数理解析研究所講究録 1227 巻 2001 年 83-95

83
Stochastic Evolutions
As Boundary Value Problems
V. P. BELAVKINSCHOOL OF MATHEMATICSNOTTINGHAM UNIVERSITYNOTTINGHAM NG7 2RD, UK
AND

V. N. KOLOKOL’TSOVDEPARTMENT OF MATHEMATICS,
STATIST1CS AND OPERATIONAL RESEARCHNOTTINGHAM TRENT UNIVERSITYBURTON STREET, NOTT1NGHAM NG1 4BU, UK

Abstract. Solutions to linear stochastic and quantum stochastic equations are proved
to provide the interaction representations for the solutions to certain Dirac type equations with boundary conditions in pseudo Fock spaces. The latter are presented as thesemi-classical limit of an appropriately dressed unitary evolutions corresponding to aboundary-value problem for general Schr\"odinger equations.

Mathematics Subject Classifications (1991). Primary$60\mathrm{H}10,81\mathrm{S}25,35\mathrm{Q}40$.

Key Words, stochastic equations, quantum stochastic equations, boundary value
problems, pseudo Fock space, stochastic limit.

1Introduction
We are going to show that the solutions to awide class of stochastic and quantum stochastic
equations can be obtained from apositive (relativistic or non-relativistic) Hamiltonian withsingular interaction as astrong limit of the input flow of quantum particles with asymptotically infinite momentum but aconstant velosity. Thus the problem of stochastic approximation is reduced to asort of quasiclassical asymptotics of aquantum mechanical boundary
value problem in extra dimension.
There exists abroad literature on the stochastic limit in quantum physics in which quantum stochastics is derived from anonsingular interaction (see monograph [1], and references
therein). Here we follow adifferent approach recently outlined in [3]: instead of rescaling theinteraction potentials we treat the singular interaction -potentials rigorously as the boundary conditions, and obtain the stochastic limit as an ultra-relativistic limit of aSchr\"odinger
boundary value problem in aHilbert space of infinite number of particles$\delta$

84
We start with ashort Section 2fixing some general notations that are used throughout

the paper.
In Section 3, we show how the solutions to linear stochastic differential equations in
Banach spaces driven by acompound Poisson process can be obtained as the interaction
picture representation for the boundary- value problem for shifts in Euclidean simplices.
This class of equations includes the Belavkin quantum filtering equations describing theaposterior dynamics of aquantum system under continuous non-demolition measurement ofcounting type [2].
Since the stochastic equations driven by aWiener process can be obtained as the limitsof equations driven by compound Poisson process, the results of Section 3allow the representation of the solutions to linear diffusion equations in Banach spaces as the limits ofcertain (deterministic) boundary value problems.
After Section 4that describes shortly the combinatorics of multiple Fock spaces, we
obtain our main results in Sections 5, where we establish aconnection between boundary
value problem and general quantum stochastic equations (and, in particular, classical diffusion equations) directly, without alimiting procedure. To this end, we develop atheory ofboundary value problems for shifts in “coloured simplices” as the restrictions of the shifts
pseudo Fock spaces, and then use the theory developed in [4], where it isin multiple
shown that working in multiple pseudo Fock spaces allows for arepresentation of stochastic
and quantum stochastic evolutions that preserves the number of particles (though changes
their colour), and consequently reduces astudy of general quantum stochastic flows to thestudy of Poisson driven evolutions in coloured simplices. Using pseudo Fock spaces is acharacteristic feature of our method that distinguishes it from an alternative way of exploiting the connection between boundary value problems and quantum stochastics initiated in[6], (see also [7] for recent developments in this direction). Unlike [6], we systematycallyconsider the evolutions in general Banach spaces (and, in particular, non-unitary boundary
conditions), which are important for applications to general stochatic equations, in particular
those decribing the models of continuous quantum measurements.
In Section 6we show how the boundary value problems for shifts can be obtained by a(which is however quite different from the usual semiclassort of semiclassical limit
sical limit for stochastic equations [8], [10], and which generalises the ultra-relativistic limitof [3] from evolutions described by general Schr\"odinger problems with abounded below
Hamiltonian.
$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$harrow \mathrm{O}$$)$2Main notations(i) Generalities. By $H$ (respectively by ) we shall always denote aHilbert (respectively a). In applications, $B$ will be the Banach(respectively
Banach) space with the norm
in $H$ . For afunction on we shall denote byalgebra of bounded linear operators
as $tarrow z$ (when it) the left (respectively the right) limit of(respectively
the indicator function of
we shall denote byexists, of course). For asubset$\inM$. We denote by $C_{B}(M)$ (resp.
$M$ that equals 1or 0respectively whenorvanishing at infinity (respectively withthe space of continuous functions
acompact support) equipped with the usual sup-norm $|| \psi||=\sup_{z}||\psi(z)||_{B}$ .
$B$$||||_{B}$$||||_{H}$$\phi$$\mathcal{L}(H)$$\phi(z_{-})$$\phi(t)$$\phi(z_{+})$$\chi_{M}(z)$$M\subset \mathrm{R}^{n}$$z$$C_{B}^{\omega mp}(M))$$\mathrm{R}$$z$$\in \mathrm{R}^{n}\backslash M$$M\vdasharrow B$

85(ii) Simplicies. We shall denote by$\Sigma_{n}$the infinite simplex:$\Sigma_{n}=\{z=(z_{1}, \ldots, z_{n})\in \mathbb{R}^{n} :z_{1}<z_{2}<\ldots<z_{n}\}$,equipped with Lebesgue measure. Clearly this simplex can be decomposed into the union of$n+1$ cells:,$\Sigma_{n}^{k}=\{z\in\Sigma_{n} : z_{k}\leq 0\leq z_{k+1}\}$ ,$k=1$ , , $n-1$ ,(1)$\Sigma_{n}^{k}$$\Sigma_{n}^{0}=\{Z \in\Sigma_{n} :z_{1}\geq 0\}$$\Sigma_{n}^{n}=\{z\in\Sigma_{n} :z_{n}\leq 0\}$$\ldots$.

