数理解析研究所講究録
1193 巻 2001 年 126-136

126

The L\’evy Laplacian and the Number Operator
Kenjiro Nishi (西健次郎) and Kimiaki Sait\^o (齊藤公明)
Department of Information Sciences
Meijo University
Nagoya 468-8502, Japan
Abstract

In this paper, we introduce a homeomorphism to connect the L\’evy Laplacian to the Number
operator. Moreover we also give a relationship between a stochastic process generated by somefunction of the L\’evy Laplacian and the semi-group generated by the Number operator.

1. Introduction

An infinite dimensional Laplacian, the L\’evy Laplacian, was introduced by P. L\’evy [17]. This
Laplacian was introduced into the framework of white noise analysis initiated by T. Hida [4]. L.
Accardi et al. [1] obtained an important relationship between this Laplacian and the Yang-Mills
equations. It has been studied by many authors (see [1, 2, 3, 5, 7, 8, 13, 15, 16, 18, 21, 22, 23,
24 etc]).
In the previous papers $[25,26]$ we obtained stochastic processes generated by the powers of an
extended L\’evy Laplacian and also in [29] we obtained stochastic processes generated by somefunctions of the Laplacian.
The purpose of this paper is to present recent developments on stochastic processes generated
by functions of the L\’evy Laplacian acting on white noise distributions based on the idea in [29]generated byand to give a stochastic expression of an equi-continuous semigroup of classthe Laplacian related to an infinite dimensional Ornstein-Uhlenbeck process following [27].
$(C_{0})$The paper is organized as folows. In Section 2 we summarize some basic definitions andresults in white noise analysis. In Section 3 we introduce a Hilbert space as a domain of theextended L\’evy Laplacian which is self-adjoint on the domain following our previous paper [27],and we give an equi-continuous semigroup of classgenerated by some functions of theextended L\’evy Laplacian. In the last section we give a stochastic expression of a semigroup
generated by some function of the Number operator.
$(C_{0})$2. Preliminaries

In this section we assemble some basic notations of white noise analysis following [7, 12, 15,
19].

127

Let
space

be the Hilbert space of real-valued square-integrable functions on R. We take theof tempered distributions with a probability measure which satisfies$L^{2}(\mathrm{R})$$E^{*}\equiv S’(\mathrm{R})$$\mu$$\xi\in E\equiv S(\mathrm{R})$$\int_{E^{\star}}\exp\{i\langle X, \xi\rangle\}d\mu(x)=\exp(-\frac{1}{2}|\xi|_{0}^{2)},$,-norm.
andis thewhereis the canonical bilinear form on$A=-(d/du)^{2}+u^{2}+1$
is a densely defined self-adjoint operator on
The differential operator
forsuch that $Ae_{\nu}=2(\nu+1)e_{\nu}$ .
. There exists an orthonormal basis$E$for $f\in E$
defined by
be the completion of with respect to the normLetand the dual space. Thenis a real separable Hilbert space with the norm
and$E$ of $\{E_{p};p\geq 0\}$ and
space
limit
10).
(see
the
projectiveofis the same as
Ref.
With
of $\{E_{-p};p\geq 0\}$ , we have a chain of Hilbert spaces: for $0\leq p\leq q$ ,the inductive limit space$E^{*}\cross E$$\langle\cdot, \cdot\rangle$$\{e_{\nu};\nu\geq 0\}$$L^{2}(\mathrm{R})$$L^{2}(\mathrm{R})$$|\cdot|0$$L^{2}(\mathrm{R})$$|f|_{\mathrm{p}}=|A^{p}f|0$$|\cdot|_{p}$$E_{p}$$p\in \mathrm{R}$$|\cdot|_{p}$$E_{p}$$E_{p}’$$E_{-p}$$E_{p}$$E^{*}$$E\subset E_{q}\subset E_{p}\subset L^{2}(\mathrm{R})\subset E_{-p}\subset E_{-q}\subset E^{*}$We denote the complexifications of$L^{2}(\mathrm{R}),$

$E$

and

$E_{p}$

by$L_{\mathrm{C}}^{2}(\mathrm{R}),$.

