数理解析研究所講究録 1227 巻 2001 年 170-183

170White Noise Analysis Based on the L\’evy Laplacian
HUI-HSIUNG KUo
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USANOBUAKI OBATA
Graduate School of Information Sciences
Tohoku University
Sendai, 980-8579, Japanand

KIMIAKI SAIT\^O
Department of Information Sciences
Meijo University
Nagoya 468-8502, Japan

Abstract
Eigenfunctions of the Levy Laplacian with an arbitrary complex number as an eigenvalue
are constructed by means of acoordinate change of white noise distributions. The Levy
Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions andthe corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional
stochastic process related to the L\’evy Laplacian is constructed from some one-dimensional
stochastic process.

1. Introduction, an infinite dimensional Laplacian introduced by P. L\’evy [21],The Levy Laplacianhas recently attracted much attention for its peculiar and unexpected characters found in
essentially infinite dimensional analysis. For example, harmonic functions with respect to the
Levy Laplacian are related to solutions of the Yang-Mills equations [2]; solutions of the heat
equation associated with the L\’evy Laplacian is obtained from normal-0rdered white noise
differential equations with quadratic white noises [26]. In this paper we focus on infinite
dimensional stochastic processes associated with the L\’evy Laplacian formulated so as to act
on anew Hilbert space of functions on aGaussian space.
There are several natural formulations of the L\’evy Laplacian. Originally P. L\’evy [21]as an operator acting on functions on the Hilbert space $L^{2}(0,1)$ , see also $[5, 27]$ .
definedHowever for several reasons it seems more natural to consider functions on aGaussian spaceor on anuclear space. Throughout this paper we fix aGelfand triple:$\triangle_{L}$$\triangle_{L}$$E\equiv S(\mathbb{R})\subset L^{2}(\mathbb{R})\subset S’(\mathbb{R})\equiv E^{*}$

171and consider aGaussian space, whereis the Gaussian measure on. It is wellknown that acertain class of functions on the nuclear space $E$ is characterized as the imageof the -transform of “generalized” functions on the Gaussian space, which are constructedfrom aGelfand triple $(E)\subset L^{2}(E^{*}, \mu)\subset(E)^{*}$ . Then, taking acomplete orthonormal basis
for $L^{2}(T)$ with a fixed finite interval, we define the Levy Laplacian by$(E^{*}, \mu)$$E^{*}$$\mu$$\mathrm{S}$$\{\zeta_{n}\}_{n=0}^{\infty}\subset E$$T\subset \mathbb{R}$$S[ \triangle_{L}\Phi](\xi)=\lim_{Narrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}S[\Phi]’(\xi)(\zeta_{n}, \zeta_{n})$,$\Phi\in(E)^{*}$,

whenever the limit exists, for aprecise definition see Section 3. As the L\’evy Laplacianvanishes on, its natural domain is to be found in generalized white noise functions(white noise distributions). Along this idea the Lev Laplacian was first introduced in white
noise analysis by T. Hida [6] and has been discussed by many authors, see e.g., [9, 16, 18]
for general properties, [4] for Cauchy problems, [7] for related functional equations, [11] forarelation to an infinite dimensional Fourier transform, $[29, 30]$ for aconnection with the
It\^o formula, $[31, 32]$ for stochastic processes generated by powers of the L\’evy Laplacian.
While, it is also possible to formulate the L\’evy Laplacian independent of the Gaussian space[1, 19, 23].
In this paper, extending the ideas in the previous works [20, 22, 34, 35, 36], we introduce
anew class of Hilbert spaces based on eigenfunctions of the Levy Laplacian. In fact, forsome continuous function and any
we construct asubspace,$p>5/12$ , consisting of eigenfunctions of the L\’evy Laplacian with eigenvalue. Thoseeigenfunctions are constructed by means of an “exponential coordinate change” for white
noise functions. Then, for some continuous function and for any $p>5/12$ ,thedirect integral Hilbert space:$L^{2}(E^{*}, \mu)$$\mathrm{y}$$h$$\lambda\in \mathbb{R}$$\mathrm{D}_{\lambda}^{h}\subset(E)_{-p}\subset(E)^{*}$$h(\lambda)$$h$$N\in \mathrm{N}$$\mathcal{E}_{-p,N}^{h}=\int_{\mathbb{R}}^{\oplus}D_{\lambda,-p}^{h}\alpha_{N}^{h}(\lambda)d\lambda$

,

where
is the completion ofin $(E)_{-p}$ andis acertain weight function, becomesanatural domain of the Levy Laplacian. Thus the Levy Laplacian is diagonalized:$D_{\lambda,-p}^{h}$$\mathrm{D}_{\lambda}^{h}$$\alpha_{N}^{h}(\lambda)$$\triangle_{L}=\int_{\mathbb{R}}^{\oplus}h(\lambda)\alpha_{N}^{h}(\lambda)d\lambda$

and, thereby, an associated equi-continuous semigroupof class
is obtained (TheOrem 3.4). This idea traces back to [31]. Finally, we obtain astochastic expression ofinterms of an $E$ -valued stochastic process derived from aone-dimensional stochastic processwith the function (Theorem 4.1). It is noteworthy that the stochastic process generated
by the Levy Laplacian depends on the choice of eigenfunctions of the Laplacian.
$\{G_{t}^{h}\}$$(\mathrm{C}_{0})$$\{G_{t}^{h}\}$$h$