,.
,
In particular,
are usually identified with the subsets
Vectorsof theby the subsets ofreal line of cardinality $|\zeta|=n$ . The representation of the points ofcoordinates)(respectively by -dimensional vectors with orderedis more natural for defining
stochastic processes (respectively, boundary value problems) we are dealing with.
(iii) Banach valued -spaces. For anumber $p\geq 1$ , we denote bythe(more precisely, equivalence classes of such functions)
Banach space of functions
with the norm$\Sigma_{1}=\mathbb{R}$$\Sigma_{1}^{0}=\mathbb{R}_{+}$$\Sigma_{1}^{1}=\mathbb{R}_{-}$$z\in\Sigma_{n}$$\langle$$=\zeta(z)=\{z_{1}, \ldots, z_{n}\}\subset \mathbb{R}$$\Sigma_{n}$$\mathbb{R}$$n$$L^{p}$$L_{B}^{p}(n)=L_{B}^{p}(\Sigma_{n})$$\psi:\Sigma_{n}arrow B$$|| \psi||=(\int_{\Sigma_{n}}||\psi(z)||_{B}^{p}dz)^{1/p}$. By
is the Hilbert tensor productwe denote theIn particular,
corresponding space of functions on the cells (1). We shall also consider the locally convex:
with the countable setof measurable functions
topological spaces$N$( is anatural number) defined as above but with the integration overof norms$\Sigma_{n}\cap\{z : |z| \leq N\}$
(respectively) is reserved for the space. The notation).
(respectively. By $T_{n}(t)=T_{n,p,B}(t)$ we shall denote(iv) Shifts and differentiation operators into the function
which takes afunction
the shift in. This is acontinuous group, whose generator (that acts on smooth functions. In
as differentiation along the vector (1, 1, , 1) we shall denote by.
. We shall write shortly $T(t)$ foris self-adjoint inparticular, the operator(v) Dressing and jumps. Let $-iE$ be agenerator of acontinuous group $\exp\{-itE\}$ in
we. For areal-valued continuous function $f(z)$ on
be an operator in, and letformulabytheindefine an operator$H\otimes L^{2}(\Sigma_{n})$$L_{H}^{2}(n)$$L_{B}^{p,loc}(n)$$\psi$$L_{B}^{p}(\Sigma_{n}^{k})$$\Sigma_{n}\vdasharrow H$$||\psi||_{N}$$L_{B}^{p,loc}$$L_{B}^{p}$$L_{B}^{p}(\mathbb{R})$$L_{B}^{p,loc}(\mathbb{R})$$L_{B}^{p}(n)$$(T_{n}(t)\varphi)(z_{1}, \ldots, z_{n})=$$\varphi\in L_{B}^{p}(n)$$L_{B}^{p}(n)$$\varphi(z_{1}+t, \ldots, z_{n}+t)$$\partial_{z}=\partial_{z_{1}}+\ldots+\partial_{z_{n}}$$)$$\ldots$$L_{H}^{2}(n)$$i\partial$

$B$

$A$

$T_{1}(t)$$L_{B}^{p}(n)$$\Sigma_{n}$$L_{B}^{p}(n)$$A_{Ef(z)}\varphi(z)=e^{iEf(z)}Ae^{-iEf(z)}\varphi(z)$(2).

, then $A_{Ef(z)}$ is
Clearly, if $E$ is aself-adjoint operator in $H$ , and if is self-adjoint in.
also self-adjoint in
we define abounded operator
and aBorel subset
For afunction
by the formulain$A$$L_{H}^{2}(n)$$L_{H}^{2}(n)$$s\subset\Sigma_{n}$$A:\Sigma_{n}\vdasharrow \mathcal{L}(B)$$A^{s}$$L_{B}^{p}(n)$$(A^{s}\varphi)(z)=\chi_{s}(z)A(z)\varphi(z)+(1-\chi_{s}(z))\varphi(z)$(3)). Clearly, the operator
would be of course(more correct, but more heavy notation for
Clearly $T(t)A^{s}=$zero.
measureLebesgueofonasetchangedifisremains the sameis$A^{s-t}T(t)$ and $(A^{s})_{Ef(z)}=(A_{Ef(z)})^{s}$ , which implies, in particular, that the notation
not ambiguous$A^{s}$$A^{s}$$A^{\chi_{s}}$$s$$A_{Ef(z)}^{s}$

86
Stochastic equations driven by aPoisson noise as
boundary value problems

3

be an arbitrary (ordered) family of operators fromLet
be agenerator of acontinuous group of linear operators in . For agiven
consider the linear multiple kick equation$\sigma=\{\sigma_{1}, \ldots, \sigma_{n}\}$$\mathcal{L}(B)$$B$$d \eta+iErjdt=\sum_{j=1}^{n}(\sigma_{j}-1)\eta d\chi_{(z_{j},\infty)}$,$\eta\in B$,$t\geq 0$

and let $-iE$, let us$z\in\Sigma_{n}^{0}$.

(4)

As in the case of stochastic equations, this equation should be understood rigorously as thecorresponding integral equation, where for any function $f(t)$ having everywhere right andis defined by the formulaleft limits the integral with respect to$d\chi(z,\infty)(t)$$\int_{0}^{t}f(\tau)d\chi_{(z,\infty)(\mathcal{T})=}\{$

0,$t\leq z$$f(z_{-})$

Ifis arandom variable on aprobability spacestochastic equation$z\in\Sigma_{n}^{0}$$d\eta+iE\eta dt=(\sigma_{n_{t}}-1)\eta dn_{t}$,$\Omega$

,

,

(5)

$t>z$ .