$E_{\mathrm{C}}$and$E_{\mathrm{C},p}$, respectively.

Let $(L^{2})=L^{2}(E*, \mu)$ be the Hilbert space of complex-valued square-integrable functionals
can be. By the Wiener-It\^o theorem every inwith norm denoted bydefined on
represented uniquely by$E^{*}$$||\cdot||_{0}$$\varphi$$\varphi=\sum_{n=0}^{\infty}\mathrm{I}n(f_{n})$,$f_{n}\in L^{2}\mathrm{c}(\mathrm{R})\otimes n\wedge$

denotes the multiple Wiener integrals of order
where. Moreover, for thesymmetric tensor product of$n\in \mathrm{N}$$\mathrm{I}_{n}$$(L^{2})$$L_{\mathrm{C}}^{2}(\mathrm{R})$$|| \varphi||0=(_{n=}\sum_{0}^{\infty}n!|f_{n}|_{0}^{2})^{1}/2$

where
For

$|\cdot|_{0}$is the norm of$p\in \mathrm{R}$defined on$L_{\mathrm{C}}^{2}(\mathrm{R})\otimes\wedge n$$(L^{2})$

,

. Let

-norm

$L_{\mathrm{C}}^{2}(\mathrm{R})^{\otimes}\wedge n$$||\varphi||_{0}$of$\varphi$denote the n-fold

we have

,

.

, let
by

$||\varphi||_{p}=||\Gamma(A)^{p}\varphi||0$, whereis the second quantization operator densely$\Gamma(A)$$(L^{2})$$\Gamma(A)\varphi=\sum_{=n0}^{\infty}\mathrm{I}n(A^{\otimes}nf_{n})$,$\varphi=\sum_{n=0}^{\infty}\mathrm{I}n(f_{n})$.

withbe the completion ofIf $p\geq 0$ , let $(E)_{p}=\{\varphi\in(L^{2});||\varphi||_{p}<\infty\}$ . If $p<0$ , let$(E)_{p},$
.
easy
It
is
norm
space
with
the
,
a
Hilbertis.
Thenrespect to the norm,
is given by $(E)_{-p}$ . Moreover, for any
to see that for $p>0$ , the dual space
we have the decomposition$(L^{2})$$(E)_{p}$$||\cdot||_{p}$$p\in \mathrm{R}$$||\cdot||_{p}$$p\in \mathrm{R}$$(E)_{p^{\mathrm{O}}}^{*}\mathrm{f}(E)_{p}$$(E)_{p}= \bigoplus_{n=0}H_{n}^{()}\infty \mathrm{p}$

where
$n$

$H_{n}^{(p)}$is the completion of$\{\mathrm{I}_{n}(f);f\in E_{\mathrm{C}^{\wedge}}^{\otimes n}\}$-fold symmetric tensor product of$E_{\mathrm{C}}$,with respect to. In fact we have$||\cdot||_{p}$. Here$E_{\mathrm{C}^{\wedge}}^{\otimes n}$$H_{n}^{(p)}=\{\mathrm{I}_{n}(f);f\in E_{\mathrm{C}^{\wedge},p}^{\otimes n}\}$is thefor any128$p\in \mathrm{R}$, where$E_{\mathrm{C}^{\wedge},p}^{\otimes n}$

is also the -fold symmetric tensor product ofis given by$n$$E_{\mathrm{C},p}$. The norm$||\varphi||_{p}$of$\varphi=\sum_{n=0n}^{\infty}\mathrm{I}(f_{n})\in(E)_{p}$$|| \varphi||_{p}=(n\sum_{0=}^{\infty}n!|f_{n}|^{2}p)^{1}/2$,$f_{n}\in E_{\mathrm{C},p}^{\wedge}\otimes n$,.
is denoted also by
where the norm of$(E)_{p},$$(E)$
is a nuclear space. The inductive limitof spaces
The projective limit spaceis called
is nothing but the dual space of $(E)$ . The space
spaceof spaces$(E)^{*}\cross(E)$isbilinearform
on
the space of generalized white noise functionals. The canonicalsdenoted by
Then we have$|\cdot|_{p}$$E_{\mathrm{C},p}^{\otimes n}\wedge$$p\in \mathrm{R}$$(E)^{*}$$(E)^{*}$$(E)_{p},p\in \mathrm{R}$$<<\cdot,$ $\cdot\gg$.