2. Preliminaries
In this section we assemble some basic notations of white noise analysis following [9, 15,
18, 24].
We take the spaceof tempered distributions with the standard Gaussian
measure which satisfies$E^{*}\equiv S’(\mathbb{R})$$\mu$$\int_{E^{*}}\exp\{i\langle x, \xi\rangle\}d\mu(x)=\exp(-\frac{1}{2}|\xi|_{0}^{2})$:$\xi\in E\equiv S(\mathbb{R})$,172
where , is the canonical bilinear form on
andis thenorm
Let $A=-(d/du)^{2}+u^{2}+1$ . This is adensely defined self-adjoint operator on
andthere exists an orthonormal basis
forsuch that $Ae_{\nu}=2(\nu+1)e_{l/}$ . We
define the norm
by $|f|_{p}=|A^{p}f|_{0}$ for $f\in E$ and, and letbe the completion ofwith respect to the norm
. Thenis areal separable Hilbert space with the norm
and the dual spaceofis the same as(see [13]). The space is the projective limit
space of $\{E_{p};p\geq 0\}$ andis the inductive limit space of $\{E_{-p};p\geq 0\}$ . Then $E$ becomes
anuclear space with the Gel’fand triple. We denote the complexifications
,
of
by
and
,
, respectively.
andThe space $(L^{2})=L^{2}(E^{*}, \mu)$ of complex-valued square-integrable functionals defined on
admits the well-known Wiener-It\^o decomposition:$\langle\cdot$$E^{*}\cross E$$\cdot\rangle$$|\cdot$$L^{2}(\mathbb{R})$$|_{0}$$L^{2}(\mathbb{R})$$\{e_{\nu}; \nu\geq 0\}$$|\cdot$$L^{2}(\mathbb{R})$$p\in \mathbb{R}$$|_{p}$$E$$|\cdot|_{p}$$E_{p}’$$E_{p}$$E_{p}$$|\cdot|_{p}$$E$$E_{-p}$$E_{p}$$E^{*}$$E\subset L^{2}(\mathbb{R})\subset E^{*}$$L^{2}(\mathbb{R})$$E$$E_{p}$$L_{\mathbb{C}}^{2}(\mathbb{R})$$E_{\mathbb{C}}$$E_{p,\mathbb{C}}$$E^{*}$$(L^{2})=\oplus H_{n}n=0\infty$

,

where
is the space of multiple Wiener integrals of order
and
. Let
denote the -fold symmetric tensor product of. If
has the representation,, then thenorm
is given by$H_{n}$$n\in \mathrm{N}$$n$$L_{\mathbb{C}}^{2}(\mathbb{R})^{\otimes n}\wedge$$\varphi\in(L^{2})$$L_{\mathbb{C}}^{2}(\mathbb{R})$$(L^{2})$$f_{n}\in L_{\mathbb{C}}^{2}(\mathbb{R})^{\otimes n}\wedge$$\varphi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(f_{n})$$H_{0}=\mathrm{C}$$||\varphi||_{0}$$|| \varphi||_{0}=(\sum_{n=0}^{\infty}n!|f_{n}|_{0}^{2})1/2$

,

where
is the
norm
, let
For
, whereis the second quantization operator of .
If $p\geq 0$ , let
be the domain of. If $p<0$ , let
be the completion ofwithrespect to the norm
. Then,, is aHilbert space with the norm
. It is$(E)_{-p}$easy to see that for $p>0$ , the dual spaceofis given by. Moreover, for any, we have the decomposition$L_{\mathbb{C}}^{2}(\mathbb{R})^{\otimes n}\wedge$$|\cdot|_{0}$$||\varphi||_{p}=||\Gamma(A)^{p}\varphi||_{0}$$p\in \mathbb{R}$$(E)_{p}$$A$$\Gamma(A)$$(E)_{p}$$\Gamma(A)^{p}$$||\cdot$$(E)_{p}$$||_{p}$$(L^{2})$$p\in \mathbb{R}$$(E)_{p}^{*}$$||\cdot||_{p}$$(E)_{p}$$p\in \mathbb{R}$$(E)_{p}=\oplus H_{n}^{(p)}n=0\infty$,

is the completion ofwherewith respect to. We also have-fold symmetric tensor product of, whereis also the -fold symmetric tensor product ofis given by$H_{n}^{(p)}$$\{\mathrm{I}_{n}(f);f\in E_{\mathbb{C}}^{\otimes n}\}\wedge$$n$$p\in \mathbb{R}$$E_{\mathbb{C}}^{\otimes n}\wedge$$H_{n}^{(p)}=\{\mathrm{I}_{n}(f);f\in E_{p,\mathbb{C}}^{\otimes n}.\}$$E_{\mathbb{C}}$$n$$E_{p,\mathbb{C}}^{\otimes^{\wedge}n}$. Hereis thefor any. The normof$||\cdot||_{p}$$E_{p,\mathbb{C}}$$||\varphi||_{p}$$\varphi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(f_{n})\in(E)_{p}$$|| \varphi||_{p}=(\sum_{n=0}^{\infty}n!|f_{n}|_{p}^{2})1/2$,$f_{n}\in E_{p,\mathbb{C}}^{\otimes^{\wedge}n}$,.
where the norm on .is denoted also by,
is anuclear space. The inductive
The projective limit space (E) of spaces, is nothing but the strong dual space of (E). The
limit space (E)’ of spaces$|\cdot|_{p}$$E_{p,\mathbb{C}}^{\otimes n}$$(E)_{p}$$(E)_{p},p\in \mathbb{R}$$p\in \mathbb{R}$173
space (E)’ is called the space of generalized white noise functionate. We denote bycanonical bilinear form on (E)’. Then we have$\langle\langle\cdot, \cdot\rangle\rangle$

the$\cross(E)$$\langle\langle\Phi, \varphi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$

for any
form onSince

andis denoted also by$\Phi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(F_{n})\in(E)^{*}$$(E_{\mathbb{C}}^{\otimes n})^{*}\cross(E_{\mathbb{C}}^{\otimes n})$$\varphi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(f_{n})\in(E)$$\phi_{\xi}(\cdot)\equiv\exp(\langle\cdot, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)\in(E)$$\langle\cdot, \cdot\rangle$.

