数理解析研究所講究録 1266 巻 2002 年 59-81

59
Segal-Bargmann Transform of White Noise Operators
and White Noise Differential Equations
UN CIG JI
DEpARTMENT OF MATHEMATICSCHUNGBUK NATIONAL UNIVERSITYCHEONGJU, 361-763 KOREA*ANDNOBUAKI OBATA \dag er
GRADUATE SCHOOL OF INFORMATION SCIENCES
Tohoku UNIVERSITYSENDAI, 980-8579 JAPAN
Abstract The Segal-Bargmann transform is applied to characterization for symbols
of white noise operators. Ageneral formulation of an initial value problem for white
noise operators is given and unique existence of asolution is proved by means ofsymbols. Regularity properties of the solution is discussed by introducing Fock spaces
interpolating the space of white noise distributions and the original Boson Fock space.
Keywords: Bargmann-Segal space, Segal-Bargmann transform, white noise theory,
Gaussian analysis, operator symbol, white noise differential equation, weighted Fockspace, Wick product1Introduction
An interesting framework for nonlinear stochastic analysis is offered by white noise operator theory or quantum white noise calculus, where singular noises such as higher powers ofquantum white noises are discussed systematically. In particular, as an extension of quantum stochastic differential equations (quantum It\^o theory) white noise differential equations(WNDEs) has become acentral topic for substantial development of white noise theory [4],[5], [27], [28]. Among others, investigation of regularity properties of solutions is important
In this paper we show that the Segal-Bargmann
but has not yet achieved
transform, which has been extensively studied, see e.g., [9], [10], [22], [30], is naturally extended for white noise operators and can be anew clue to answer this question.
describes a. In quantum physics,
Consider the Boson Fock spacequantum field theory on the 1-dimensional space ;while, in quantum stochastic calculus
is understood as atime axis. Then, field operators at each time point[23], [29] thisare considered as noise generators. In particular, the pair of annihilation and creation
is called aquantum white noise and plays afundamental role in quantum
operators
is to smear thewhite noise calculus. One traditional way of giving ameaning of ,$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}1\mathrm{y}_{\backslash }$$\Gamma(L^{2}(\mathrm{R}))$$\Gamma(L^{2}(\mathrm{R}))$$\mathrm{R}$$\mathrm{R}$$t\in \mathrm{R}$$\{a_{t}, a_{t}^{*}\}$$a_{t}$

Supported by the Brain Korea 21 Project.

\dagger Supported by JSPS Grant-in-Aid for Scientific Research (No. 12440036)$a_{t}^{*}$

60
time, i.e., such field operators are formulated as (unbounded) operator-valued distributionsin. Then, the time parameter disappears and observation of the time evolution isalways indirect. Another is to introduce aGelfand triple:$t\in \mathrm{R}$$t$$\mathcal{W}\subset\Gamma(L^{2}(\mathrm{R}))\subset \mathcal{W}^{*}$,

where such field operators at apoint are formulated as continuous operators from
into
$W$ . Prom the Fock space viewpoint, these are not proper operators
but something like
distribution (or generalized operators). The white noise theory is based on the latter idea,
see e.g., [12], [14], [21]. In this paper we adopt the recent framework proposed by Cochran,Kuo and Sengupta [6]. In general, acontinuous operators frominto
is called awhite
noise operator and we denote by
the space of such operators. Asystematic studyof white noise operators has been launched out in [24] and developed extensively along withthe symbol calculus, see e.g., [3], [26].
It is our long-range project to develop atheory of differential equations for white noise
operators. During the last years our main attention has been paid to anormal-0rdered white
noise differential equation:$\mathcal{W}$$\mathcal{W}$$\mathcal{W}^{*}$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\frac{\Pi-}{dt}=L_{t}\mathrm{o}---$

where

,

$—|_{t=0}=---0$

,

(1.1)

is acontinuous map (also called aquantum stochastic pr0cess). Such alinear equation arises from avariety of mathematical models in theoreticalphysics. For example, aquantum stochastic differential equation introduced by Hudson andParthasarathy [15] is equivalent to anormal-0rdered white noise differential equation withinvolving only lower powers (i.e., linear terms) of quantum white noises. In the seriesof papers [4], [5], [27], [28], we proved unique existence of asolution in the space of white
noise operators and established amethod of examining its regularity properties in terms ofweighted Fock spaces. Moreover, in the recent paper [18] we started an approach on thebasis of infinite dimensional holomorphic functions.
The next step is to discuss anonlinear equation beyond the normal-0rdered white noise
equations (1.1). In this paper we focus on an initial value problem for white noise operators:$t\vdash*L_{t}\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\{L_{t}\}$$\frac{\mathcal{L}-}{dt}=F(t, ---)$,$—|_{t=0}=---0$

,

,

(1.2)