$<<\Phi,$ $\varphi>>=\sum n\infty=0n!\langle Fn’ fn\rangle$$\Phi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(F_{n})\in(E)^{*}$

where the canonical bilinear form on
inequality takes the form:, the$\phi_{\xi}(\cdot)=\exp(\langle\cdot, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)\in(E)$-transfo$S$$S[\Phi](\xi)=<<\Phi,$ $\psi_{\xi}>>$3. An equi-continuous semigroup of class
LaplacianLet$\Phi$

be in$(E)^{*}$. Then the -transform$S$$S[\Phi]$$\varphi=\sum_{n=0}^{\infty}\mathrm{I}n(f_{n})\in(E)$is denoted also by$(E_{\mathrm{C}}^{\otimes n})^{*}\cross E_{\mathrm{C}}^{\otimes n}$$|<<\Phi,$ $\varphi\gg|\leq||\Phi||-p||\varphi||_{\mathrm{P}}$

Since

,

,

$7m$

,

$p\in \mathrm{R}$$o(\eta)$$(E)^{*}$

by.

$(C_{0})$

generated by a function of the L\’evyofis Fr\’echet differentiable, i.e.

$\Phi$$S[\Phi](\xi+\eta)=S[\Phi](\xi)+s[\Phi]’(\xi)(\eta)+o(\eta)$

where. The Schwarz.

is defined on$\xi\in E_{\mathrm{C}}$$\langle\cdot, \cdot\rangle$,

means that there exists $p\geq 0$ depending on$\xi$such that,$o(\eta)/|\eta|_{\mathrm{p}}arrow 0$

as$|\eta|_{p}arrow 0$.

for $L^{2}(T)$ satisfying theFix a finite interval in R. Take an orthonormal basis
denote theequally dense and uniform boundedness property (see [7,15,16,18,24, etc]). Let$\Phi\in(E)^{*}$
set of all
such that the limit$T$$\{\zeta_{n}\}_{n=0}^{\infty}\subset E$$D_{L}$$\overline{\Delta}_{L}S[\Phi](\xi)=\lim_{Narrow\infty}\frac{1}{N}n\sum^{N}S[\Phi]\prime\prime(\xi)(\zeta n’\zeta_{n})=0-1$

exists for any$\xi\in E_{\mathrm{C}}$and is in $S[(E)^{*}]$ . The L\’evy Laplacian$\Delta_{L}\Phi=S-1\overline{\Delta}_{L}s\Phi$$\Delta_{L}$is defined by

129

for $\Phi\in D_{L}$ . We denote the set of all functionals $\Phi\in D_{L}$ such that $S[\Phi](\eta)=0$ for all $\eta\in E$
.
by
with
$D_{L}^{T}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\eta)\subset T^{c}$Take a generalized white noise functional$\Phi=\int_{T^{n}}f$( $u1,$$\ldots,$un) :$e^{ia_{1}x(u_{1})\ldots ax}e^{i()}nu_{n}$$a_{k}\wedge\in$f\in L_{\mathrm{C}}^{2}(\mathrm{R})^{\otimesn},$\mathrm{R},$$k=1,2,$$\ldots,$:$d\mathrm{u}\in D_{L}^{T}$

$n$

,

(3.1)

,

where: . : means the Wick ordering. Its -transform is given by$S$$S[ \Phi](\xi)=\int_{T^{n}}f(\mathrm{u})eedia1\xi(u1)\ldots ia_{n}\xi(un)\mathrm{u}$Theorem $1.[27]$A generalized white noise functional(3.2)

as in (3.1) satisfies the equation$\Phi$$\triangle_{L}\Phi=-\frac{1}{|T|}\sum_{k=1}^{n}a_{k}^{2}\Phi$

.

.