, where the canonical bilinear, we can define the$S[\Phi](\xi)=\langle\langle\Phi, \phi_{\xi}\rangle\rangle$,$\xi\in E_{\mathbb{C}}$$S$-transform on (E)’ by.

Acomplex-valued function $F$ on
is called a -functional if for every ,, thefunction $arrow F(\xi+z\eta)$ ,, is an entire function of and there exist non-negative
constants $K$ , and such that$U$

$E_{\mathbb{C}}$$\xi$$z\in \mathbb{C}$

$z$

$a$

$\eta\in E_{\mathbb{C}}$

$z$

$p$

$|F(\xi)|\leq K\exp\{a|\xi|_{p}^{2}\}$,$\xi\in E_{\mathbb{C}}$.

Theorem(see e.g. [9, 18, 24, 28]) A complex-valued function $F$ onof an element in (E)’ if and only if $F$ is a U-functional.

$E_{\mathbb{C}}$is the S-transform

3. Asemigroup generated by the L\’evy Laplacian
Let $F\in S[(E)^{*}]$ . Then, by Theorem 2.1, we see that for any ,the function$F(\xi+z\eta)$ is an entire function of. Hence we have the series expansion:$\langle$$\eta\in E_{\mathbb{C}}$$z\in \mathrm{C}$$F( \xi+z\eta)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}F^{(n)}(\xi)(\eta, \ldots, \eta)$

,

:
where
is acontinuous -linear functional.
We fix afinite interval of R. Take an orthonormal basis
for $L^{2}(T)$ satisfying
the equal density and uniform boundedness property (see e.g., [9, 18, 19, 23, 30]). Let
denote the set of all $\Phi\in(E)^{*}$ such that the limit$F^{(n)}(\xi)$$E_{\mathbb{C}}\cross\cdots\cross E_{\mathbb{C}}arrow \mathbb{C}$

$n$

$T$

$\{\zeta_{n}\}_{n=0}^{\infty}\subset E$$D_{L}$$\overline{\triangle_{L}}S[\Phi](\xi)=\lim_{Narrow\infty}\frac{1}{N}\sum S[\Phi]’(\xi)(\zeta_{n}, \zeta_{n})N-1$$n=0$

exists for any$\xi\in E_{\mathbb{C}}$and is in $S[(E)^{*}]$ . The L\’evy Laplacian$\triangle_{L}$is defined by$\triangle_{L}\Phi=S^{-1}\overline{\triangle_{L}}S\Phi$for $\Phi\in D_{L}$ . We denote by.
all $\eta\in E$ with

$D_{L}^{T}$

the set of all functionals (I)$\in D_{L}$such that $S[\Phi](\eta)=0$ for$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\eta)\subset T^{c}$Lemma 3.1. For $n\geq 1$ ,$a_{1}$,$\ldots$,$a_{n}\in \mathbb{C}$and$f\in L_{\mathbb{C}}^{1}(T^{n})$, let$F( \xi)=\int_{T^{n}}f(u_{1}, \ldots, u_{n})e^{a_{1}\xi(u_{1})+\cdot\cdot+a_{n}\xi(u_{n})}$du,$\xi\in E_{\mathbb{C}}$

(3. 1

174
where
for all$d\mathrm{u}=du_{1}\cdots$$p> \frac{5}{12}$dun. Then there exists $\Phi\in(E)^{*}$ such that $S[\Phi]=F$ and (I is in $(E)_{-p}$.

Proof. We can estimate $|F(\xi)|$ as follows:$|F(\xi)|$

du$\leq$$\int_{T^{n}}|f(u_{1}, \ldots, u_{n})|e^{|a_{1}||\xi(u_{1})|+\cdots+|a_{n}||\xi(u_{n})|}$$\leq$$|f|_{L^{1\max}}\{e^{|a_{1}||\xi(u_{1})|+\cdots+|a_{n}||\xi(u_{n})|} ; u_{1}, \ldots, u_{n}\in T\}$$\leq$$|f|_{L^{1}}e^{(|a_{1}|+\cdots+|a_{n}|)|\xi|_{\infty}}$

,

where
is the-norm of and $| \xi|_{\infty}=\max\{|\xi(u)|;u\in \mathbb{R}\}$ . Since for any
there exists aconstant $M_{p}>0$ such thatfor all(see the next Remark),
we get.
$|f|_{L^{1}}$$f$$L_{\mathbb{C}}^{1}(T^{n})$$p> \frac{5}{12}$$|\xi|_{\infty}\leq M_{p}|\xi|_{p}$$\xi\in E_{\mathbb{C}}$$|F(\xi)|\leq|f|_{L^{1}}e^{(|a_{1}|+\cdots+|a_{n}|)M_{p}|\xi|_{p}}$Hence we have$|F(\xi)|\leq|f|_{L^{1}}e^{2^{-1}[(|a_{1}|+\cdots+|a_{n}|)^{2}+M_{p}^{2}|\xi|_{p}^{2}]}$.

