}}^{-}T_{t-t_{2k}}^{+}b(0)$

,

$=\{$
$T_{t_{1}-t_{0}}^{+}T_{t_{2}-t_{1}}^{-}\cdots T_{t_{2k-1}-t_{2k-2}}^{+}T_{t-t_{2k-1}}^{-}b(0)$,if $t_{2k}\leq t<t_{2k+1}$
if $t_{2k-1}\leq t<t_{2k}$Noting $b(x)=\int_{[0,\infty)}1_{[0,y]}(x)\mu(dy)$ , we have by Lemma 3.1 the following:$E[b(\xi(t))]$$\int_{0}^{\infty}\mu(dy)(\hat{T}_{t-t_{2k}}^{0}\hat{T}_{t_{2k}-t_{2k-1}}^{+}\cdots\hat{T_{t_{2}-t_{1}}}^{+}\hat{T}_{t_{1}-t_{0}}^{0}1_{[0,\infty)})(y)$,if $t_{2k}\leq t<t_{2k+1}$
$=$

$\{$

$\int_{0}^{\infty}\mu(dy)(\hat{T}_{t-t_{2k-1}}^{+}\hat{T}_{t_{2k-1}-t_{2k-2}}^{0}\cdots\hat{T_{t_{2}-t_{1}}}^{+}\hat{T_{t_{1}-t_{0}}}^{0}1_{[0,\infty)})(y)$,if $t_{2k-1}\leq t<t_{2k}$

If the random set
is defined by (1.7), it is not diﬃcult to deduce,
ﬀom the last expression of $E[b(\xi(t))]$ , that $\int_{0}^{1}E[b(\xi(t))]dt$ coincides with. Then $P(\overline{S}\cap F=\emptyset)=P(S_{X}^{acc}\cap F=\emptyset)$ by (3.8). Since.
this holds for every elementary set $F$ , we can conclude that$\overline{S}$$P(\overline{S}\cap F=\emptyset)$$S_{X}^{acc}=d\overline{S}$$\blacksquare$First, we prove (3.5). For $\lambda>0$ , let
be the resolvent operators associated with the semigroups

Proof of Lemma 3.1.
and$\hat{U}_{\lambda}^{0}$$U_{\lambda}^{+}$$T_{t}^{+}$

J. Warren and S. Watanabe

366

and

respectively. Let be continuous and compactly supported in. Thensolves Poisson’s equation$\hat{T}_{t}^{0}$$(0, \infty)$$f$$u=U_{\lambda}^{+}f$$Lu-\lambda u=-f$ ,with the boundary conditions $u’(0+)=u(\infty)=0$ . Define functionsand via$g$$v$$g(y)=\int_{0}^{y}\frac{f(x)}{a(x)}dx$

and$v(y)=\int_{0}^{y}\frac{u(x)}{a(x)}dx$,

where $a(x)=(1-b(x))$ . Dividing Poisson’s equation through by $a(x)$and integrating, we obtain$\hat{L}v-\lambda v=-g$.

Moreover and are bounded and $v(0)=0$ . Thus we must have $v=$
. Letting approach a delta function we may write the relationshipbetween and as:$v$$\hat{U}_{\lambda}^{0}g$

$g$

$f$

$u$

$v$

$\frac{1}{a(z)}\hat{U}_{\lambda}^{0}1_{[z,\infty)}(y)=\int_{0}^{y}\frac{u_{\lambda}^{+}(x,z)}{a(x)}dx$

,

where
is the continuous version of the resolvent density corresponding. Recalling the symmetry relation,to$u_{\lambda}^{+}$$U_{\lambda}^{+}$$\frac{1}{a(x)}u_{\lambda}^{+}(x, z)a(z)=u_{\lambda}^{+}(z, x)$,

we obtain$\hat{U}_{\lambda}^{0}1_{[z,\infty)}(y)=\hat{U}_{\lambda}^{+}1_{[0,y]}(z)$,$bom$ which the first equality of (3.5) follows by uniqueness of Laplacetransforms. The second equality may be proved by a similar method.

On spectra of noises associated with Harris flflows

367(3.6) and (3.7) can be proved by applying (3.5) successively: For
example,$T_{t_{1}-t_{0}}^{+}$.