(3.3)

We set:$\mathrm{D}_{n}=\{\int_{T^{n}}f(\mathrm{u}):\prod_{1}eix(\nu=nu_{\nu})$$d\mathrm{u}\in D_{L}^{T};f\in E_{\mathrm{C}^{\wedge}\}}^{\otimes n}$. Thenis a linear subspace of $(E)_{-p}$ for any $p\geq 1$ . We definefor eachand set,. Then for eacha spacein $(E)_{-p}$ with respect toby the completion of$(E)_{-p}$operator
,
.
the
of
For
each
a
with
the
inner
space
product
becomes Hilbert
into itself satisfying
becomes a continuous linear operator from$n\in \mathrm{N}$$\mathrm{D}_{0}=\mathrm{C}$$\overline{\mathrm{D}}n$$\mathrm{D}_{n}$$n\in \mathrm{N}\cup\{0\}$$||\cdot||_{-p}$$\mathrm{D}_{n}$$n\in \mathrm{N}\cup\{0\}$$\overline{\mathrm{D}}_{n}$$\overline{\mathrm{D}}_{n}$$\Delta_{L}$$|| \Delta_{L}\Phi||_{-}p=\frac{n}{|T|}||\Phi||_{-_{\mathrm{P}}}$

The operator$\overline{\Delta L}$

is a self-adjoint operator on

Proposition 2. [27] Let
are insuch that
and
.
each
for
$\Phi_{n}$$\Psi_{n}$$\overline{\mathrm{D}}_{n}$for any
for each$\Phi\in\overline{\mathrm{D}}_{n}$.

$n\in \mathrm{N}\cup\{0\}$.

be generalized white noise functionals, then. Ifinin$\Phi=\sum_{n=0n}^{\infty}\Phi,$ $\Psi=\sum_{n=0^{\Psi}n}^{\infty}$$\overline{\mathrm{D}}_{n}$for each$n\in \mathrm{N}\cup\{0\}$$\Phi=\Psi$$(E)^{*}$$\Phi_{n}=\Psi_{n}$$(E)^{*}$$n\in \mathrm{N}\cup\{0\}$Letmined as an element ofby$\alpha_{N}(n)=\sum_{f}^{N}=0(\frac{n}{|T|})2\ell$$(E)^{*}$. Proposition 2 says that
. Therefore, for any$\mathrm{E}_{-p,N}=\{_{n}\sum_{=0}^{\infty}\Phi_{n}\in(E)^{*};$

with the norm$|||\cdot|||_{-p,N}$$N\in \mathrm{N}\cup\{0\}$$||\Phi\infty n||_{-p}^{2}<\infty,$$\sum_{n=0}\alpha_{N(}n)$, is uniquely deter, we can define a space$\sum_{n=0}^{\infty}\Phi_{n},$$\Phi_{n}\in\overline{\mathrm{D}}_{n}$$\Phi_{n}\in\overline{\mathrm{D}}_{n},$$\mathrm{E}_{-p,N}$$n=0,1,2,$

given by$||| \Phi|||-p,N=(_{n=}\sum^{\infty}\alpha_{N}(n)||\Phi_{n}||_{-}2\mathrm{I}^{1}\mathrm{o}p/2,$$\Phi=\sum_{n=0}^{\infty}\Phi n\in \mathrm{E}_{-p},N$$\ldots\}$

130

for eachthe norm$N\in \mathrm{N}\cup\{0\}$$|||\cdot|||_{-p,N}$

and $p\geq 1$ . For any$N\in \mathrm{N}\cup\{0\}$and $p\geq 1,$$\mathrm{E}_{-p,N}$

is a Hilbert space with.

Put
by the completion of$\mathrm{E}_{-p,\infty}=\bigcap_{N\geq 1p}\mathrm{E}_{-},N$$\mathrm{E}_{-p,\infty}$

with the projective limit topology. Define a Hilbert spacewith respect to the norm $|||\cdot|||_{-p,-N}$ given by$||| \Phi|||_{-p,N}-=(_{n=}\sum^{\infty}\alpha_{N}(n)^{-1}||\Phi_{n}||_{-p}2)^{1}0/2$