Similarly we use the same argument as above to show that$\sum_{\nu_{1},\ldots,\nu_{n}=0}^{\infty}\prod_{j=1}^{n}\frac{(|a_{j}||z|)^{\nu_{j}}}{\nu_{j}!}\int_{T^{n}}|f(u_{1}, \ldots, u_{n})|\prod_{j=1}^{n}(|\eta(u_{j})|^{\nu_{j}}|e^{a_{j}\xi(u_{\mathrm{J}})}|)$$\leq$$\leq$

du$\sum_{\nu_{1},\ldots,\nu_{n}=0}^{\infty}\prod_{j=1}^{n}\frac{(|a_{j}||z||\eta|_{\infty})^{\nu_{j}}}{\nu_{j}!}|f|_{L^{1}}e^{(|a_{1}|+\cdots+|a_{\mathfrak{n}}|)|\xi|_{\infty}}$$|f|_{L^{1}}e^{(|a_{1}|+\cdots+|a_{n}|)(|\xi|_{\infty}+|z||\eta|_{\infty})}<\infty$.

Therefore we obtain$F(\xi+z\eta)$$=$$\sum_{\nu_{1\prime}\ldots,\nu_{n}=0}^{\infty}\int_{T^{n}}f(u_{1}, \ldots, u_{n})\prod_{j=1}^{n}(\frac{(a_{j}z)^{\nu_{j}}}{\nu_{j}!}\eta(u_{j})^{\nu_{j}}e^{a_{j}\xi(u_{j})})$$=$$\sum_{\ell=0}^{\infty}\frac{z^{\ell}}{\ell!}F_{\ell}(\xi;\eta)$

du

,

where
$F_{\ell}( \xi;\eta)=\int_{T^{n}}f(u_{1}, \ldots, u_{n})(a_{1}\eta(u_{1})+\cdots+a_{n}\eta(u_{n}))^{\ell}\prod_{j=1}^{n}e^{a_{j}\xi(u_{\mathrm{j}})}$

This implies that $F(\xi+z\eta)$ is an entire function ofassertion follows from Theorem 2.1.

$z$$\in \mathbb{C}$for each ,
$\xi$

du.

$\eta\in E_{\mathbb{C}}$. Thus the$\square$. Since $\max_{t}|e_{j}(t)|=$. [25]) We have(see$O(j^{-1/12})$ (see e.g. [12]), we can check that the above series converges if. Therefore,.
for allthere exists aconstant $M_{p}>0$ such thatfor any
Remark,$| \delta_{t}|_{-p}^{2}=\sum_{j=0}^{\infty}e_{j}(t)^{2}(2j+2)^{-2p}$$\mathrm{e}.\mathrm{g}$$p> \frac{5}{12}$$|\delta_{t}|_{-p}\leq M_{p}$$p> \frac{5}{12}$$t\in \mathbb{R}$Using the Wick ordering : :, we can write ageneralized white noise functional (I) whose-transform is given as in (3.1) by$\cdot$$S$$\Phi(x)=\int_{T^{n}}f(u_{1}, \ldots, u_{n})$:$\prod_{\nu=1}^{n}e^{a_{\nu}x(u_{\nu})}$

: du.

(3.1)

175
The functional (I) belongs toand is important as an eigenfunction of the operator
In fact, we have the following result.
$D_{L}^{T}$$\triangle_{L}$.

Theorem 3.2. [34] A generalized white noise functional (I as in (3.2) satisfies the equation$\triangle_{L}\Phi=\frac{1}{|T|}(\sum_{\nu=1}^{n}a_{\nu}^{2})\Phi$

.

(3.3)

Let
be the set of all complex-valued continuous functions
conditions:$\mathcal{F}$$h$satisfying the following

1) $h(0)=0$ ,2) there exists astochastic process $\{X_{t};t\geq 0\}$ such that $e^{th(z)}=E[e^{izX_{t}}]$ for all $t\geq 0$ and,$z\in \mathbb{R}$3)$A_{\lambda,n}^{h}\equiv\{(a_{1}, a_{2}, \ldots, a_{n})\in \mathbb{C}^{n}$and$n\in \mathrm{N}$;$\sum_{\nu=1}^{n}a_{\nu}=\sqrt{|T|}\lambda$,$\sum_{\nu=1}^{n}a_{\nu}^{2}=|T|h(\lambda)\}\neq\phi$$\in \mathbb{R}$.

For example, afunction $h(z)=-|z|^{\gamma}$ ,For each A
,
, and
, let
$n\in \mathrm{N}$$\in \mathbb{R}$$\mathrm{D}_{\lambda,n}^{h}$$1\leq\gamma\leq 2$, is an element of$=$$LS \{\int_{T^{n}}f(\mathrm{u})$:$\prod_{\nu=1}^{n}e^{a_{\nu}x(u_{\nu})}$

.

,

: du;

$(a_{1}, a_{2}, \ldots, a_{n})\in A_{\lambda,n}^{h}\}$where $LS$ means the linear span.
From now on,
will be an arbitrarily fixed function. Set$h\in \mathcal{F}$$\mathrm{D}_{\lambda}^{h}=LS\{\mathrm{D}_{\lambda,n}^{h};n\in \mathrm{N}\}$Thenis alinear subspace of $(E)_{-p}$ for all
operator from
into itself such thatwill use the following notation$\mathrm{D}_{\lambda}^{h}$$\mathrm{J}_{\mathrm{a}}[f](x)=\int_{T^{n}}f(\mathrm{u})$:by Lemma 3.1, andis alinear(I). For convenience, we
for any$\triangle_{L}$$\in \mathrm{D}_{\lambda}^{h}$: du.