$T_{t_{2}-t_{1}}^{-}1_{[0,y]}(x)$$=$$\int_{[0,\infty)}T_{t_{1}-t_{O}}^{+}(x, du)T_{t_{2}-t_{1}}^{-}1_{[0,y]}(u)$$=$$\int_{[0,\infty)}T_{t_{1}-t_{0}}^{+}(x, du)\hat{T}_{t_{2}-t_{1}}^{+}1_{[u,\infty)}(y)$$=$$\int\int_{0\leq u\leq v<\infty}T_{t_{1}-t_{0}}^{+}(x, du)\hat{T}_{t_{2}-t_{1}}^{+}(y, dv)$$=$$\int_{[0,\infty)}\hat{T_{t_{2}-t_{1}}}^{+}(y, dv)T_{t_{1}-t_{0}}^{+}1_{[0,v]}(x)$$=$$\int_{[0,\infty)}\hat{T_{t_{2}-t_{1}}}^{+}(y, dv)\hat{T}_{t_{1}-t_{O}}^{0}1_{[x,\infty)}(v)$$=$$\hat{T}_{t_{2}-t_{1}}^{+}\cdot\hat{T}_{t_{1}-t_{O}}^{0}1_{[x,\infty)}(y)$.

This proves a particular case of (3.6). In the same way, the general casecan be proved easily by induction.

$\blacksquare$

Remark 3.1. We remark that an alternative proof of (3.5) is possible by means of the time reversal of stochastic flflows on the half line.
is defifined similarly by replacing
A stochastic flflow on the half linein (cf. 1.2. A key idea in the proof is to
the whole line $R$ by
whose one-point moconstruct a stochastic flflow $X=(X_{s,t})$ on$t\mapsto
X_{0,t}(x)$
$x\in
R$
,
,
, is given by the absorbing -diﬀusiontion, and then show that its time
, the diﬀusion with the semigroup$\hat{X}_{s,t}=(X_{-t,-s})^{-1}$, has the one, defifined byreversed flow
, the diﬀusion,
point motion given by the reflflecting -diﬀusion. Here, for a right-continuous and non-decreasing
with the semigroupis the right,:
such that
continuous inverse of $\varphi:\varphi^{-1}(x)=\inf\{y|\varphi(y)>x\}$ . This is connectedto the fact that and , when written in H\"ormander form, diﬀer onlyin the sign of the drift term. The corresponding fact in the case ofstochastic flflows of homeomorphisms is well-known (cf. $[K]$ p. 131, [IW]p. 265).
$[0, \infty)$$[0, \infty)$$[0, \infty)$$L$$i.e.$$\xi^{-}(t)$$T_{t}^{-}$$\hat{X}=(\hat{X}_{s,t})$$\hat{L}$$\hat{\xi}^{+}(t)$$i.e.$$\hat{T}_{t}^{+}$$[0, \infty)$$\varphi$

$L$

\S 4.

$\lim_{x\nearrow\infty}\varphi(x)=\infty$$\rightarrow[0, \infty)$$\varphi^{-1}$$\hat{L}$Proof of Th. 1.3.
Consider a Harris flow $Xsatis\mathfrak{h}^{r}ing(1.4)$ , (1.6) and (1.11).
It is suﬃcient to show that the set of zeros ofhas the Hausdorﬀ dimension $(1-\alpha)/(2-\alpha),\hat{P}_{0}$ -almosy

Proof of $Cor$ . 1.1.
$\hat{L}$-diﬀusion$\hat{\xi}(t)$J. Warren and S. Watanabe368

is the range of the inverse local time $l^{-1}(t)$surely. The set of zeros of, which is a subordinator with exponent:
at 0 of$\hat{\xi}(t)$$\Psi(\lambda)=g_{\lambda}(0,0)^{-1}$$\hat{\xi}(t)$$E(e^{-\lambda l^{-1}(t)})=e^{-t\Psi(\lambda)}=e^{-t/gx(0,0)}$Here,$g_{\lambda}(x, y)$.