for each $N\geq 0$ . With the inductive limit spacehave the following inclusion relations:,$\mathrm{E}_{-p,-N}$$\Phi=\sum_{n=0}^{\infty}\Phi n\in \mathrm{E}-p,\infty$$\mathrm{E}_{-p,-\infty}\equiv\bigcup_{N\geq 1}\mathrm{E}_{-p,-N}$, for any $N\geq 0$ , we$\mathrm{E}_{-p,\infty}\subset \mathrm{E}_{-p,N+1}\subset \mathrm{E}_{-p,N}\subset \mathrm{E}_{-p,1}\subset \mathrm{E}_{-p,0}\subset \mathrm{E}_{-p,-1}\subset \mathrm{E}_{-p,-N}\subset \mathrm{E}_{-p,-N-1}\subset \mathrm{E}_{-p,-\infty}$.

The spaceincludesfor any $n\in \mathrm{N}\cup\{0\}$ . The
can be extended to acontinuous linear operator defined on, denoted by the same notation, satisfying$N\in$,, for eachis alsoZ. Any restriction ofdenoted by the same notation. Using the similar method of Theorem 2 in [27], we have thefollowing:$\overline{\mathrm{D}}_{n}$$\mathrm{E}_{-p,\infty}$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{\mathrm{o}\mathrm{r}}\overline{\triangle_{L}}$$\overline{\Delta_{L}}$$\mathrm{E}_{-p,-\infty}$$|||\overline{\triangle_{L}}\Phi|||-p)N\leq|||\Phi|||_{-p,N+1}$$\Phi\in \mathrm{E}_{-p,N+1}$$\overline{\triangle_{L}}$$\overline{\Delta L}$

Theorem 3.

on$\mathrm{E}_{-p,N}$The
for each$operato\Gamma\overline{\Delta_{L}}$$N\in \mathrm{Z}$restricted on$\mathrm{E}_{-p,N+1}$

and $p\geq 1$ .

is a self-adjoint operator densely defined

Let $c(t, z)$ be a complex-valued bounded function onwhich is differentiable in andcontinuous in . There exists a constant $M>0$ such that $\sup_{t,z}|c(t, Z)|\leq M$ . For each $t\geq 0$
we consider an operator
ondefined by$\mathrm{R}^{2}$

$t$

$z$

$G_{t}$$\mathrm{E}_{-p,-\infty}$$G_{t} \Phi=\sum_{=n0}\infty c(t,$for$\Phi=\sum_{n=0^{\Phi_{n}}}^{\infty}\in \mathrm{E}_{-p,-\infty}$$\Phi\in \mathrm{E}_{-p,N}$. Then, for any. For any$t\geq 0,p\geq 1$$|||G_{t}\Phi|||_{-p,N}^{2}$

where$\alpha_{N}(n)$meansThus the operator$\alpha_{-N}(n)^{-1}$$G_{t}$$\frac{n}{|T|})\Phi_{n}$$\Phi=\sum_{n=0}^{\infty}\Phi_{n}$, the normin

, there exists
such thatis estimated as follows:$N\in \mathrm{Z}$$\mathrm{E}_{-p,-\infty}$$|||G_{\#}\Phi|||_{-p,N}$$=$$\sum_{n=0}^{\infty}\alpha_{N}(n)||c(t,$ $\frac{n}{|T|})\Phi_{n}||_{-p}^{2}$$\leq$$M^{2} \sum_{n=0}^{\infty}\alpha N(n)||\Phi_{n}||^{2}-p|=M|2|\Phi|||2-p,N$’

for $N\leq 0$ .

is a continuous linear operator from$\mathrm{E}_{-p,-\infty}$

For any $p\geq 1$ and complex-valued continuous function $h(z),$(P) there exists a polynomial $r(z)$ of$z\in \mathrm{R}$into itself.

$z\in \mathrm{R}$satisfying the condition:such that $|h(z)|\leq r(|z|)$ for all$z\in \mathrm{R}$

,

131

the operator$h(-\overline{\triangle_{L}})$on$\mathrm{E}_{-p,-\infty}$

is given by$h(- \overline{\triangle_{L}})\Phi=\sum_{n=0}^{\infty}h(\frac{n}{|T|})\Phi_{n}$

, for

.