$0\leq t\leq T$

where
is acontinuous function. The usual characterization theorem for operator symbols is powerful to solve (1.2), however, is not sufficient toclaim regularity properties of the solution. To overcome this situation, in the recent paper
Ji-Obata [17], anew aspect of operator symbols is introduced from the viewpoint of theSegal-Bargmann transform. In this paper, we show that the new idea helps to investigate aproper Fock space in which the solution acts as ausual (unbounded) operator rather thanageneralized operator. We hope that the main result stated in Theorem 7.2, which needs
more mature consideration, is asmall step toward our goal.
$F:[0, T]\cross \mathcal{L}(\mathcal{W}, \mathcal{W}^{\cdot})arrow \mathcal{L}(\mathcal{W}, \mathcal{W}^{\cdot})$

2Preliminaries
2.1

Boson Fock space and weighted Fock space

Let $H$ be areal or complex Hilbert space with norm
. For $n\geq 0$ let
denotethe -fold symmetric tensor power of aHilbert space $H$ . Their norms are denoted by the$|\cdot|$$n$$H^{\hat{\Phi}n}$61common symbol |.| for simplicity. Given apositive sequence$\Gamma_{\alpha}(H)=\{\phi=(f_{n})_{n=0}^{\infty};f_{n}\in H^{\otimes n}\wedge$,$\alpha=\{\alpha(n)\}_{n=0}^{\infty}$

we put$|| \phi||_{+}^{2}\equiv\sum_{n=0}^{\infty}n!\alpha(n)|f_{n}|^{2}<\infty\}$.

Thenbecomes aHilbert space and is called aweighted Fock space with weight sequence.
. The Boson Fock spaceis aspecial case of$\Gamma_{\alpha}(H)$$\alpha(n)\equiv 1$$\Gamma(H)$$\alpha$is densely imbedded in another Hilbert space
Lemma 2.1 Assume that a Hilbert space
be a positive sequenceis a contraction. Let
and the inclusion map$\inf\alpha(n)>0$. Then we have continuous inclusions with dense images:
such that$H_{2}$$\alpha=\{\alpha(n)\}$$H_{2}\mapsto H_{1}$$H_{1}$$\Gamma_{\alpha}(H_{2})\ovalbox{\tt\small REJECT} \mathrm{e}arrow\Gamma(H_{2})‘arrow\Gamma(H_{1})$.

Moreover the second inclusion is a contraction.
2.2

Rigged Hilbert space constructed from aselfadjoint operator

This is astandard construction, see e.g., [2], [8]. Let $H$ be acomplex Hilbert space and. We
aselfadjoint operator with dense domain Dom $(T)\subset H$ such that
becomes abounded operator on $H$ and put
note that$T$$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)>0$$T^{-1}$$p_{T}=||T^{-1}||_{\mathrm{o}\mathrm{P}}=( \inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T))^{-1}$Then, for each $p\geq 0$ , the dense subspace$D_{p}\equiv \mathrm{D}\mathrm{o}\mathrm{m}$.
becomes aHilbert space$(T^{p})\subset H$

equipped with the norm$|\xi|_{T,p}=|T^{p}\xi|_{0}$,$\xi\in \mathrm{D}\mathrm{o}\mathrm{m}$$(T^{p})$,to be the completion of $H$ withis the norm of $H$ . Furthermore, we define
whererespect to the norm $|\xi|_{T,-p}=|T^{-p}\xi|_{0}$ , $\xi\in H$ . In view of astraightforward inequality:$D_{-p}$$|\cdot|_{0}$$|\xi|_{T\mathrm{p}}\leq\rho_{T}^{q-p}|\xi|_{T,q}$,$\xi\in D_{q}$,$-\infty<p\leq q<+\infty$ ,we come to aHilbert riggings:.

$..\subset D_{q}\subset\cdots\subset D_{p}\subset\cdots\subset D_{0}=H\subset\cdots\subset D_{-p}\subset\cdots\subset D_{-q}\subset\cdots$

,

(2.1)

the
where each inclusion is continuous and has adense image. Moreover, for any ,(2.1)
weobtain.
Prom
ontoisometryfromasanoperator
is naturally considered$p$$T^{p-q}$

Obviously,
such that,$D_{\infty}^{*}= \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty}D_{-p}$.

is acountable Hilbert space. It is nuclear if and only if there exists $p>0$
is of Hilbert-Schmidt type

$D_{\infty}$$T^{-p}$$D_{q}$$D_{p}$$D_{\infty}= \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim_{\infty parrow}D_{p}$$q\in \mathrm{R}$

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2.3

Riggings of Fock spacesLet
be apositive sequence such that $\inf\alpha(n)>0$ . Based on arigged Hilbert
space (2.1), we obtain achain of weighted Fock spaces:$\alpha=\{\alpha(n)\}$$\ldots\subset\Gamma_{\alpha}(D_{q})\subset\cdots\subset\Gamma_{\alpha}(D_{p})\subset\cdots$$\subset$$\Gamma_{\alpha}(D_{0})=\Gamma_{\alpha}(H)\subset\Gamma(H)$where Lemma 2.1 is taken into account. By definition the norm ofby$|| \phi||_{p,+}^{2}=\sum_{n=0}^{\infty}n!\alpha(n)|f_{n}|_{p}^{2}$