, one can rewrite equation (4) as the$\eta\in B$,$t\geq 0$,(6)

driven by the counting process $n_{t}(\zeta)=n_{t}(\zeta(z))=|\zeta\cap[0, t)|$ .
between thesatisfying (4) evolves according to the free equation
Clearly.
jump-times , and at the times $t=z_{k}$ the wave function experiences the jump
This proves the following$\Delta\partial\partial t=-iE\eta$$\eta(t)$$\eta-*\sigma(z_{k})\eta$$z_{k}$, the operator $V(t, z)$ , which gives the solution $V(t, z)\eta_{0}$ to
Proposition 1For any
and hasthe Cauchy problem for equation (4) with the initial function , belongs to
the following explicit form: for $z_{k}\leq t<z_{k+1}$$z$$\in\Sigma_{n}^{0}$$\mathcal{L}(B)$$\eta_{0}$$V(t, z)=\exp\{-iE(t-z_{k})\}\sigma_{k}\exp\{-iE(z_{k}-z_{k-1})\}\sigma_{k-1}\ldots\sigma_{1}\exp\{-iEz_{1}\}$

If

$B$

is a Hilbert space $H$ and if all$\sigma_{j}$.

(7)

are unitary, then $V(t, z)$ is also a unitary operator.

We are going to give arepresentation for this operator in terms of the boundary value, more precisely, in terms of the solutions to the equation
problem for shifts in$L_{B}^{p}(n)$$i\partial_{t}\varphi=(i\partial_{z}+E)\varphi=(i(\partial_{z_{1}}+\partial_{z_{2}}+\ldots+\partial_{z_{n}})+E)\varphi$,$\varphi\in L_{B}^{p}(n)$,(8)

combined with the boundary conditions,$\varphi(z_{1}, \ldots, z_{k-1},0_{-}, z_{k+1}, \ldots, z_{n})=\sigma_{k}\varphi(z_{1}, \ldots, z_{k-1},0_{+}, z_{k+1}, \ldots, z_{n})$$k=1$ ,$\ldots$

, .
$n$

(9)

with thedenote the dense subspace of functions
Let
properties:
has acontinuous version such that on all lines(i) for each $k=0$ , , the restriction
.
is
,
1)
this restiction absolutely continuous andparallel to the vector (1,(ii) the boundary conditions (9) are satisfied.
acts on .in the obvious way. The
The differentiation operator
inspectionbyisobtainedfollowing result$\varphi\in L_{B}^{p}(n)$$D_{\sigma}=D_{\sigma_{1},\ldots,\sigma_{n}}(p, B)$$\ldots$$n$$\varphi|_{\Sigma_{n}^{k}}$$\partial_{z}\varphi|_{\Sigma_{n}^{k}}\in L_{B}^{p}(n)$$\ldots$$\partial_{z}^{\sigma}=\partial_{z_{1}}^{\sigma_{1}}+\ldots+\partial_{z_{n}}^{\sigma_{n}}$$D_{\sigma}$

87
Proposition 2For an arbitrary p$\geq 1$, the operators$T^{\sigma}(t)$defined by the formulas(10)$T^{\sigma}(t)=\sigma_{n}^{\{z_{n}\in[-t,0]\}}\ldots\sigma_{1}^{\{z_{1}\in[-t,0]\}}T_{n}(t)=T_{n}(t)\sigma_{n}^{\{z_{n}\in[0,t]\}}\ldots\sigma_{1}^{\{z_{1}\in[0,t]\}}$

form a continuous semigroup of operators inoperator
defined initially on
action of the semigroup.

$L_{B}^{p}(n)$$\partial_{z}^{\sigma}=\partial_{z_{1}}^{\sigma_{1}}+\ldots+\partial_{z_{n}}^{\sigma_{n}}$$D_{\sigma}$, whose generator is the closure of the. The spaceis invariant under the$D_{\sigma}$gives the solution to the Cauchy problem of equation (8), (9) for
The semigroup$E$ . To include anon-trivial $E$ , let us introduce the domain
vanishing$T^{\sigma}$$D_{\sigma}^{E}=D_{\sigma}\cap\{\varphi :which is clearly dense in$L_{B}^{p}(n)$E\varphi\in L_{B}^{p}(n)\}$,for any . The operator
$p$

(11)

$\partial_{z}^{\sigma}-iE=\partial_{z_{1}}^{\sigma_{1}}+\ldots+\partial_{z_{n}}^{\sigma_{n}}-iE$

in the obvious way. By usual abuse of notations, we shall denote by the sameacts onthe closure of this operator. The following statement is again proved bysymbolinspection.
$D_{\sigma}^{E}$$\partial_{z}^{\sigma}-iE$

Proposition 3(i) The operator (11) generates a continuous semigroup$U_{E}^{\sigma}(t)=(\sigma_{n}^{\{z_{n}\in[-t,0]\}})_{Ez_{n}}\ldots(\sigma_{1}^{\{z_{1}\in[-t,0]\}})_{Ez_{1}}\exp\{-iEt\}T_{n}(t)$

(12)

,

(where we used notations (2), (3)) which solves the Cauchy problem for equations (8), (9).
Moreover,(13).
$(T_{n}(t))^{-1}U_{E}^{\sigma}(t)=e^{-iEt}(\sigma_{n}^{\{z_{n}\in[0,t]\}})_{Ez_{n}}\ldots(\sigma_{1}^{\{z_{1}\in[0,t]\}})_{Ez_{1}}$(ii) The operators (12) are invertible for all $t\geq 0$ if and only if all , $k=1$ , , ,has
are similar. More precisely, if
are invertible. Operators (11) with invertible
$k=1$
,
,
then
,
,
continuous inverse$\sigma_{k}$$\sigma_{k}$$\sigma_{k}$$\sigma_{k}^{-1}$$\ldots$$\ldots$

$n$

$a$

$n$

$U_{E}^{\sigma}(t)=(\sigma_{n}^{\{z_{n}<0\}})_{Ez_{n}}\ldots(\sigma_{1}^{\{z_{1}<0\}})_{Ez_{1}}$

$x$

(14)

$\exp\{-iEt\}T_{n}(t)((\sigma_{1}^{\{z_{1}<0\}})_{Ez_{1}})^{-1}\ldots((\sigma_{n}^{\{z_{n}<0\}})_{Ez_{n}})^{-1}$

are unitary, the operator(Hi) If $B$ is a Hilbert space $H$ and all.
in(iv) All statements of the Proposition remain valid in the spaces$\sigma_{k}$$i\partial_{z}^{\sigma}+E$

is self-adjoint$L_{H}^{2}(n)$$L_{B}^{p,loc}(n)$.