$\Phi=\sum_{n=0}^{\infty}\Phi_{n}\in \mathrm{E}_{-p,-\infty}$

such that $h(z)$ is
Theorem 4. Let $h(z)$ be a complex-valued bounded function $h(z)$ ofcontinuous, $h(\mathrm{O})=0$ and $c(t, z)=e^{h(z)t}$ for all $t\geq 0$ . If $h(z)$ satisfies the condition (P), then.
generated by
the family $\{G_{t}; t\geq 0\}$ is an equi-continuous semigroup of class$z\in \mathrm{R}$$h(-\overline{\Delta_{L}})$$(C_{0})$

Proof. If there exists a complex-valued continuous function $h(z)$ of, then it is easily checked that $G_{0}=I,$ $G_{t}G_{s}=G_{t+s}$ for eachestimate that$e^{h(z)t}$$|||G_{t}\Phi-G_{t}\Phi|0||_{-p,N}^{2}$for eachtheorem, we get that$t,$$t_{0}\geq 0,$ $N\in \mathrm{Z}^{*}$

andsuch that $c(t, z)=$. Moreover we can$s\geq 0$$\frac{n}{|T|})|^{2}||\Phi_{n}||2-p$$\sum_{n=0}^{\infty}\alpha_{N()}n|c(t,$$\leq$$4M^{2} \sum_{n=0}^{\infty}\alpha_{N}(n)||\Phi_{n}||^{2}-\mathrm{P}|=4M||\Phi|||_{-}2<N..\infty 2\mathrm{P}$,$\Phi=\sum_{n=0}^{\infty}\Phi_{n}\in \mathrm{E}_{-p,N}$

in. Therefore, by the Lebesgue convergence$\mathrm{E}_{-p,\infty}$. Thus the family $\{G_{t}; t\geq 0\}$ is an equi-continuous semigroup
andsuch that. Then, there exists. Let $p\geq 1$ and let
that
we
note
(P).
Then
conditiontheinpolynomialof
thebe the degree. Let$t_{0}\geq 0$$\Phi\in \mathrm{E}_{-p,-\infty}$$N\in \mathrm{Z}$$\Phi=\sum_{n=0}^{\infty}\Phi n\in \mathrm{E}_{-p,-\infty}$$(C_{0})$$\Phi\in \mathrm{E}_{-p,N}$

$t,$

$=$

$\lim_{tarrow t_{0}}c_{t}\Phi=Gt\mathrm{o}\Phi$for eachof class$\frac{n}{|T|})-C(t_{0},$$z\in \mathrm{R}$$r$$d_{r}$$||| \frac{G_{t}\Phi-\Phi}{t}-h(-\overline{\triangle L})\Phi|||_{-_{\mathrm{P},r}}^{2}N-dr=n=0\sum^{\infty}\alpha_{N-d}(n)||(\frac{e^{h(\frac{n}{|T|})}t-1}{t}-h(^{\frac{n}{|T|}})_{\mathrm{I}}\Phi_{n}||_{-p}^{2}$

By the mean value theorem, for any $t>0$ there exists a constant $\theta\in(0,1)$ such that$| \frac{e^{h(\frac{n}{|T|})}t-1}{t}|=|h(\frac{n}{|T|})|e)]t\theta\leq MrRe[h(^{\frac{n}{|T|}}(^{\frac{n}{|T|})}$.

Therefore we get that$|| \frac{e^{h(\frac{n}{|T|})t}-1}{t}\Phi_{n}-h(\frac{n}{|T|})\Phi_{n}|\frac{e^{h(\frac{n}{|T|})t}-1}{t}-h(^{\frac{n}{|T|}})|^{2}||\Phi n||_{-p}^{2}$$\leq$$2(M+1)r( \frac{n}{|T|})^{2}||\Phi n||^{2}-p$

.

(3.4)

132

Since there exists a positive constant
we have$C_{r}$

depending on such that$r$$\grave{\sum_{n=0}^{\infty}}\alpha_{N-}b(n)r(\frac{n}{|T|})2||\Phi n||_{-p}^{2}<\infty$$\alpha_{N-d_{r}}(n)r(\frac{n}{|T|})^{2}\leq c_{r}\alpha_{N}(n)$

.