Identifying$\Gamma(H)$,,$0\leq p\leq q$$\Gamma_{\alpha}(D_{p})$$\phi=(f_{n})\in\Gamma_{\alpha}(D_{p})$,, $p\geq 0$ , is given.

with its dual, we have$\Gamma_{\alpha}(D_{p})^{*}\cong\Gamma_{\alpha^{-1}}(D_{-p})$where the norm of$\Gamma_{\alpha^{-1}}(D_{-p})$is defined by$|| \Phi||_{-p,-}^{2}=\sum_{n=0}^{\infty}\frac{n!}{\alpha(n)}|F_{n}|_{-p}^{2}$

The canonical complex bilinear form on
and$\Phi=(F_{n})\in\Gamma_{\alpha}(D_{p})^{*}$,,$\Phi=(F_{n})$$\Gamma_{\alpha}(D_{p})^{*}\cross\Gamma_{\alpha}(D_{p})$$\phi=(f_{n})\in\Gamma_{\alpha}(D_{p})$.

is denoted by$\langle\langle\cdot, \cdot\rangle\rangle$. Then for

it holds that$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$.

With these notations we come to arigging of the Fock space$\Gamma(H)$:$...\subset\Gamma_{\alpha}(D_{q})\subset\cdots\subset\Gamma_{\alpha}(D_{p})\subset\cdots\subset\Gamma(H)\subset\cdots\subset\Gamma_{\alpha^{-1}}(D_{-p})\subset\cdots\subset\Gamma_{\alpha^{-1}}(D_{-q})\subset\cdots$,

where $0\leq p\leq q$ . Furthermore, we obtain$\Gamma_{\alpha}(D)=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim_{\infty parrow}\Gamma_{\alpha}(D_{p})\subset\Gamma(H)\subset\Gamma_{\alpha}(D)^{*}=\mathrm{i}\mathrm{n}_{\mathrm{P}}\underline{\mathrm{d}}\lim_{\rangle\infty}\Gamma_{\alpha^{-1}}(D_{-p})$

,

where
is acountable Hilbert space. The canonical bilinear form is denoted by thesame symbol.
$\Gamma_{\alpha}(D)$$(\langle\cdot, \cdot\rangle\rangle$

3Two Riggings of Fock Space
From now on we denote by andthe space of complex valuedof real valued ones, respectively. We shall construct two riggings of$\mathcal{H}$$\mathcal{H}_{\mathrm{R}}$$L^{2}$-functions and that:$\Gamma(\mathcal{H})$$\mathcal{W}_{\alpha}\equiv \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim\Gamma_{\alpha}(\mathcal{E}_{p})\subset\Gamma(\mathcal{H})\subset \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty parrow\infty}\Gamma_{\alpha^{-1}}(\mathcal{E}_{-p})=W_{\alpha}$,.

$\mathcal{G}_{\infty}\equiv \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim\Gamma(D_{p})\subset\Gamma(?t)\subset \mathrm{i}\mathrm{n}\underline{\mathrm{d}}\mathrm{h}.\mathrm{m}\Gamma(D_{-p})=\mathcal{G}_{\infty}^{*}parrow\infty \mathrm{P}^{\rangle\infty}$

The former will be referred as aCKS-space and the latter as aFock chain

63
CKS-space

3.1

Consider the famous selfadjoint operator:.

$A=1+t^{2}- \frac{d^{2}}{dt^{2}}$As is well known, there exists an orthonormal basis
and$(2k+2)e_{k}$ , $k\geq 0$ . In particular,$\{e_{k}\}_{k=0}^{\infty}\subset \mathcal{H}_{\mathrm{R}}$of 7{ such that $Ae_{k}=$$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A)=2$$\rho\equiv||A^{-1}||_{\mathrm{o}\mathrm{P}}=\frac{1}{2}$,$||A^{-q}||_{\mathrm{H}\mathrm{S}}^{2}= \sum_{k=0}^{\infty}\frac{1}{(2k+2)^{2q}}<\infty$,$q> \frac{1}{2}$.