$\prod_{\nu=1}^{n}e^{a_{\nu}x(u_{\nu})}$be anumber arbitrarily fixed. Define aspace,. Then for eachin $(E)_{-p}$ with respect towith the inner product of $(E)_{-p}$ and the Levy Laplacian
operator from
into itself satisfying$n\in \mathrm{N}\cup\{0\}$by the completion of
becomes aHilbert spacebecomes acontinuous linear$D_{\lambda,-p}^{h}$$D_{\lambda,-p}^{h}$$\triangle_{L}$$D_{\lambda,-p}^{h}$$\triangle_{L}\Phi=h(\lambda)\Phi$for any (I)and let.

$p> \frac{5}{12}$$||_{-p}$,$\mathrm{D}_{\lambda,0}^{h}=\mathbb{C}$$p> \frac{5}{12}$$\triangle_{L}\Phi=h(\lambda)\Phi$$\mathrm{D}_{\lambda}^{h}$$||\cdot$$\mathcal{F}$$h\in \mathcal{F}$$f\in E_{\mathbb{C}}^{\otimes n}\wedge$Letfor all A$\in D_{\lambda,-p}^{h}$.

$\mathrm{D}_{\lambda}^{h}$

176
The Levy Laplacian$\triangle_{L}$is aself-adjoint operator on$D_{\lambda,-p}^{h}$and
Proposition 3.3. (cf. [34]) Let (I)for each A
functionals such that and are in$=\Psi$ in (E)’, then
(E)’
all A
almostinfor$= \int_{\mathbb{R}}\Phi_{\lambda}d\lambda$$\Phi$$A_{\lambda}^{h}= \bigcup_{n}A_{\lambda,n}^{h}$$\in \mathbb{R}$$\in \mathbb{R}$$\Phi_{\lambda}=\Psi_{\lambda}$

Proof. Let
forms:. Then, for almost all,$\Phi_{\lambda}=\lim_{Narrow\infty}\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}]$$\in \mathbb{R}$and$p> \frac{5}{12}$.

be generalized white noiseand strongly mesurable in A. If$= \int_{\mathbb{R}}\Psi_{\lambda}d\lambda$$D_{\lambda,-p}^{h}$$\Psi_{\lambda}$$\Phi_{\lambda}$$\Psi$

for each A$\lambda\in \mathbb{R}$,$\Phi_{\lambda}$.

and$\Psi_{\lambda}$

can be expressed in the,$\Psi_{\lambda}=\lim_{Narrow\infty}\sum_{\in \mathrm{a}^{[N]}A_{\lambda}^{h}}\mathrm{J}_{\mathrm{a}^{[N]}}[g_{\mathrm{a}^{[N]}}]$

means asum of finitely many terms on
where
Then, taking the -transform, we have$\mathrm{a}^{[N]}\in A_{\lambda}^{h}$$\sum_{\mathrm{a}^{[N]}}\in A_{\lambda}^{h}$. Suppose (I) $=\Psi$ in (E)’.

$S$$\int_{\mathbb{R}}\lim_{Narrow\infty}\sum_{\mathrm{a}1^{N}\mathrm{J}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}-g_{\mathrm{a}^{[N]}}])(\xi)d\lambda=0$

for alland$\xi$. Take. Then we get$\xi_{T}\in E_{\mathbb{C}}$$\in E_{\mathbb{C}}$$\eta\in E_{\mathbb{C}}$such that $\xi_{T}=|T|^{-1/2}$ on$T$and put $\xi=a\xi_{T}+\eta$ with

$a\in \mathbb{C}$$\int_{\mathbb{R}}e^{\lambda a}\lim_{Narrow\infty}\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}-g_{\mathrm{a}^{[N]}}])(\eta)d\lambda=0$

for alland$a\in \mathbb{C}$$\eta\in E_{\mathbb{C}}$. Therefore,$\lim_{Narrow\infty}\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}-g_{\mathrm{a}^{[N]}}])(\eta)=0$

for almost all$\lambda\in \mathbb{R}$and all$\eta\in E_{\mathbb{C}}$. This implies that$\Phi_{\lambda}=\Psi_{\lambda}$in (E)’ for almost all$\lambda\in \mathbb{R}$.

$\square$For any$N \in \mathrm{N},p>\frac{5}{12}$$\int_{\mathbb{R}}^{\oplus}D_{\lambda,-p}^{h}\alpha_{N}^{h}(\lambda)d\lambda$, we can define aspace$\mathcal{E}_{-p,N}^{h}$:,$\Phi_{\lambda}\in D_{\lambda,-p}^{h}\forall\lambda\in \mathbb{R}\}$$\mathcal{E}_{-p,N}^{h}=\{(\Phi_{\lambda})_{\lambda};\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda<\infty$

where$\alpha_{N}^{h}(\lambda)$

by the direct integral space,is given by$\alpha_{N}^{h}(\lambda)=\sum_{\ell=0}^{N}|h(\lambda)|^{2\ell}$Define anorm$|||\cdot$$|||_{-p,N}$on$\mathcal{E}_{-p,N}^{h}$by$||| \Phi|||_{-p,N}=(\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda)1/2$

Then the space$\mathcal{E}_{-p,N}^{h}$

(3.4)

.

is a Hilbert space with the norm,$\Phi$$=(\Phi_{\lambda})_{\lambda}\in \mathcal{E}_{-p,N}^{h}$$|||\cdot|||_{-p,N}$for each.

$N\in \mathrm{N}$and$p> \frac{5}{12}$.