,

(3.5)

By (3.4), (3.5) and$\lim_{tarrow 0}|\frac{e^{h(\frac{n}{|T|})}t-1}{t}-h(\frac{n}{|T|})|=0$,

the Lebesgue convergence theorem admits$\lim_{tarrow 0}|||\frac{G_{t}\Phi-\Phi}{t}-h(-\overline{\Delta L})\Phi|||_{-p,N-}^{2}k=0$

Thus the proof is completed.

.

$\square$

4. A relationship to the Number operatorLet$\beta(p)=\sum_{k=\mathrm{o}(2k}^{\infty}+2)^{-2p}|ek|_{0}2$and set$[E]_{p,N}= \{\varphi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(f_{n})\in(L^{2});\sum_{n=0}^{\infty}\alpha N(n)e|n\beta 2(\mathrm{P})fn|_{0}^{2}<\infty,$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f_{n})\subset T,$

for $p\geq 0$ and $N\geq 0$ . The space $[E]_{p,N}$ is a Hilbert space with norm

$n=0,1,2,$$||\cdot||_{[E]_{\mathrm{p},N}}$$\ldots\}$

given by$|| \varphi||_{[E]_{\mathrm{p},N}}=(_{n=0}\sum^{\infty}o_{N(}n)e|n^{2}\beta(p)fn|_{0}^{2})^{1}/2$

for$\varphi=\sum_{n=0}^{\infty}\mathrm{I}_{n}(f_{n})\in(L^{2})$.

Define an operator $K$ on $[E]_{p,N}$ by$K[\Phi]=s^{-1}[s[\Phi](e^{i\xi})]$.

Then we have the following:

Proposition 4. Let $p\geq 1$ and let $N\geq 1$ . Then the operator $K$ is a continuous linear operator.
from $[E]_{p,N}$ into$\mathrm{E}_{-p,N}$

Proof.

of$K[\varphi]$

Let $p\geq 1$ and let $N\geq 1$ . Then for eachfor
I $(f_{n})\in[E]_{p,N}$ as follows:$\ell\geq 1$

we can calculate the norm$|||K[\varphi]|||_{-p,N}^{2}$$\varphi=\sum_{n=0}^{\infty}$$|||K[\varphi]|||_{-p}^{2},N$$=$$\sum_{n=0}^{\infty}\alpha_{N}(n)||\langle:(e^{ix})^{\otimes}n:, f_{n}\rangle||_{-p}2$$\leq$$\sum_{n=0}^{\infty}\alpha_{N}(n)\sum\ell\infty!$$\sum\infty$

$l=0$

.

$\prod(2kj+2)f-2p|_{|\nu|=}\sum_{l}\frac{1}{\nu!}\langle F_{\nu’ k}e\otimes\cdots\otimes e_{k\ell}\rangle 1|^{2}$$k_{1},\ldots,k_{\ell}=0j=1$

,

133

where

$\nu=(\nu_{1}, \ldots, \nu_{n})\in \mathrm{N}\cup\{0\},$$|\nu|=\nu_{1}+\cdots+\nu_{n},$ $\nu!=\nu_{1}!,$$.\nu_{n}!$
$,$

and

$F_{\nu}= \int_{\mathrm{R}^{n}}f(\mathrm{u})\otimes^{n}j=1\delta^{\otimes}u_{j}d\mathrm{u}\wedge\sim\nu_{j}$Since there exists $q\geq 0$ such that$\sum_{k_{1},\ldots,k\ell=0}^{\infty}\prod_{j=1}l(2k_{j}+2)^{-}2p|_{|\nu|\ell}\sum_{=}\frac{1}{\nu!}\langle F\nu’ e_{k_{1}}\otimes\cdots\otimes e_{k\ell}\rangle|^{2}$$|f_{n}|_{0}^{2}( \sum_{|\nu|=\ell}\frac{1}{\nu!}\mathrm{I}2(_{k=0}\sum^{\infty}(2k+2)^{-}2p|ek|^{2)}0l$$\leq$,

we get that$|||K[\varphi]|||_{-p}^{2},N$

Thus the proof is completed.