is the Green function (resolvent density) with respect to$thespeedmeasuredxofreflflecting\hat{L}- diffffusionwhere\hat{L}=\frac{d}{dxin}(1-b(x))\frac{d}{\infty)dx}Ifweintroducethescale\xi=\int_{0}^{x}(l-b(y))^{-1}dyasthecoordateof[0,,\cdot$

where $\tilde{b}(\xi)=b(x(\xi))$ , so that the speed measurein the new coordinate is given by $ d\tilde{m}(\xi)=a(\xi)d\xi$ with $a(\xi)=1-\overline{b}(\xi)$ .
. Let
It is easy to deduce from (1.11) thatasbe the Green function for -diﬀusion with respect to the speed. By Th.2.3 in p.243 of [KW], we havemeasure so thatthen$\hat{L}=(1-\overline{b}(\xi))^{-1}\frac{d^{2}}{d^{2}\xi}$$a(\xi)\wedge\vee\xi^{\alpha/(1-\alpha)}$$\xi\rightarrow 0$$\hat{L}$$\tilde{g}_{\lambda}(\xi, \eta)$$\tilde{g}_{\lambda}(0, O)=g_{\lambda}(0,0)$as$\Psi(\lambda)=\overline{g}_{\lambda}(0,0)^{-1}\wedge\vee\lambda^{1/(2+\frac{\alpha}{1-\alpha})}=\lambda^{\frac{1-\alpha}{2-\alpha}}$$\lambda\rightarrow\infty$.

Then we can conclude that the range of the subordinator $l^{-1}(t)$ has thealmost surely, by a result of Blumenthal and
Hausdorﬀ dimension
Getoor (cf. [B], p. 94, Th. 16).
$\frac{1-\alpha}{2-\alpha}$$\blacksquare$

Now we proceed to prove Th. 1.3. We need several lemmas.
Lemma 4.1. (i)linear combination

Let$\Phi_{1}$,and consider their$\Phi_{2}\in L_{2}^{us}(\mathcal{F}_{-\infty,\infty}^{X})$. If $A\in B(C)$ satisfifies$P(S_{\Phi_{1}}\in A)=P(S_{\Phi_{2}}\in A)=1$ , then it holds that $P(S_{\Phi}\in A)=1$ .
(ii) Let, $n=1,2$ ,, constitute a dense family in. If $A\in B(C)$ satisfifies $P(S_{\Phi_{n}}\in A)=1$ for all , then it$\Phi=\alpha\Phi_{1}+\beta\Phi_{2}\in L_{2}^{us}(F_{-\infty,\infty}^{X})$$\Phi_{n}\in L_{2}^{us}(F_{-\infty,\infty}^{X})$$\ldots$$L_{2}^{us}(\mathcal{F}_{-\infty,\infty}^{X})$$n$holds that $P(S_{\Phi}\in A)=1$

for all$\Phi\in L_{2}^{us}(F_{-\infty,\infty}^{X})$.

Proof. According to Theorem $3d12$ of $[T5]$ , every $A\in B(C)$ is associated with a closed subspacemeasureof any satisfiesof$\prime \mathcal{H}_{A}$$L_{2}(F_{-\infty,\infty}^{X})$such that the spectral$\Phi$$\mu_{\Phi}$$||P_{A}\Phi||^{2}=\mu_{\Phi}(A)$

,

where
denotes the orthogonal projection ontolemma are immediate consequences.
$P_{A}$

Lemma 4.2.
that

. Both parts of this$\blacksquare$

Let $t_{1}<t_{2}<t_{3}$ andand. Then$\Phi_{1}\in L_{2}^{us}(\mathcal{F}_{t_{1},t_{2}}^{X})$$\Phi=\Phi_{1}\Phi_{2}\in L_{2}^{us}(\mathcal{F}_{t_{1},t_{3}}^{X})$$S_{\Phi}\cap[t_{1}, t_{2}]$such$\Phi_{2}\in L_{2}^{us}(\mathcal{F}_{t_{2},t_{3}}^{X})$$S_{\Phi}\cap[t_{1}, t_{2}]=dS_{\Phi_{1}}$Furthermore,$?\{_{A}$and,$S_{\Phi}\cap[t_{2}, t_{3}]=dS_{\Phi_{2}}$$S_{\Phi}\cap[t_{2}, t_{3}]$.

ore mutually independent.

369

On spectra of noises associated with Hams flflows

The proof is easy and omitted.