177
Proposition 3.3 implies thatelement of (E)’. We note that$\ovalbox{\tt\small REJECT} 4$$”\ovalbox{\tt\small REJECT}_{p_{\rangle}N}$$\mathrm{E}_{-p,N}^{h}=\{\int_{\mathbb{R}}\Phi_{\lambda}d\lambda$$\in(E)^{*};$

with the norm induced from$\mathcal{E}_{-p,N}^{h}$dA with, p is uniquely determined as an
is isomorphic to aHilbert space$+\cdot$$+_{2}\mathrm{c}=$$\mathrm{I}\supset\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\int_{1\mathrm{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\wedge)d\lambda$$<\infty$,$\Phi_{\lambda}\in D_{\lambda,-p}^{h}\forall\lambda\in \mathbb{R}\}$,by abijection$( \Phi_{\lambda})_{\lambda}arrow\int_{\mathbb{R}}\Phi_{\lambda}d\lambda$.

by the same notation
We denote the norm on.
Putwith the projective limit topology. Then, for any $N\geq 1$ , wehave the following inclusion relations:$|||\cdot|||_{-p,N}$$\mathrm{E}_{-p,N}^{h}$$\mathrm{E}_{-p,\infty}^{h}=\bigcap_{N\geq 1}\mathrm{E}_{-p,N}^{h}$.

$\mathrm{E}_{-p,\infty}^{h}\subset \mathrm{E}_{-p,N+1}^{h}\subset \mathrm{E}_{-p,N}^{h}\subset \mathrm{E}_{-p,1}^{h}\subset(E)_{-p}$and is acontinuous linear operator fromThe Laplaciancan be defined oninto$|||\triangle_{L}\Phi|||_{-p,N}\leq|||\Phi|||_{-p,N+1}$
for alland. Any restriction
satisfying
.
by
same
notation
theofis also denoted$\triangle_{L}$$\mathrm{E}_{-p,2}^{h}$$\mathrm{E}_{-p,2}^{h}$$N\in \mathrm{N}$$\Phi\in \mathrm{E}_{-p,N+1}^{h}$$\mathrm{E}_{-p,1}^{h}$$\triangle_{L}$$\triangle_{L}$Let$h\in \mathcal{F}$. For each $t\geq 0$ we consider an operator$G_{t}^{h}$on$\mathrm{E}_{-p,\infty}^{h}$defined by$G_{t}^{h} \Phi=\int_{\mathbb{R}}e^{th(\lambda)}\Phi_{\lambda}d\lambda$

for. Then we have the following:$\Phi=\int_{\mathbb{R}}\Phi_{\lambda}d\lambda\in \mathrm{E}_{-p,\infty}^{h}$is an equi-continuous semigroup of
the family
Theorem 3.4. For each
operator.
linearas a continuous
generated byclass
defined on$\{G_{t}^{h};t\geq 0\}$$h\in \mathcal{F}$$(C_{0})$$\mathrm{E}_{-p,\infty}^{h}$$\triangle_{L}$ia acharacteristic function, we have $Reh(z)$Proof. Since,
for (1), the normandbe estimated as follows:$e^{th(z)}$$|||G_{t}^{h}\Phi|||_{-p,N}$$N\in \mathrm{N}$$|||G_{t}^{h}\Phi|||_{-p,N}^{2}$$= \int_{\mathbb{R}}\Phi_{\lambda}d\lambda\in \mathrm{E}_{-p,\infty}^{h}$$\leq 0$. For any $t\geq 0$ ,, $n=0,1,2$ ,$\Phi_{\lambda}\in D_{\lambda,-p}^{h}$$=$$\int_{\mathit{1}\mathrm{R}}||e^{th(\lambda)}\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda$$\leq$$\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}^{2}e^{2tReh(\lambda)}\alpha_{N}^{h}(\lambda)d\lambda$$=$$|||\Phi|||_{-p,N}^{2}$$G_{t+s}^{h}$

$t$

$t$

$|||G_{t}^{h}\Phi-G_{t_{0}}^{h}\Phi|||_{-p,N}^{2}$$\ldots$can.

is equi-continuous in . It is easily checked that $G_{0}^{h}=I$ ,Hence the family
for each , $s\geq 0$ . We can also estimate that$\{G_{t}^{h};t\geq 0\}$$p> \frac{5}{12}$$=$$\int_{\mathbb{R}}|e^{th(\lambda)}-e^{t_{0}h(\lambda)}|^{2}||\Phi_{n}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda$$\leq$

4$=$$4|||\Phi|||_{-p,N}^{2}<\infty$$\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda$$G_{t}^{h}G_{s}^{h}=$

178
0, CU
convergence theorem, we getfor each t,$\ovalbox{\tt\small REJECT} \mathrm{V}$$\mathrm{a}\mathrm{n}\mathrm{d}+\ovalbox{\tt\small REJECT} f\ovalbox{\tt\small REJECT}$$\mathrm{t}_{0}\ovalbox{\tt\small REJECT}$$+_{\mathit{2}}dA$

C$\mathrm{E}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}p\ovalbox{\tt\small REJECT}$. Therefore, by the Lebesgue dominatedin$\lim_{tarrow t_{0}}G_{t}^{h}\Phi=G_{t_{0}}^{h}\Phi$$\mathrm{E}_{-p,\mathrm{o}\mathrm{o}}^{h}$. Thus the family
is an equi-continuous semigroup
andfor each. We next prove that the infinitesimal generator of the semigroup is given byof class, we see thatand
. For any$t_{0}\geq 0$$\{G_{t}^{h}; t\geq 0\}$$\Phi\in \mathrm{E}_{-p,\infty}^{h}$$(C_{0})$$N\in \mathrm{N}$$\triangle_{L}$$p> \frac{5}{12}$$||| \frac{G_{t}^{h}\Phi-\Phi}{t}-\triangle_{L}\Phi|||_{-p,N}^{2}$$= \int_{\mathbb{R}}||\frac{e^{th(\lambda)}-1}{t}\Phi_{\lambda}-h(\lambda)\Phi_{\lambda}||_{-p}^{2}\alpha_{N}^{h}(\lambda)d\lambda$

Since

.