By the standard method mentioned in \S 2.2 we obtain aGelfand triple:$\mathcal{E}\equiv \mathrm{p}\mathrm{r}$$p arrow\infty \mathrm{o}\mathrm{j}\lim \mathcal{E}_{p}\subset \mathcal{H}\subset \mathcal{E}^{*}\cong \mathrm{i}\mathrm{n}\mathrm{d}\lim \mathcal{E}_{-p}parrow\infty,\cdot$

For simplicity, we write$|\xi|_{p}=|A^{p}\xi|_{0}$,$\xi\in \mathcal{E}_{p}$.

of is also defined by asimilar method for
The real partnote the topological isomorphisms:$\mathcal{E}$$A$$\mathcal{E}_{\mathrm{R}}$$\mathcal{E}_{\mathrm{R}}\cong \mathrm{S}(\mathrm{R})$,$\mathcal{E}_{\mathrm{R}}^{*}\cong S’(\mathrm{R})$is areal operator. We,

the space of temperedis the space of rapidly decreasing functions andwhere
distributions.
For our purpose we choose aweight sequence $\alpha=\{\alpha(n)\}$ satisfying the following four
conditions:$\mathrm{S}’(\mathrm{R})$$S(\mathrm{R})$(A1) $\mathrm{a}(0)=1$ and there exists some(A2)$\lim_{narrow\infty}\{\frac{\alpha(n)}{n!}\}^{1/n}=0$$\sigma\geq 1$such that;;is equivalent to apositive sequencesuch that(A4) there exists aconstant(A3)$\inf_{n\geq 0}\alpha(n)\sigma^{n}>0$such that $\{\gamma(n)/n!\}$ is log-concave;$\alpha(m)\alpha(n)\leq C_{1\alpha}^{m+n}\alpha(m+n)$ for all $m$ ,.

$\gamma=\{\gamma(n)\}$$\alpha$$C_{1\alpha}>0$$n$Given such aweight sequence , we obtain$\alpha$$\mathcal{W}_{\alpha}\equiv \mathrm{p}\mathrm{r}$$p arrow\infty \mathrm{o}\mathrm{j}\lim\Gamma_{\alpha}(\mathcal{E}_{p})\subset\Gamma(\mathcal{H})\subset \mathrm{i}\mathrm{n}\mathrm{d}\lim\Gamma_{\alpha^{-1}}(\mathcal{E}_{-p})=\mathcal{W}_{\alpha}^{*}parrow\infty,$

is anuclear space.
which is referred to as a $CKS$-space. Recall thatis defined by
The generating function of$\mathcal{W}_{\alpha}$$\{\alpha(n)\}$,$G_{\alpha}(t)= \sum_{n=0}^{\infty}\frac{\alpha(n)}{n!}t^{n}$

’

(3.1)

64
which is entire holomorphic by condition (A2). Moreover, we have
$G_{\alpha}(0)=1$$G_{\alpha}(s)\leq G_{\alpha}(t)$,$\gamma[G_{\alpha}(t)-1]\leq G_{\alpha}(\gamma t)-1$,$0\leq s\leq t$,$\gamma\geq 1$

,
,

$t$

$\geq 0$.

(3.2)

It is known [1] that condition (A3) is necessary and sufficient for the power series.
$\tilde{G}_{\alpha}(t)=\sum_{n=0}^{\infty}\frac{n^{2n}}{n!\alpha(n)}\{\inf_{s>0}\frac{G_{\alpha}(s)}{s^{n}}\}t^{n}$to have apositive radius of convergence.
Concrete examples ofsatisfying conditionsare (i)with, called the Bell numbers of order , defined by$R_{\alpha}>0$$\{\alpha(n)\}$$(n!)^{\beta}$$(\mathrm{A}1)-(\mathrm{A}4)$$\mathrm{a}(\mathrm{n})\equiv 1;(\mathrm{i}\mathrm{i})\mathrm{a}(\mathrm{n})=$$0\leq\beta<1;(\mathrm{i}\mathrm{i}\mathrm{i})\alpha(n)=\mathrm{B}\mathrm{e}11_{k}(n)$$k$$= \sum_{n=0}^{\infty}\frac{\alpha(n)}{n!}t^{n}$.

The corresponding CKS-spaces in the case of (i) and (ii) are called the $Hida-Kubo$ -Takenaka
space [20] and Kondratiev-Streit space [19], respectively. CKS-spaces are also constructedby means of infinite dimensional holomorphic functions, see Gannoun-Hachaichi-OuerdianeRezgui [7].

3.2Fock chain
Let $K$ be aselfadjoint operator in

(i)
(ii)

(iii)

Here

$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(K)\geq 1$$\mathcal{H}$satisfying the following conditions:;is invariant under $K$ ;is densely and continuously imbedded in$\mathcal{E}_{\mathrm{R}}$$\mathcal{E}$$D_{p}$

for all $p\geq 0$ .

stands for the Hilbert space obtained from Dom$|\xi|_{K,p}=|K^{p}\xi|_{0}$ . We set$D_{p}$$\mathcal{G}_{p}=\Gamma(D_{p})$By definition, the norm of$\mathcal{G}_{p}$,$p\in \mathrm{R}$$(K^{p})$equipped with the norm.

is given by$|| \phi||_{K,p}^{2}=\sum_{n=0}^{\infty}n!|f_{n}|_{K\mathrm{p}}^{2}$$\phi=(f_{n})$,$f_{n}\in D_{p}^{\hat{\Phi}n}$

.