$\leq$$\sum_{n=0}^{\infty}\alpha N(n)e|n^{2}\beta(p)f_{n}|_{0}^{2}$$=$$||\varphi||_{[E]_{\mathrm{p},N}}$.

$\square$Regarding $K$ as an operator from $[E]_{p,N}$ onto $K[[E]_{p,N}]$ , it is a bijection. Define a
on $K[[E]_{p,N}]$ byRemark:norm [$\cdot \mathrm{I}-p,N$$[\Phi \mathrm{J}_{-p,N}=||K^{-1}\Phi||p,N$

for $\Phi\in K[[E]_{p,N}]$ . Let $\mathcal{K}_{-p,N}=\{\Phi\in K[[E]_{p,N}];[\Phi \mathrm{J}_{-p},N<\infty\}$ . Then this becomes a Hilbert
space with the norm [and the operator $K$ is a homeomorpism from.
onto$[E]_{p_{)}N}$$\cdot \mathrm{J}_{-p,N}$Let
be the projective limit space of spaces $[E]_{p,N}$ ,continuous linear operator from
into
. Let
the characteristic function of
given by$\{X_{t}\cdot, t\geq 0\}$$\mathrm{E}_{-p,\infty}$. Then the operator $K$ is a
be a stochastic process with

$N\in \mathrm{N}$$[E]_{p,\infty}$$[E]_{p,\infty}$$\mathcal{K}_{-p,N}$$X_{t}$$E[e^{izX_{l}}]=e^{th(z)}$and let $\{G_{t};t\geq 0\}$ be an equi-continuous semigroup of classgenerated by
Theorem 4. Take a smooth function
withon . Define an operatoronby$(C_{0})$$\eta_{T}\in E$$T$$\eta_{T}=\frac{1}{|T|}$in
acting$h(-\overline{\triangle_{L}})-$$G_{t}$$S[\mathrm{E}_{-p,\infty}]$$\overline{G_{t}}=SG_{t}s-1$.

Here the spaceis endowed with the topology induced fromby the S-transform.
Then by Theorem 3.5,
is an equi-continuous semigroup of classgenerated by, wherethe operator. Then we have the following theorem.
$S[\mathrm{E}_{-}]p,\infty$$\mathrm{E}_{-p,\infty}$$\{\overline{G_{t}};t\geq 0\}$$h(-\overline{\triangle_{L}})$

Theorem$5.[\mathrm{c}\mathrm{f}.

26,29]$$(C_{0})$$\overline{\triangle_{L}}=S\overline{\triangle_{L}}S^{-}1$

For all$F\in S[\mathrm{E}_{-p,\infty}]$, the following equality holds$\overline{G_{t}}F(\xi)=\mathrm{E}[F(\xi+X_{\mathrm{t}}^{\gamma}\eta_{T})]$

.

.

134

The operator $K$ implies a relationshipby$\mathrm{b}\mathrm{e}\mathrm{t}_{\mathrm{W}\mathrm{e}\mathrm{e}}\mathrm{n}\overline{\Delta_{L}}$$N \Phi=\sum_{n=0}^{\infty}n\mathrm{I}_{n}(f_{n})$Proposition 6.

forand the number operator $N$ on$\Phi=\sum_{n=0^{\mathrm{I}}n}^{\infty}(f_{n})\in(E)^{*}$

given

.

For any $\Phi\in[E]_{p,N}$ we have$\overline{\Delta_{L}}K[\varphi]=-\frac{1}{|T|}K[N[\varphi]]$

Theorem 7.

$(E)^{*}$.

For any $\varphi\in[E]_{p,N}\cap(E)$ we have,$\overline{K}[e^{th}(-\frac{1}{|T|}N)]\varphi(\xi)=E[\overline{K}[\varphi](\xi+Xt\eta_{T})\gamma]$

where$\overline{K}[\varphi](\xi)=S[\varphi](e^{i})\xi,$$\xi\in E_{\mathrm{C}}$.

Acknowledgements

The authors would like to express deepest gratitude to Professor I. D\^oku for organizing
the stimulating workshop “ Mathematical Models and Stochastic Processes Arising in Natural
Phenomena and Their Applications”.
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