(3.5)

, we have$\Phi=\int_{\mathbb{R}}\Phi_{\lambda}d\lambda\in \mathrm{E}_{-p,\infty}^{h}$$\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{N+1}^{h}(\lambda)d\lambda<\infty$

.

(3.6)

By the mean value theorem, for any $t>0$ there exists aconstant$| \frac{e^{lh(\lambda)}-1}{t}|=|h(\lambda)|e^{t\theta Reh(\lambda)}\leq|h(\lambda)|$$\mathit{0}\in(0,1)$

such that.

Therefore we can estimate each term in (3.5) as follows:$\alpha_{N}^{h}(\lambda)||\frac{e^{th(\lambda)}-1}{t}\Phi_{\lambda}-h(\lambda)\Phi_{\lambda}||_{-p}^{2}$$=$$\leq$$\alpha_{N}^{h}(\lambda)|\frac{e^{th(\lambda)}-1}{t}-h(\lambda)|^{2}||\Phi_{\lambda}||_{-p}^{2}$$4\alpha_{N+1}^{h}(\lambda)||\Phi_{\lambda}||_{-p}^{2}$.

Note that$\lim_{tarrow 0}|\frac{e^{th(\lambda)}-1}{t}-h(\lambda)|=0$.

Thus by (3.6) we can apply the Lebesgue dominated convergence theorem to obtain$\lim_{tarrow 0}|||\frac{G_{t}^{h}\Phi-\Phi}{t}-\triangle_{L}\Phi|||_{-p,N}^{2}=0$.

Hence the proof is completed.

Remark: For each$N\in \mathrm{N}$$\square$, we can write$\triangle_{L}$and$G_{t}^{h}$acting on$\triangle_{L}=\int_{\mathbb{R}}^{\oplus}h(\lambda)\alpha_{N}^{h}(\lambda)d\lambda$$\mathcal{E}_{-p,N}^{h}$

as

179

$G_{t}^{h}= \int_{\mathbb{R}}^{\oplus}e^{th(\lambda)}\alpha_{N}^{h}(\lambda)d\lambda$

,

where
is given in Equation (3.4). These formulations can be regarded as the diag0.
andnalizations of the operators$\alpha_{N}^{h}(\lambda)$$G_{t}^{h}$$\triangle_{L}$4. Astochastic process generated by the L\’evy LaplacianIn this section, we will give astochastic process generated by the L\’evy Laplacian by.
considering the stochastic expression of the operator
given bybe astochastic process with the characteristic function ofLet$G_{t}^{h}$$\{X_{t}^{h}; t\geq 0\}$

$X_{t}^{h}$$E[e^{izX_{t}^{h}}]=e^{th(z)}$Take asmooth function$S[\mathrm{E}_{-p,\infty}^{h}]$$\eta_{T}\in E$

with

,

$h\in \mathcal{F}$.

on . Define an operator$T$$\eta_{T}=\frac{1}{\sqrt{|T|}}$$G_{t}^{h}-$acting onby$G_{t}^{h}=SG_{t}^{h}S^{-1}-$.

by the Sis endowed with the topology induced fromtransform. Then by Theorem 3.4, $\{G_{t}^{h};-t\geq 0\}$ is an equi-continuous semigroup of class.
generated by the operator$E$ -valued stochastic process defined by
be an
Let

Here the space$S[\mathrm{E}_{-p,\infty}^{h}]$$\mathrm{E}_{-p,\infty}^{h}$$(C_{0})$$\triangle_{L}$$\{\mathrm{X}_{t}^{h},\cdot t\geq 0\}$$\mathrm{X}_{t}^{h}=\xi+iX_{t}^{h}\eta_{T}$,$\xi\in E$.

Then we have the following theorem.
Theorem 4.1. Let$h\in \mathcal{F}$. Then for all$F\in S[\mathrm{E}_{-p,\infty}^{h}]$, the following equality holds$G_{t}^{h}F(\xi)-=\mathrm{E}[F(\mathrm{X}_{t}^{h})|\mathrm{X}_{0}^{h}=\xi]$Proof. First consider the case when$F\in S[\mathrm{E}_{-p,\infty}^{h}]$.

is given by$F( \xi)=S(\mathrm{J}_{\mathrm{a}}[f])(\xi)=\int_{T^{n}}f(u_{1}, \ldots, u_{n})\prod_{\nu=1}^{n}e^{a_{\nu}\xi(u_{\nu})}$with$\sum_{\nu=1}^{n}a_{\nu}=\sqrt{|T|}\lambda$,$\sum_{\nu=1}^{n}a_{\nu}^{2}=|T|h(\lambda)$$\mathrm{E}[F(\mathrm{X}_{t}^{h})|\mathrm{X}_{0}^{h}=\xi]$du,. Then we have$=$$\mathrm{E}[F(\xi+iX_{t}^{h}\eta_{T})]$$=$$\int_{T^{n}}f(u_{1}, \ldots, u_{n})\prod_{n=1}^{n}e^{a_{\nu}\xi(u_{\nu})}\mathrm{E}[e^{i\lambda X_{t}^{h}}]$$=$$e^{th(\lambda)}F(\xi)$$=$$G_{t}^{h}F(\xi)-$

.

du

180
Next, letfollowing form:$F\in S[\mathrm{E}_{-p,\infty}^{h}]$be represented by$F= \int_{\mathbb{R}}F_{\lambda}d\lambda$

with$F_{\lambda}( \xi)=\lim_{Narrow\infty}\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}])(\xi)$$F_{\lambda}$

being expressed in the.