(3.3)

Then we come to$\mathcal{G}_{\infty}\equiv \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim \mathcal{G}_{p}\subset\Gamma(\mathcal{H})\subset \mathrm{i}\mathrm{n}\mathrm{d}\lim \mathcal{G}_{-p}=\mathcal{G}_{\infty}^{*}parrow\infty parrow\infty$

’

(3.4)

where
becomes a countable Hilbert space equipped with the Hilbertian norms definedby (3.3). In general,
is not anuclear space$\mathcal{G}_{\infty}$$\mathcal{G}_{\infty}$

65
Lemma 3.1 For any weight sequence satisfying conditions
continuous inclusions with dense images:$(\mathrm{A}1)-(\mathrm{A}3)$$\alpha$$\mathcal{W}_{\alpha}\subset \mathcal{G}_{p}\subset\Gamma(\mathcal{H})\subset(i$ $-p\subset \mathcal{W}_{\alpha}^{*}$

Proof.

Since$\mathcal{E}\mapsto D_{p}$and p$\geq 0$.

is continuous, there exist $C\geq 0$ and $p’\geq 0$ such that$|\xi|_{K,p}\leq C|\xi|_{\theta}\leq C\rho^{q-p’}|\xi|_{q}$,,$\xi\in \mathcal{E}$

Hence for asufficiently large $q\geq 0$ we haveis acontraction. It then follows from Lemma 2.1 that$|\xi|_{K,p}\leq|\xi|_{q}$,$q\geq p’$$\xi\in \mathcal{E}$.

;in other words,
is continuous.

$\cdot$,$\mathcal{E}_{q}\mapsto D_{p,1}$$\Gamma_{\alpha}(\mathcal{E}_{q})\mathrm{L}arrow\Gamma(D_{p})$It is reasonable by Lemma 3.1 to denote the canonical complex bilinear form on
by , . Then, obviously,$\langle\langle\cdot$

we have$\mathcal{G}_{\infty}^{*}\cross \mathcal{G}_{\infty}$$\rangle\rangle$$|\langle\langle\Phi, \phi\rangle\rangle|\leq||\Phi||_{K,-p}||\phi||_{K,p}$,$\Phi\in \mathcal{G}_{\infty}^{*}$,$\phi\in \mathcal{G}_{\infty}$.

4Gaussian Space and Bargmann-Segal Space
4.1

Gaussian space
Recall the Gelfand triple:

.

(4.1)

$\mathcal{E}_{\mathrm{R}}=S(\mathrm{R})\subset \mathcal{H}_{\mathrm{R}}=L^{2}(\mathrm{R}, dt)_{\mathrm{R}}\subset \mathcal{E}_{\mathrm{R}}^{*}=S’(\mathrm{R})$By the Bochner-Minlos theorem, for each $\sigma>0$ there exists aprobability measuresuch that$\mu_{\sigma^{2}}$on$\mathcal{E}_{\mathrm{R}}^{*}$,$\xi\in \mathcal{E}_{\mathrm{R}}$$\exp\{-\frac{\sigma^{2}}{2}\langle\xi, \xi\rangle\}=\int_{\mathcal{E}_{\mathrm{R}}^{*}}e^{:\langle x,\xi)}\mu_{\sigma^{2}}(dx)$

We put $\mu=\mu_{1}$ for simplicity. Then the probability space
Gaussian space. Define aprobability measure on$\mathcal{E}^{*}=\mathcal{E}_{\mathrm{R}}^{*}+i\mathcal{E}_{\mathrm{R}}^{*}$$\nu$$\nu(dz)=\mu_{1/2}(dx)\cross\mu_{1/2}(dy)$Following Hida [13] the probability space
space associated with (4.1).
4.2,$(\mathcal{E}^{*}, \nu)$$(\mathcal{E}_{\mathrm{R}}^{*}, \mu)$$z=x+iy$ ,

$x$

,

.

is called the (standard)in such away that$y\in \mathcal{E}_{\mathrm{R}}^{*}$.

is called the (standard) complex Gaussian

Wiener-It\^o-Segal isomorphism

Theorem 4.1 (Wiener-It\^o-Segal) There exists a unitary isomorphism between, which is uniquely determined by the correspondence:and$L^{2}(\mathcal{E}_{\mathrm{R}}^{*}, \mu)$$\Gamma(\mathcal{H})$$e^{(x,\xi\rangle-(\xi,\xi\rangle/2}$

where 4runs over$\mathcal{E}$–$\phi_{\xi}\equiv(1,$

$\xi$

,

$\frac{\xi^{\otimes 2}}{2!}$,$\ldots$,$\frac{\xi^{\mathfrak{H}n}}{n!}$,$\ldots$

),

(4.2)

.

The above is called an exponential vector or acoherent vector. We often use the same
symbol for the left hand side of (4.2)$\phi_{\xi}$

66
4.3

Bargmann-Segal space

The Bargmann-Segal space, denoted bytions
such that$E^{2}(\nu)$, is by definition the space of entire func-$g:\mathcal{H}arrow \mathrm{C}$$||g||_{E^{2}(\nu)}^{2} \equiv\sup_{P\in \mathcal{P}}\int_{\mathcal{E}}$.