Lemma 4.3. Let be afor $0<\beta<1$ and $K>0$ , that
$S$

$C_{[0,1]^{-}}$$P(S\cap[t, t+\epsilon]\neq\emptyset)\leq K\epsilon^{\beta}$

valued random variable and assume,
and $t\in[0,1]$ .

for all $0<\epsilon<1$

Then, $P(dimS\leq 1-\beta)=1$ .

Proof.

For every $ a>1-\beta$ , we have$E(\sum_{k=1}^{n}1_{\{S\cap[\frac{k-1}{n},\frac{k}{n}]\neq\emptyset\}}\cdot(\frac{1}{n})^{a})$($S\cap[\frac{k-1}{n},$

).

$=$

$\sum_{k=1}^{n}P$$\leq$$nK(\frac{1}{n})^{\beta}\cdot(\frac{1}{n})^{a}=K\cdot n^{1-(\beta+a)}\rightarrow 0$$\frac{k}{n}]\neq\emptyset$

Hence, there exists a subsequence$(\frac{1}{n})^{a}$as$\sum_{k=1}^{n_{\nu}}1_{\{S\cap[\frac{k-1}{n_{\nu}},\frac{k}{n_{\nu}}]\neq\emptyset\}}\cdot(\frac{1}{n_{\nu}})^{a}\rightarrow 0$$C_{l/}$$ n\rightarrow\infty$$lJ$$\rightarrow\infty$$S$$a.s.$.

, $k=1$ ,, , which. Thenis a covering of$E_{k}=[\frac{k-1}{n_{\nu}}, \frac{k}{n_{\nu}}]$$\sum_{E_{k}\in C_{\nu}}(diam ^{E_{k}})^{a}\rightarrow 0$.

such that, almost surely,$ n_{\nu}\rightarrow\infty$Let
be the collection of intervals
have nonempty intersections with the set
andas$\ldots$$n_{lJ}$$S$$C_{\nu}$

, as

$lJ$

$\rightarrow\infty$.

Hence, $dimS\leq a$ , $a.s.$ , implying that $dimS\leq 1-\beta$ , $a.s$ .

Proof of Th. 1.3.
(4.1)$\blacksquare$

It is suﬃcient to show that
$dimS_{\Phi}\leq\frac{1-\alpha}{2-\alpha}$

$a.s$

.

, then byfor. Indeed, if (4.1) is true forthe stationarity of the flow, it is also true for. By
Lemma 4.2, (4.1) is true for a finite product of such ’s. Since linear, we can conclude
combinations of such products are dense in.
by Lemma 4.1 that (4.1) is true for any
First, we consider the case whenis given by$\Phi\in L_{2}^{us}(\mathcal{F}_{0,1}^{X})$$\Phi\in L_{2}^{us}(\mathcal{F}_{0,1}^{X})$$\Phi\in L_{2}^{us}(\mathcal{F}_{n,n+1}^{X})$$\Phi$$L_{2}(F_{-\infty,\infty}^{X})$$\Phi\in L_{2}^{us}(\mathcal{F}_{-\infty,\infty}^{X})$$\Phi\in L_{2}^{us}(F_{0,1}^{X})$$\Phi=f(X_{0,1}(x_{1}), \ldots, X_{0,1}(x_{n}))$,$x_{1}$,$\ldots$, $x_{n}\in R$ ,

370

J. Warren and S. Watanabe

and a function is uniformly Lipschitz-continuous on.
Let $F=[t, t+\epsilon]$ , 0 $\leq t<t+\epsilon\leq 1$ , and let $(X, X’)$ be a$(1^{-}, F)$ -joining.
Then we know by Lemma 2.2 that $ 2P(S_{X}^{acc}\cap F\neq$$\emptyset)=E(|X_{0,1}(0)-X_{0,1}’(0)|^{2})$ and similarly, we have $2P(S_{\Phi}^{acc}\cap F\neq\emptyset)=$$E(|\Phi-\Phi’|^{2})$ where $\Phi’=f(X_{0,1}’(x_{1}), \ldots, X_{0,1}’(x_{n}))$ .
Therefore, notingthat $E(|X_{0,1}(x)-X_{0,1}’(x)|^{2})$ is independent of , we have$R^{n}$$f$$x$$P(S_{\Phi}^{acc}\cap F\neq\emptyset)=\frac{1}{2}E(|\Phi-\Phi’|^{2})$(4.2)$\leq$$KE(|X_{0,1}(0)-X_{0,1}’(0)|^{2})=2KP(S_{X}^{acc}\cap F\neq\emptyset)$

where a constant $K$ depends on and the Lipschitz constant of .
be the reflecting -diﬀusion on. As in theLet
proof of Cor. 1.1, take a canonical scale as the coordinate so that.
and we have
as
as
and
Let $\mu(d\xi)=da(\xi)$ . By what we have shown above,
$f$