Comparing formulas (7) and (13) yields the following result.
Theorem 1Solution (7) to the Cauchy problem of equation (4) can be written in the form
$V(t, z)\eta_{0}=((T_{n}(t)^{-1}U_{E}^{\sigma}(t)\varphi)(z),$uthere$\varphi(z)=\eta_{0}$for$z\in\Sigma_{n}^{0}$and vanishes otherw $ise$ .

(15)

88
which equals to aconstant vector 770 for all positive does
Remark. The function, which was the main reason for introducing these spaces.
, but only to
not belong to
In physical language, one interpretes formula (15) by saying that the stochastic evolution$V(t, z)$ gives the solutions of equation (8), (9) in the interaction representation with respect to. Notice that stochastic linear equations driven by acompound Poisson
the “free” shiftnoise, in particular the quantum filtering equations describing the aposterior dynamics ofquantum states under continuous observations of counting tyPe, can be reduced to equationof type (4) or (6) pathwise, because aPoisson process has almost surely afinite number ofjumps on each bounded time interval. Therefore, the solutions to these stochastic equations
can be obtained as the interaction representation of the solutions of problem (8), (9) with. Theorem 4at the end of the paper shows that the model
respect to the “free” shift(8), (9) in its turn can be obtained as asemiclassical limit of Schrodinger evolutions with abounded below Hamiltonian.
$\varphi(z)$$z$$L_{B}^{p,loc}$$L_{B}^{p}$$T_{n}(t)$$T_{n}$

4Combinatorics of the multiple Fock space
This is an auxiliary section describing the combinatorics of secondly quantised operators in
multiple Fock spaces in away that is convenient for our purposes.
By acoloured simplex of particle having $m$ colours we understand the set$n$$CS_{n,m}= \bigcup_{n_{\grave{1}}+\ldots+n_{m}=n^{\Sigma}n_{1}}\cross\ldots\cross\Sigma_{n_{m}}$,where the (disjoint) union is taken over all partitions of the integer number in the sum of$m$ non-negative numbers (the order is relevant), and where it is assumed that the product isover all non-vanishing . The points of $CS_{n,m}$ can be parametrised either by ordered chainsof labeled variables$n$$n_{j}$$z$$=\{z^{\alpha}\}=\{z_{1}^{\alpha(1)}<\ldots<z_{n}^{\alpha(n)}\}$

,

(16)

(that label the variables in astandard
being functions
with), or by the families of $m$ vector variables
simplex$\alpha:\{1, \ldots, n\}\vdasharrow\{1, \ldots, m\}$$\alpha$$\Sigma_{n}$$\zeta=\{\zeta^{1}=(z_{1}^{1}, \ldots, z_{n_{1}}^{1}), \ldots, \zeta^{m}=(z_{1}^{m}, \ldots, z_{n_{m}}^{m})\}$

,

(17)

(each can be thus considered either
where the entries of each ar ordered:), and where the subsets (;’ areor as asubset of of cardinality
as avector indisjoint.
, whichThere is anatural projection from $CS_{n,m}$ to the standard (uncoloured) simplex$pr(z)$
(16)
point
of
the
) the image(orsimply “forgets” the colour. We shall denote by(or (17)) under this projection.
allows one to present any, $j=1$ , , $m$ , in
. Choosing abasis
Let. As usual, the tensor productfunction $f\in l^{m}$ as the sum $f= \sum f_{j}e_{j}$ with allparametrised by
basis
with
the
space
O...
is defined as a $mn$ -dimensional vector. The (algebraic) symmetric tensor productfunctionsgenerated by the monomials
can be defined as the space of functions$\zeta^{j}$$z_{1}^{j}<\ldots<z_{n_{j}}^{j}$$\mathrm{e}$$\zeta^{j}$$|\zeta^{j}|=n_{j}$$\mathbb{R}$$\Sigma_{n_{j}}$$\Sigma_{n}$$pr(\zeta)$$\{e_{j}\}$$l^{m}=C_{\mathbb{C}^{m}}^{comp}(\mathbb{R})$$\mathbb{C}^{m}$$\ldots$$(\mathbb{C}^{m})^{\otimes n}$$f_{j}\in l^{1}$$\otimes e_{\alpha(n)}$$e_{\alpha(1)}$$\alpha:\{1, \ldots, n\}\vdasharrow\{1, \ldots, m\}$$l_{n}^{m}=(l^{m})_{sym}^{\otimes n}$$\Sigma_{n}\vdash\Rightarrow(\mathbb{C}^{m})^{\otimes n}$$f_{1}(z_{1})\ldots f_{n}(z_{n})e_{\alpha(1)}\otimes\ldots\otimes e_{\alpha(n)}$,$f_{j}\in l^{1}$

.