Hence we have$\int_{\mathbb{R}}\mathrm{E}[|F_{\lambda}(\xi+iX_{t}^{h}\eta_{T})|]d\lambda$$=$$\int_{\mathbb{R}}\mathrm{E}[\lim_{Narrow\infty}|\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}])(\xi+iX_{t}^{h}\eta\tau)|]d\lambda$$=$$\int_{\mathbb{R}}\mathrm{E}[\lim_{Narrow\infty}|\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}])(\xi)\prod_{\nu=1}^{n}e^{ia_{\nu}^{[N]}X_{t}^{h}}|]d\lambda$$=$$\int_{\mathbb{R}}\lim_{Narrow\infty}|\sum_{\mathrm{a}^{[N]}\in A_{\lambda}^{h}}S(\mathrm{J}_{\mathrm{a}^{[N]}}[f_{\mathrm{a}^{[N]}}])(\xi)|d\lambda$$=$$\int_{\mathrm{R}}|F_{\lambda}(\xi)|d\lambda$.

, there exists somesuch thatSince$\xi\inE$
any
By
inequality,
.
and
the
we see thatfor
Schwarz$F_{\lambda}\in S[\mathrm{E}_{-p,\infty}^{h}]$$F_{\lambda}(\xi)=S[\Phi_{\lambda}](\xi)=\langle\langle\Phi_{\lambda}, \phi_{\xi}\rangle\rangle$$\Phi_{\lambda}\in \mathrm{E}_{-p,\infty}^{h}$$\lambda\in \mathbb{R}$$\int_{\mathbb{R}}|F_{\lambda}(\xi)|d\lambda$$\leq$$\leq$$<$$\int_{\mathbb{R}}||\Phi_{\lambda}||_{-p}||\phi_{\xi}||_{p}d\lambda$$\{\int_{\mathbb{R}}\alpha_{M}^{h}(\lambda)^{-1}d\lambda\}^{1/2}\{\int_{1\mathrm{R}}||\Phi_{\lambda}||_{-p}^{2}\alpha_{M}^{h}(\lambda)d\lambda\}^{1/2}||\phi_{\xi}||_{p}$$\infty$,and some $M\geq 1$ , wherecontinuity of
we getfor all$\xi\epsilon- E$$\alpha_{N}^{h}(\lambda)$

is given in Equation (3.4). Therefore by the$\overline{G_{t}^{h}}$$\mathrm{E}[F(\xi+iX_{t}^{\gamma}\eta_{T})]$$=$$\int_{\mathrm{R}}\mathrm{E}[F_{\lambda}(\xi+iX_{t}^{h}\eta_{\Gamma})]d\lambda$$=$$\int_{\mathbb{R}}\overline{G_{t}^{h}}F_{\lambda}(\xi)d\lambda$$=$$\overline{G_{t}^{h}}F(\xi)$.

Thus we obtain the assertion.

$\square$is genTheorem 4.1 implies that the infinite dimensional stochastic process.
for eachdefined onerated by, $x\in E^{*}$ , can be lifted to the space of generalized functions (E)’
The translation$\tau_{\eta}\Phi(x)=\Phi(x+\eta)$
is defined for $\Phi\in(E)^{*}$ . More precisely, acontinuous
whenever $\eta\in E$ , i.e.,from (E)’ into itself is uniquely specified by $S[\tau_{\eta}\Phi](\xi)=S[\Phi](\xi+\eta)$ ,linear operator$\{\mathrm{X}_{t}^{h}; t\geq 0\}$$\overline{\triangle_{L}}$$\mathrm{E}_{-p,\infty}^{h}$$x\vdasharrow x+\eta$$\tau_{\eta}$$h\in \mathcal{F}$

181
$\xi$

$\in E_{\mathbb{C}}$. Then, as is easily verified, $\tau_{\eta}\phi(x)=\phi(x+\eta)$ for $\phi\in(E)$ , which gives aground forthe above formal notation. It is also known that
, where is the Wickproduct, see e.g., [17]. Then, Theorem 4.1 is translated into the language of generalized
white noise functionals.
$\tau_{\eta}\Phi=\phi_{-\eta}0$Corollary 4.2. Let$h\in \mathcal{F}$. Then for all$\Phi\in \mathrm{E}_{-p,\infty}^{h}$$(\phi_{\eta}\Phi)$$0$, the following equality holds$G_{t}^{h}\Phi=\mathrm{E}[\tau_{iX_{t}^{h}\eta T}\Phi]$.

By Corollary 4.2 we can see thatis an operator-valued stochastic process
andis an equi-continuous semigroup of classgenerated bydefinedonfor each.
$\{\tau_{iX_{t}^{h}\eta\tau} ; t\geq 0\}$$\{E[\tau_{iX_{t}^{h}\eta\tau}];t\geq 0\}$$\mathrm{E}^{h}$$(C_{0})$$\triangle_{L}$$h\in \mathcal{F}$$-p,\infty$

Acknowledgements
This work was supported in part by the Joint Research Project “Quantum Information
Theoretical Approach to Life Science” for the Academic Frontier in Science promoted by
the Ministry of Education in Japan and was also supported in part by JSPS- PAN Joint
Research Project “Infinite Dimensional Harmonic Analysis”. The authors are grateful for
their support.

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