$n$

$\hat{L}$$\{\hat{\xi}^{+}(t),\hat{P}_{\xi}\}$$[0, \infty)$$\hat{L}=$$\xi$$ a(\xi)\wedge\xi^{\alpha/(1-\alpha)}\vee$$\frac{d^{2}}{a(\xi)d\xi^{2}}$$a(\xi)\rightarrow 1$$\xi\rightarrow 0$$\xi\rightarrow\infty$$P(S_{X}^{acc}\cap[t, t+\epsilon]\neq\emptyset)=P(\overline{S}\cap[t, t+\epsilon]\neq\emptyset)$$=$$\int_{0}^{1}\hat{P}_{\mu}$($\hat{\xi}^{+}(u-s)=0$$=$$\int_{t}^{1}\hat{P}_{\mu}$($\hat{\xi}^{+}(\theta)=0$$=$$o(\epsilon)+\int_{t}^{1}\hat{P}_{\mu}(\hat{\xi}^{+}(\theta)=0$

for some$s$$\in[0, u]\cap[t, t+\epsilon]$

) du

)

for some $\theta\in[(u-t -\epsilon)_{+}, u-t]$ dufor some $\theta\in[u-t, u-t+\epsilon])$ du.

We would show
(4.3)
$I(t):=\int_{t}^{1}\hat{P}_{\mu}(\hat{\xi}^{+}(\theta)=0$

as$\epsilon\rightarrow 0$

for some $\theta\in[u-t, u-t+\epsilon])$$du=O(\epsilon^{\frac{1}{2-\alpha}})$uniformly in $t\in[0,1]$ . If we can show this, then
$P(S_{X}^{acc}\cap[t, t+\epsilon]\neq\emptyset)=O(\epsilon^{1/(2-\alpha)})$asuniformly in $t\in[0,1]$ and, combining this with (4.2), we seethat, so that, by Lemma 4.3, we
can conclude that the estimate (4.1) holds for because 1-1/(2-\mbox{\boldmath $\alpha$})=$\epsilon\rightarrow 0$$P(S_{\Phi}^{acc}\cap[t, t+\epsilon]\neq\emptyset)=O(\epsilon^{1/(2-\alpha)})$$\Phi$$(1-\alpha)/(2-\alpha)$

.

371

On spectra of noises associated with Harris flflows

To obtain (4.3), we estimate$I(t)\leq\int_{0}^{1}\hat{P}_{\mu}(\hat{\xi}^{+}(\theta)=0$

for some $\theta\in[u, u+\epsilon])$ du$du\leq e\int_{0}^{1}e^{-u}\hat{E}_{\mu}(\hat{P}_{\hat{\xi}(u)}+[\hat{\sigma}_{0}\leq\epsilon])$$=$$\int_{0}^{1}\hat{E}_{\mu}(\hat{P}_{\hat{\xi}(u)}+[\hat{\sigma}_{0}\leq\epsilon])$$\leq$$e\int_{0}^{\infty}e^{-u}\hat{E}_{\mu}(\hat{P}_{\hat{\xi}(u)}+[\hat{\sigma}_{0}\leq\epsilon])$$=$$ e\int_{[0,\infty)}\mu(d\xi)\int_{[0,\infty)}\overline{g}_{1}(\xi, \eta)\hat{P}_{\eta}[\hat{\sigma}_{0}\leq\epsilon]a(\eta)d\eta$

where

du

du
,

is the first hitting time of
to 0. Since the resolvent density
is bounded, we have, for some $C>0$ ,$\hat{\xi}^{+}(t)$$\hat{\sigma}_{0}$$\tilde{g}_{1}(\xi, \eta)$$ I(t)\leq C\int_{[0,\infty)}\hat{P}_{\eta}[\hat{\sigma}_{0}\leq\epsilon]a(\eta)d\eta$.