(18)

89
It is convenient to get rid of tensors by transfering the index from the basis to the variablesof ordered labelled
and to encode the element (18) by the function
is represented as aspace ofvariables. Thus the symmetric tensor productaspacewords,
of functions on the colouredasfunctions of variables (16), or, in othersimplex $CS_{n,m}$ . We shall call this representation the functional representation for the tensor
can be identified with aspace of $H$ -valued functions
product . Similarly, the space $H$ C&
on $CS_{n,m}$ . Let$f_{1}(z_{1}^{\alpha(1)})\ldots f_{n}(z_{n}^{\alpha(n)})$$n$$l_{n}^{m}=(l^{m})_{sym}^{\otimes n}$$l_{n}^{m}$$l_{n}^{m}$$S=\{S(k)\}=\{(S_{\mu,\nu})(k) :k\in\{1, \ldots, n\}, \mu, \nu\in\{1, \ldots, m\}\}$

(19)

,

. Clearly, each matrix $S(k)$ defines anbe afamily of $m\cross m$ -matrices with entries fromthat takes the function $f(z)= \sum f_{j}(z)e_{j}$ to the function
operator in$\mathcal{L}(\mathcal{H})$$H\otimes l^{m}$$(S(k)f)(z)= \sum_{\mu,\nu}S_{\mu,\nu}(k)f_{\nu}(z)e_{\mu}$.

The tensor product $S(n)$ (&... 6 $S(1)$ is defined as the operator inelement (18) to$H\otimes l_{n}^{m}$$\sum_{\mu_{1},\ldots,\mu_{n}}S_{\mu_{n},\alpha(n)}(n)\ldots S_{\mu_{1},\alpha(1)}(1)f_{1}(z_{1})\ldots f_{n}(z_{n})e_{\mu 1}\otimes\ldots\otimes e_{\mu_{n}}$that takes the(20).

Clearly, each monomial in the sum (20) has the form
$S_{\mu_{n},\alpha(n)}(n)\ldots S_{\mu_{1},\alpha(1)}(1)f_{1}(z_{1}^{\mu_{1}})\ldots f_{n}(z_{n}^{\mu_{n}})$,in the functional representation. Since the monomials (18) form the basis for the space
we obtain the following functional representation for the operator $S(n)\otimes\ldots\otimes S(1)$ :$l_{n}^{m}$,

Proposition 4$(S(n) \otimes\ldots\otimes S(1)f)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})=\sum_{\beta}\prod_{k}S_{\alpha(k)\beta(k)}(k)f(z_{1}^{\beta(1)}, \ldots, z_{n}^{\beta(n)})$,where the sum is taken over all functions $\beta:\{1, \ldots, n\}-$ 1,
product, where the index decreases from the left to the right.
$\{$$\ldots$, $m\}$ , and$\prod$,(21)

is the ordered$k$is called the (algebraic) symmetric Fock space
The infinite (algebraic) direct sum
being the -particle subspace. In particular, if $S(j)=S$ does notover , the spaceis the restriction on the -particle subspace of the second quantization
depend on , thenof the operator .
$\oplus_{n=0}^{\infty}l_{n}^{m}$$l^{m}$$l_{n}^{m}$$S^{\otimes n}$

$j$

$n$

$n$

$S$

5General stochastic evolutions as boundary value
problems.
In this section, we shall study the boundary value problem in coloured simplices, whenjumps
may not only change the value of afunction in apoint, but also acolour of this point. Using
the combinatorics of multiple Fock spaces from the previous section yields the key conclusio$\mathrm{n}$90that the interaction representation for the evolutions given by the boundary value problems
for shifts in coloured simplices is given by the solutions to the purejump stochastic equations
in multiple pseudo Fock spaces. This will be the key ingredient needed for the representationof general quantum stochastic evolutions as boundary value problems for shifts in pseudo
Fock spaces.
First we describe cetain topologies on algebraic Fock spaces from the previous sectionthat are parametrised by $m$-tuples $p=\{p_{1}\leq\ldots\leq p_{m}\}$ of positive numbers, whereandis allowed to be +00. To be concrete, we suppose that $p_{1}\leq\ldots\leq p_{m-1}<p_{m}=\infty$). Let us define anorm on the space(see [5] for the caseconsidered in thefunctional representation (see previous section), i.e. as aspace of functions:$p_{1}\geq 1$$p_{m}$$p_{m}\neq\infty$$H\otimes l_{n}^{m}$$\varphi:CS_{n,m}\vdash\Rightarrow H$$|| \varphi||_{p}=\sum(\int_{\Sigma_{n_{1}}}($...

$( \max_{\Sigma_{n_{m}}}|\varphi(\zeta^{1}, \ldots, \zeta^{m})|)^{p_{m-1}}\ldots)^{p1/p_{2}}d\zeta^{1})^{1/p_{1}}$

,

(22)

where
is taken over all partition $n=n_{1}+\ldots+n_{m}$ of . We shall denote by $L_{H}^{p}(CS_{n,m})$the completion of the spacewith respect to the norm (22). By
weshall denote the corresponding locally convex space defined by the countable set of norms
parametrised by the positive integers $N$ and defined by (22) with all integrations performed
not over the whole infinite simplices but over their intersections with the balls of radius $N$
We shall use the same notation $T_{n}(t)=T_{n,p,H}(t)$ as before for the shift in $L_{H}^{p}(CS_{n,m})$ orthat shifts all variables independently of their colours. The spaces $L_{H}^{p}(CS_{n,m})$can be considered as -particle subspaces in the (multiple) Fock space$n$$\sum$$L_{l\mathrm{f}}^{p,loc}(CS_{n,m})$$H\otimes l_{n}^{m}$$L_{H}^{p,loc}(CS_{n,m})$$n$$\mathcal{F}_{H}^{\varphi}=H\oplus L_{H}^{p}(CS_{1,m})\oplus L_{H}^{p}(CS_{2,m})\oplus\ldots$

,

(23)

considered in thewhich is aBanach version of the algebraic Fock spacecan be considered as the restrictions
previous section. In particular, the shifts.
(to the -particle subspaces) of the corresponding shifts in the Fock space
Consider now the family of operators (19) under additional assumption that all $S(k)$ areor (ii) $\mu>\nu$
is allowed not to vanish only if either (i)block upper triangular (i.e.
afamilyinthe space
operatorsoflinearoperators
define
).
these
Then
but
that act by the formula$H\otimes(\Sigma_{n=0}^{\infty}l_{n}^{m})$$T^{\sigma_{1,\ldots:}\sigma_{n}}(t)$$n$$\mathcal{F}_{H}^{p}$$\mu\leq\nu$$S_{\mu,\nu}$$\sigma$$p_{\mu}=p_{\nu}$$\sigma_{k}$$H\otimes l_{n}^{m}$$(\sigma_{k}\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})$.