$|g(Pz)|^{2}\nu(dz)<\infty$,. Notewith range contained inwhere $P$ is the set of all finite rank projections on
$P\in
P$
(in
),
operator
into
fact
into
ffom
naturally
thatisextended to acontinuous
is aHilbert space
which is denoted by the same symbol. The Bargmann-Segal space
define. For
with norm$\mathcal{E}_{\mathrm{R}}$$\mathcal{H}_{\mathrm{R}}$$\mathcal{E}^{*}$$\mathcal{E}$$\mathcal{H}$$E^{2}(\nu)$$||\cdot||_{E^{2}(\nu)}$$\phi=(f_{n})_{n=0}^{\infty}\in\Gamma(\mathcal{H})$$J \phi(\xi)=\sum_{n=0}^{\infty}\langle\xi^{\Phi n}, f_{n}\rangle$,$\xi\in \mathcal{H}$

.

(4.3)

becomes
Since the right hand side converges uniformly on each bounded subset of ,an entire function on ??. Moreover, it is known (e.g., [9], [10]) that becomes aunitary. In fact, forisomorphism fromwe have
onto$J\phi$$\mathcal{H}$$J$$\Gamma(\mathcal{H})$$E^{2}(\nu)$.

$||J \phi||_{E^{2}(\nu)}^{2}=\sup_{P\in \mathcal{P}}\int_{\mathcal{E}}$$\phi\in\Gamma(\mathcal{H})$$| \langle\langle\phi, \phi_{Pz}\rangle\rangle|^{2}\nu(dz)=\sup_{P\in \mathcal{P}}||\Gamma(P)\phi||_{0}^{2}=||\phi||_{0}^{2}$.

The map defined in (4.3) is called the duality transform and is related with the S-transform(see (4.5) below) in an obvious manner:$J$$J\phi|v_{\infty}=S\phi$,$\phi\in\Gamma(\mathcal{H})$,which follows from (4.5) and (4.3).
4.4

White noise functions

The riggings obtained from (3.1) and (3.4) through the Wiener-It\^o-Segal isomorphism
are denoted respectively by,$\mathcal{W}_{\alpha}\subset \mathcal{W}_{p}\subset L^{2}(\mathcal{E}_{\mathrm{R}}^{*}, \mu)\subset \mathcal{W}_{-p}\subset \mathcal{W}_{\alpha}^{*}$$\mathcal{G}_{\infty}\subset \mathcal{G}_{p}\subset L^{2}(\mathcal{E}_{\mathrm{R}}^{*}, \mu)\subset \mathcal{G}_{-p}\subset \mathcal{G}_{\infty}^{*}$,and ofare called awhite noise test function
where $p\geq 0$ . In this context, elements ofand awhite noise distribution, respectively. We note also$\mathcal{W}_{\alpha}$$\mathcal{W}_{\alpha}^{*}$,$\mathcal{W}_{\alpha}\subset \mathcal{G}_{\infty}\subset \mathcal{G}_{p}\subset L^{2}(\mathcal{E}_{\mathrm{R}}^{*},\mu)\subset \mathcal{G}_{-p}\subset \mathcal{G}_{-\infty}\subset \mathcal{W}_{\alpha}^{*}$$p\geq 0$,

which is proved in Lemma 3.1. When there is no danger of confusion, we write
simplicity(4.4)$\mathcal{W}=\mathcal{W}_{\alpha}$

for

67
4.5

S-transforms
For$\Phi\in \mathcal{W}^{*}$, the-transform is defined by$S$$S\Phi(\xi)=\langle\langle\Phi, \phi_{\xi}\rangle\rangle$,$\xi\in \mathcal{E}$

.

(4.5)

span adense subspace of , each is uniquely
Since the exponential vectors
specified by the -transform. Obviously, the -transform $F=S\Phi$ possesses the following
properties:$\mathrm{S}$$\mathrm{S}$(F1) for each ,$\xi$$\Phi$$\mathcal{W}$$\{\phi_{\xi} ; \xi\in \mathcal{E}\}$$\eta\in \mathcal{E}$, the function$z$}$arrow F(z\xi+\eta)$

is entire holomorphic on$\mathrm{C}$;(F2) there exist $C\geq 0$ and $p\geq 0$ such that$|F(\xi)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})$,$\xi\in \mathcal{E}$.