, $\eta>0$ , and in the coordinate , is obtained
The process
underfrom a one-dimensional Brownian motion $B(t)$ with $B(0)=0$ by$\hat{\xi}^{+}(t)$$\hat{P}_{\eta}$$\hat{\xi}^{+}(t)=|\eta+B(A^{-1}(t))|$

$\xi$

where

$A(t)=\int_{0}^{t}a(|\eta+B(s)|)ds$ .

Hence,$\hat{P}_{\eta}(\hat{\sigma}_{0}\leq\epsilon)=P(\int_{0}^{\sigma_{0}}a(|\eta+B(s)|)ds\leq\epsilon)$

where
and, noting$a(\xi)\geq K^{-1}\cdot\xi^{\alpha/(1-\alpha)}\wedge 1$$\sigma_{0}=\min\{s|\eta+B(s)=0\}$.

for some $K>0$ ,$\hat{P}_{\eta}(\hat{\sigma}_{0}\leq\epsilon)\leq P(\int_{0}^{\sigma_{0}}(|\eta+B(s)|^{\alpha/(1-\alpha)}\wedge 1)ds\leq K\epsilon)$.

The scaling property of $B(t)$ combined with an easy inequality
for $a>0$ and $1\geq\epsilon>0$ yields that the RHS is dominated by, where$(\epsilon a)\wedge 1\geq$$\epsilon(a\wedge 1)$$\phi(\epsilon^{-(1-\alpha)/(2-\alpha)}\eta)$$\phi(\eta)=P(\int_{0}^{\sigma_{0}}(|\eta+B(s)|^{\alpha/(1-\alpha)}\wedge 1)ds\leq K)$

.

372

J. Warren and S. WatanabeThen,$ I(t)\leq C\int_{[0,\infty)}\phi(\epsilon^{-(1-\alpha)/(2-\alpha)}\eta)a(\eta)d\eta$$\leq$$ K’\int_{[0,\infty)}\phi(\epsilon^{-(1-\alpha)/(2-\alpha)}\eta)\eta^{\alpha/(1-\alpha)}d\eta$$=$$ K’\epsilon^{1/(2-\alpha)}\int_{[0,\infty)}\phi(\eta)\eta^{\alpha/(1-\alpha)}d\eta$

and we have otained (4.3).
,
In the same way, we have the estimate (4.1) for $\Phi=f(X_{s,t}(x_{1}),$$X_{s,t}(x_{n}))$ ,$<t$ , where,, $x_{n}\in R$ ,is uniformly Lipschitz con. Then, by Lemma 4.2, we have the estimate (4.1) for
tinuous on
if $t_{0}=0<t_{1}<t_{2}<\cdots<t_{m}=1$ ,, $k=1,2$ ,, $m$ , is given in the form
and$\ldots$$x_{1}$$f$$s$$\ldots$$R^{n}$$\Phi=\Phi_{1}\Phi_{2}\cdots\Phi_{m}\in L_{2}^{us}(\mathcal{F}_{0,1}^{X})$$\Phi_{k}\in ub[L^{2}(\mathcal{F}_{t_{k-1},t_{k}}^{X})]$$f_{k}$($X_{t_{k-1},t_{k}}(x_{1}^{(k)})$,$X_{t_{k-1},t_{k}}(x_{n_{k}}^{(k)})$$\ldots,$$\Phi_{k}=$$\ldots$),$x_{1}^{(k)}$,$\ldots$,$x_{n_{k}}^{(k)}\in R$, where$f_{k}$

is. By Lemma 4.1 (i), the estimate
uniformly Lipschitz continuous on(4.1) still holds for a finite linear combination of such functionals andthis class of functionals is dense in.
$R^{n_{k}}$$L_{2}^{us}(F_{0,1}^{X})$$\blacksquare$

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Jon Warren
Department of Statistics, University of Warwick
Coventry, CV47AL, United Kingdom

Shinzo Watanabe
527-10, Chaya-cho, Higashiyama-ku
Kyoto, 605-0931, Japan