$= \sum_{\nu=\alpha(k)}^{m}S_{\alpha(k),\nu}(k)\varphi(z_{1}^{\alpha(1)}, \ldots, z_{k-1}^{\alpha(k-1)}, z_{k}^{\nu}, z_{k+1}^{\alpha(k+1)}, \ldots, z_{n}^{\alpha(n)})$(24). However, since the
These operators may not be continuous in the spaces, the standard
norm of any function
matrices $S(k)$ are triangular and since for any
norm, the following statement holds.
on acompact set can be estimated by its$L_{H}^{p}(CS_{n,m})$$L^{p1}$$p_{1}\leq p_{2}$$L^{p2}$

Proposition 5The operators$\sigma_{k}$are continuous in$L_{f\acute{f}}^{ploc}(CS_{n,m})$for all .
$p$Generalising the boundary value problem (8), (9), we are going to consider the equation(8) in $L_{H}^{p}(CS_{n,m})$ combined with the boundary conditions$\varphi(z_{1}^{\alpha(1)}, \ldots, z_{k-1}^{\alpha(k-1)}, 0_{-}^{\alpha(k)}, z_{k+1}^{\alpha(k+1)}, \ldots, z_{n}^{\alpha(n)})$91$=(\sigma_{k}\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{k-1}^{\alpha(k-1)}, 0_{+}^{\alpha(k)}, z_{k+1}^{\alpha(k+1)}, \ldots, z_{n}^{\alpha(n)})$,$k=1$ ,$\ldots$

, .

(25)

$n$

To deal with this problem in the same way as with the problem (8), (9), let us decompose
using the decomposition (1)the coloured simplex $CS_{n,m}$ into the union of $n+1$ cells:of the underlying uncoloured simplex$CS_{n,m}^{k}$$\Sigma_{n}$$CS_{n,m}^{k}=\{z\in CS_{n,m} :pr(z)\in\Sigma_{n}^{k}\}$$k=0$ ,,$\ldots$, ,$n$) of functions
and then define the subspaces $D_{S}=D_{S}(p, H)$ (respectively
properties:
)
with
the
(respectively from(i) for each $k=0$ , , and each particion $n=n_{1}+\ldots+n_{m}$ the restriction of$D_{S}^{loc}$

from$\varphi(z)$$L_{H}^{p,loc}(CS_{n,m})$$L_{H}^{p}(CS_{n,m})$$\ldots$$n$$\varphi$on$CS_{n,m}^{k}\cap\Sigma_{n_{1}}\cross\ldots\cross\Sigma_{n_{m}}$has acontinuous version such that it is absolutely continuous on all lines parallel to thebelongs to $L_{H}^{p}(CS_{n,m})$vector (1, 1, , 1) and such that
(respectively $L_{H}^{p,loc}(CS_{n,m})$ ),(iii) the boundary conditions (25) are satisfied.
for the closures of the operator defined on the domains
Let us use the same notation. We introduced the notations for coloured simplices in such away that the mainorformulas of the previous section still make sense in this new framework. It remains only
to assume that the use of the operator-valued functions of the variables without acolour
means that the colour is preserved. For example, the action of the operator $\exp\{-iEz_{j}\}$ ,say, is given by the formula$(\partial_{z}\varphi)(z)=((\partial_{z_{1}}+\ldots+\partial_{z_{n}})\varphi)(z)$$\ldots$$\partial_{z}$$\partial_{z}^{S}$$D_{S}$$D_{S}^{loc}$$z$$(\exp\{-iEz_{j}\}\varphi)(\{z_{1}^{\alpha(1)}<\ldots<z_{n}^{\alpha(n)}\})=\exp\{-iEz_{j}^{\alpha(j)}\}\varphi(\{z_{1}^{\alpha(1)}<\ldots<z_{n}^{\alpha(n)}\})$.

quite similarly to the case without colours.
At last, we can define the operator
Moreover, due to Proposition 5we get the following$i\partial_{z}^{S}+E$

Proposition 6Propositions 2and 3remain valid for spaces
for all$S_{mm}(j)$.
are identical operators inan additional assumption that all elements$L_{H}^{p,loc}(CS_{n,m})$

$p$

under

$\mathcal{L}(H)$

Remark. The last assumption was necessary, because jumps destroy the continuity ofspaces, because shifts are not continuous there). The
afunction (and one can not use.
assumption $S_{mm}(j)=1$ ensures that there will be no discontinuity in the variables
shiftsComparing formula (21) with the formula for the boundary value problem fordefined in (24)) and by straightdescribed in Proposition 6(see, in particular, (10) withforward generalisation of Proposition 1one obtains the following result, which connects shifts
in coloured simplices with pure jump stochastic equations and with secondly quantised operators in multiple (Banach) Fock spaces.
$L^{\infty}$$\zeta^{m}$$\sigma_{k}$Theorem 2Let us introduce a time dependent version of the operator (21) which acts only
“till the time $t”$ , $i.e$ . the operator$(S_{t}^{\otimes}\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})=(S(k(t))\otimes\ldots\otimes S(1)\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})$

,

(26)

92
where $k(t)$ is the largest$k$such that $pr(z)_{k}\leq t$ . Then$((T_{n}(t))^{-1}T^{\sigma_{1},\ldots,\sigma_{n}}(t)\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})=(S_{t}^{\otimes}\varphi)(z_{1}^{\alpha(1)}, \ldots, z_{n}^{\alpha(n)})$Moreover, the $r.h.s$ . of (27) gives the solution to the Cauchy problem forof the multiple-kick equation (4) with $E=0$ , $i.e$ . to the equation$d \varphi+iE\varphi dt=\sum_{j=1}^{n}(\sigma_{j}-1)\varphi d\chi_{(pr(z)_{j},\infty)}$

,

$a$

.