It is emphasized in white noise theory that the converse assertion is also true. This famous-Takenaka spacecharacterization theorem for -transform was first proved for the
by Potthoff and Streit.
$\mathrm{H}\mathrm{i}\mathrm{d}\mathrm{a}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}$$\mathrm{S}$Theorem 4.2
Then is the -transfom of some(F2). In that case, for any $q>1/2$ with

Let $F$ be a complex valued function on .
if and only if $F$ satisfies conditions (Fl) and
we have$\mathcal{E}$$(\mathrm{C}\mathrm{o}\mathrm{c}\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{n}-\mathrm{K}\mathrm{u}\mathrm{o}-\mathrm{S}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{p}\mathrm{t}\mathrm{a}[6])$

$S$

$F$

$\Phi\in \mathcal{W}^{*}$$||A^{-q}||_{\mathrm{H}\mathrm{S}}^{2}<R_{\alpha}$$||\Phi||_{-(p+q),-}^{2}\leq C\overline{G}_{\alpha}(||A^{-q}||_{\mathrm{H}\mathrm{S}}^{2})$.

plays an essential role. While,
In the proof of Theorem 4.2, the nuclearity of the space
is not nuclear and hence the method of those usedin general the countable Hilbert space.
in the proof of Theorem 4.2 is not applicable to characterize -transforms of elements ofofelements
ofHowever we have the following characterization theorem for -transforms
by using Bargmann-Segal space, see [10], [17].
$\mathcal{W}$$\mathcal{G}_{\infty}$$S$$\mathcal{G}_{\infty}^{*}$$S$

is the S-transform. Then a complex valued function onand
if and only if can be extended to a continuous function on

Theorem 4.3 Letof some$\Phi\in \mathcal{G}_{p}$$g\mathrm{o}K^{p}\in E^{2}(\nu)$$\mathcal{G}_{\infty}^{*}$$g$$p\in \mathrm{R}$$D_{\infty}$$D_{-p}$$g$. In this case,$||\Phi$$||_{K,p}=||g$ $\mathrm{o}K^{p}||_{E^{2}(\nu)}$.

5White Noise Operators. Note thatis called awhite noise
Moreover,(4.4).
,
seeof
are subspaces
by duality. Ageneral theory for white noise operators
is isomorphic tohas been extensively developed in [3], [24], [26]. In this section we shall focus on regularity
properties of awhite noise operator in terms of Fock riggings.
Acontinuous linear operator
and,$\mathcal{L}(\mathcal{W}, \mathcal{W})$$\mathcal{L}(\Gamma(\mathcal{H}), \Gamma(\mathcal{H}))$$operator^{\mathrm{t}}$$—\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$\mathcal{L}(\mathcal{W}, \mathcal{W})$$\mathcal{L}(\mathcal{W}^{*}, \mathcal{W}^{*})$

1 In general, for two locally convex spaces ,. We always assume thatis denoted by$X$$\mathcal{L}(X, \mathfrak{Y})$on every bounded subset, the space of all continuous linear operators from into
is equipped with the topology of uniform convergence$X$$\mathfrak{Y}$$\mathcal{L}(X,\mathfrak{Y})$$\mathfrak{Y}$

68
5.1

Integral kernel operatorsLet
we have$a_{t}$

and$a_{t}^{*}$

be the annihilation and creation operators at apoint$a_{t} \phi(x)=\lim_{\thetaarrow 0}\frac{\phi(x+\theta\delta_{t})-\phi(x)}{\theta}$,$t\in \mathrm{R}$,$x\in \mathcal{E}_{\mathrm{R}}^{*}$$t\in \mathrm{R}$. For$\phi\in \mathcal{W}$,where the limit always exists. It is known thatand
. Moreover, the maps
andare both infinitely many times differentiable. The pairis referred to as the quantum white noise process.
Let , $m\geq 0$ be integers. Given
we define an integral kernel operator by$a_{\ell}\in \mathcal{L}(\mathcal{W},\mathcal{W})$$t|arrow a_{t}$$a_{t}^{*}\in \mathcal{L}(\mathcal{W}^{*}, \mathcal{W}^{*})$$t\succ\nu a_{t}^{*}$$\{a_{\mathrm{t}},a_{t}^{*}\}$$l$$\kappa$$\in(\mathcal{E}^{\Phi(l+m)})^{*}$$–l,m-( \kappa)=\int_{\mathrm{R}^{l+m}}\kappa(s_{1}, \cdots, s_{l}, t_{1}, \cdots, t_{m})a_{s_{1}}^{*}\cdots a_{s_{l}}^{*}a_{t_{1}}\cdots a_{t_{m}}ds_{1}\cdots ds_{l}dt_{1}\cdots dt_{m}$where the integral is understood in aformal sense. To be more precise, for
we defineby,$\phi=(f_{n})\in \mathcal{W}$$\underline{=}_{l,m}(\kappa)\phi=(g_{n})$$g_{n}=0$,$0\leq n<l$ ;$g_{l+n}= \frac{(n+m)!}{n!}\kappa\otimes_{m}f_{n+m}$,$n\geq 0$,