(27)

“coloured” version$\varphi\in L_{B}^{p,loc}(CS_{n,m})$

.

(28)

with vanishing $E$ .

As in the previous section, equation (28) can be written as astochastic equation, if thetimesof jumps are random variables. In fact, in terms of the counting process $n_{t}=$$|pr(\zeta)\cap[0, t)|$ equation (28) takes the form
$pr(\zeta)$,$d\varphi+iEipdt=(\sigma_{n_{t}}-1)\varphi dn_{t}$$\varphi\in L_{B}^{p,loc}(CS_{n,m})$

.

(29)

In particular, since the number of jumps of aPoisson process is almost surely finite on eachin (29) to be astandard Poisson
finite interval of time, one can consider the processprocess.
Theorem 2expresses the solutions to pure jump stochastic equations in multiple Fockspaces in terms of the boundary value problems for shifts. As was proven in [4], the general
stochastic and even quantum stochastic linear equations can be obtained as the epimorphic
projection of such pure jump stochastic equations. Let us recall now how this projection isconstructed. To this end, one uses the Fock space (23) with $p_{1}=1$ , $p_{2}=\ldots=p_{m-1}=2$ ,$p_{m}=\infty$ constructed over the one-particle Banach space$n_{t}$$L_{H}^{p}(CS_{1,m})=L_{H}^{1}(\mathbb{R})$ $\oplus L_{H}^{2}(\mathbb{R})\oplus\ldots\oplus L^{2}(\mathbb{R})\oplus C_{H}(\mathbb{R})$To work in this space, it is convenient to index the variables
by the formulas:connected with our general notations. The formula
, and
with

as

$\zeta^{-}=\zeta^{1}$$\zeta^{j}$$\zeta^{0,j}=\zeta^{j+1}$$\langle$,.
$\zeta^{-}$(30),$\zeta^{0}$,$\zeta^{+}$

which are$\zeta^{0}=(\zeta^{0,1}, \ldots, \zeta^{0,m-1})$$\zeta^{+}=\zeta^{m}$,$((f^{-}, f^{0}, f^{+})|(g^{-}, g^{0}, g^{+}))= \int((f^{-}, g^{+})_{H}(z)+\sum_{j=1}^{m-1}(f_{j}^{0}, g_{j}^{0})_{H}(z)+(f^{+}, g^{-})_{H}(z))dz$

defines apseudo scalar product in space (30), which is then naturally lifted to the whole Fock
space
. Then one defines the linear (pseudo) isometry operator :and its (psedo) adjoint by the formulas$J$$\mathcal{F}_{H}^{p}$$(J(\psi))(\zeta^{-}, \zeta^{0}, \zeta^{+})=\delta_{\emptyset}(\zeta^{-})\psi(\zeta^{0})1(\zeta^{+})$,$H\otimes \mathcal{F}^{2}\mapsto\not\simeq H\otimes \mathcal{F}^{1,2,\infty}$$J^{\star} \psi(\zeta)=\int\psi(\zeta^{-}, \zeta, \emptyset)d\zeta^{-}$

(31)

,

is empty andis the indicator function of the vacuum (i.e. it equals one if
where
. The
all
equalsforoneis the constant function which
vanishes otherwise), and 1.
integral over (means the sum of the integrals over all finite dimensional simplices$\langle$$\delta_{\emptyset}(\zeta^{-})$$(\zeta^{+})$$\zeta^{+}$$\Sigma_{n}$93Consider now the family of matrices (19) (which, for simplicit , will be supposed not todepend explicitly on ) with the structure$1\mathrm{y}$$k$$S(k)=(_{0}^{1}0$

It turns out (see [4]) that the operator$J^{\star}S_{t}^{\otimes}J$$S_{0}^{-}S_{0}^{0}0$$S_{+}^{-}S_{+}^{0}1)$

.

(32)

solves the linear quantum stochastic equation$d\eta+S_{+}^{-}\eta dt=S_{0}^{-}\eta dA^{+}(t)+S_{+}^{0}\eta dA^{-}+S_{0}^{0}\eta d\Lambda$

,

(33)

where
with $j=1$ , , $m-1$ are the creation and annigilation quantum martingales
respectively and is the gauge process.
Therefore, the following result holds.
$A^{\pm}=A_{j}^{pm}$$\ldots$$\Lambda$

Theorem 3The solution operator for equation (33) is given by the formuladefined in (31) and defined by (32), (26),(27).
$J^{\star}$$J^{\star}S_{t}^{\otimes}J$

with J,$S_{t}$

Thus the solution to ageneral quantum stochastic equation is expressed in terms of theboundary value problem in acoloured (pseudo) Fock space. Due to awell known result from[9], any L\’evy process can be represented in aFock space and thus any stochastic equation
driven by such aprocess can be written in the form of aquantum stochastic equation given
above.

6Stochastic dynamics as semi-classical limit
The aim of this section is to show that the evolutions defined by the boundary value problems for shifts can be obtained as asort of semiclassical limit of the evolutions defined byaboundary value problem for rather general Schr\"odinger equations. This completes thedescription of stochastic evolutions as certain limits of boundary value problems for the
standard (deterministic) quantum mechanical equations with physical (real and bounded
below) Hamiltonians.
We first recall some notations related to pseud0-differential operators (&DO) with operatorvalued symbols.
Recall first that if is ameasurable function on with values in linear operators in $H$ ,inacts asthe pseud0-differential operator $(\Psi DO)$$\mathbb{R}$$\gamma$$\gamma(-i\partial_{z})$$L_{H}^{2}$(34)$( \gamma(-i\partial_{z})\varphi)(z)=\int_{-\infty}^{+\infty}e^{ikz}\gamma(k)f(k)dk$

on the functions$\varphi$given by their Fourier transforms as$\varphi(z)=\int_{-\infty}^{+\infty}e^{ikz}f(k)dk$,$f\in L_{H}^{2}$

(35)

.

consists of the functions of form (35) with fromThe domain of the operator
is called the. The function
the domain of the operator of multiplication by. Choosing apositive parameter , we denot
symbol of the DO$\gamma(-i\partial)$$\Psi$$f$$\varphi$$\gamma=\gamma(p)$$\gamma(k)$$h$$\gamma(-i\partial_{z})$$\hat{\gamma}=\hat{\gamma}(h)=h^{-1}\gamma(-ih\partial_{z})$.

$\mathrm{e}$94be an even function on R with values in aset of commuting non-negative selfLet
adjoint operators in H defined on the same dense domain DCH. Suppose also thatexist as selfadjoint operators on D for all p-j 0, and for arbitary $4>0$ and vC $D$$\ovalbox{\tt\small REJECT}(p)$$\ovalbox{\tt\small REJECT}’(p)$(36)$||[\epsilon(\xi+p)-\epsilon(\xi)-p\epsilon’(\xi)]v||=O(|p|^{2})$uniformly for from an arbitrary compact interval. Next, let us fix aunitary operator ain$H$ . The operatorsand describe respectively the free continuous evolution and the jumps
of aquantum system.
For an arbitrary selfadjoint operator in $H$ , which is defined on $D$ and commutes within
, and an arbitrary positive number 4, we define the operators
allby the formula
(37)
.
$p$

$\hat{\epsilon}$$\sigma$$E$$\omega_{E,\xi}^{\mp}=\omega_{E,\xi}^{\mp}(h)$$\epsilon(p)$$L_{H}^{2}$$\omega_{E,\xi}^{\mp}(h)=\frac{1}{h}e^{\pm i(E+\xi/h)z}(\epsilon(-hi\partial_{z})-\epsilon(\xi))e^{\mp i(E+\xi/h)z}$

are DO with symbols $(\epsilon(h(p\mp E)\mp\xi)-\epsilon(\xi))/h$ . Moreover,
The operators
are selfadjoint and generate the unitary evolutions$\omega_{E,\xi}^{\mp}(h)$$\Psi$$( \exp\{-it\omega_{E,\xi}^{\mp}(h)\}\varphi)(z)=\int_{-\infty}^{+\infty}e^{ikz}\exp\{-it[\epsilon(h(p\mp E)\mp\xi)-\epsilon(\xi)]/h\}f(k)dk$

for$\varphi$$\omega_{E,\xi}^{\mp}(h)$(38)

given by (35).

Theorem 4[5] For any $\xi>0$ , and T $>0$ the evolutions (38) converge strongly to theunifomly for t $\in[0,$T].
as hevolutions$\exp\{-it\epsilon’(\xi)(E\pm i\partial_{z})\}$$arrow \mathrm{O}$is obtained as alimit
Thus the Dirac type evolution with the unbounded generator
Hamiltonians. As
below
with
boundedof arather general Schr\"odinger evolutionassatisfying the assumptions of the theorem one can take the symbols
an example of.or $p^{2}/2m$ of the standard relativistic or non-relativistic Schr\"odinger operators.
We deduced this limit only for the case of asingle-kick equation. The generalisations to a
multi-dimensional case are straightforward.
Concluding remark. The paper is based on alecture delivered on aconference in RIMS
Kyoto, November 2000. An extended version of this paper will be published elsewhere (see[5] .
$\partial_{z}$$harrow \mathrm{O}$$\epsilon(p)$$.\sqrt{p^{l}+m^{l}}$$)$References[1] L. Accardi, Y. G. Lu and I. Volovich. Quantum Theory and its Stochastic Limit.
Springer-Verlag, Texts and Mopnographs in Physics, 2000.

[2] V. P. Belavkin. Acontinuous counting observation and posterior quantum dynamics.
J. Phys. A Math. Gen. 22 (1989), L1109-L1114.
[3] V. P. Belavkin. Quantum Stochastic Dirac Boundary Value Problem and the Ultra
Relativistic Limit. Rep. Math. Phys. 46:3 (2000), 359-386.
[4] V. P. Belavkin. AQuantum Nonadapted Ito Formula and Stochastic Analysis in Fock
Scale. J. Funct. Anal. 102:2 (1991), 414-447

95[5] V. P. Belavkin and V. N. Kolokoltsov. Stochastic evolution as asemiclassical limit ofabouondary value problem for Schrodinger equation. Dep. Math. Stat. Nottingham
Trent University Preprint October 2000.
[6] A. M. Chebotarev. The quantum stochastic equation is equivalent to aboundary value
problem for the Schr\"odinger equation. Mathematical Notes 61:4 (1997), 510-518.
[7] M. Gregoratti. On the Hamiltonian Operator Associated to Some Quantum Stochastic Differential Equations. Infinite Dimensional Analysis, Quantum Probability and
Related Topics, 3:4 (2000), 483-504.
[8] V. N. Kolokoltsov. Semiclassical Analysis for Diffusions and Stochastic Processes
Springer Lecture Notes Math. 1724, 2000.
[9] R. F. Streater. Current commutation relations, continuous tensor products, and infinitely divisible group representations. In: Local Quantum Theory (Ed. R. Jost),
Academic Press, 1969, 247-263.

[10] A. Truman, Z. Zhao. The stochastic H-J equations, stochastic heat equations andSchrodinger equations. In: (Eds. ATruman et al) Stochastic Analysis and Application.
World Scientific Press (1996), 441-464