where
is the right $m$ -contraction, see [5]. An integral kernel operator is always awhite
noise operator, that is,for an arbitrary kernel. Moreover,
if the weight sequence, thenfulfills conditions
if andonly if.
It is an interesting question to characterize the integral kernel operators belonging tofor some. We here only mention the following$\otimes_{m}$$\kappa\in(\mathcal{E}^{\Phi(l+m)})^{*}$$–l,m-(\kappa)\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\{\alpha(n)\}$$(\mathrm{A}1)-(\mathrm{A}4)$$–l,m-(\kappa)\in \mathcal{L}(\mathcal{W}, \mathcal{W})$$\kappa\in \mathcal{E}^{\Phi l}\otimes(\mathcal{E}^{\Phi m})^{*}$$p\in \mathrm{R}$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$Theorem 5.1 (Chung-Ji-Obata [4]) Let $\alpha=\{\alpha(n)\}$ be a weight sequence satisfying
conditions $(Al)-(A\mathit{4})$ and. Then for
if and only
only
.
if
if andif$\kappa\in(\mathcal{E}^{\Phi(l+m)})^{*},$$p\in \mathrm{R}$$\kappa\in D_{p}^{\Phi l}\otimes(\mathcal{E}^{\Phi m})^{*}$

5.2$–l,m-(\kappa)\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$\kappa\otimes_{m}\in \mathcal{L}(\mathcal{E}^{\Phi m},D_{p}^{\Phi l})$Operator symbols from the viewpoint of Segal-Bargmann transform

The symbol, which is an operator version of the famous Segal-Bargmann transform, of awhite noise operator
is acomplex valued function on
defined by$—\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\mathcal{E}\cross \mathcal{E}$$-(\xi, \eta)=\langle\langle_{-}^{-}\underline{\underline{\wedge}}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$

,

$\xi$

,

$\eta\in \mathcal{E}$.

Every white noise operator is uniquely determined by its symbol. By definition the symbol
and the -transforms are related as$\mathrm{S}$$-(\xi, \eta)=S(_{-}^{-}\underline{\underline{\wedge}}-\phi_{\xi})(\eta)=S(_{-}^{-*}-\phi_{\eta})(\xi)$

It is straightforward to see that the symbol
possesses the folowing properties:(01) for any

on$\xi,\xi_{1},\eta$$\mathrm{C}\cross \mathrm{C}$;,$\eta_{1}\in \mathcal{E}$$\mathrm{e}=-\underline{\underline{\wedge}}$

,

$\xi$

,

$\eta\in \mathcal{E}$.

of awhite noise operator$—\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$

the function $(z,w)\vdash*\Theta(z\xi+\xi_{1},w\eta+\eta_{1})$ is entire holomorphic

69(02) there exist constant numbers $C\geq 0$ and $p\geq 0$ such that$|(\xi, \eta)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})G_{\alpha}(|\eta|_{p}^{2})$

,

$\xi$

,

$\eta\in \mathcal{E}$.

As in the case of -transform, the characterization theorem for symbols, which was firstproved by Obata for the-Takenaka space, is asignificant consequence of white
noise theory. The characterization in the case of CKS-space was proved by Chung-Ji-Obata[3].
We now prove the characterization theorem for symbols of operators on Fock spaces,. Then it is easily shown that the symbol ofin this connection see also [17]. Let.
is extended to an entire function on. Then there
and let $$ a complex valued function defined on
Theorem 5.2 Let
such thateists
if and only if;(i) $$ can be extended to an entire function on$\mathrm{S}$$\mathrm{H}\mathrm{i}\mathrm{d}\mathrm{a}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}$$p\in \mathrm{R}$$\mathcal{E}\cross D_{-p}$$—\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$\mathcal{E}\cross \mathcal{E}$$p\in \mathrm{R}$$=-\underline{\underline{\wedge}}$$—\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$\mathcal{E}\cross D_{-p}$(ii) there exist $q\geq 0$ and $C\geq 0$ such that$||(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq CG_{\alpha}(|\xi|_{q}^{2})$Proof. Suppose that there exists
is obvious and there exists $q\geq 0$ such thatthat,$\xi\in \mathcal{E}$.

. Then condition (i)such that. Hence there exists $C\geq 0$ such$\Theta=\mathrm{E}$$—\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$—\in \mathcal{L}(\mathcal{W}_{q}, \mathcal{G}_{p})$$||_{-}^{-}-\phi||_{K,p}\leq C||\phi||_{q,+}$,$\phi\in \mathcal{W}_{q}$.

Therefore, we have$||\Theta(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}=||_{-}^{-}-\phi_{\xi}||_{K,p}^{2}\leq C^{2}||\phi_{\xi}||_{q,+}^{2}=C^{2}G_{\alpha}(|\xi|_{q}^{2})$.

Conversely, suppose that conditions (i) and (ii) are satisfied. Let. Then by (ii),by $F_{\xi}(\eta)=(\xi, \eta)$ ,:define afunction
andsuch that
Hence by Theorem 4.3, there exists$\xi\in \mathcal{E}$$F_{\xi}$$\eta\in D_{-p}$$D_{-p}arrow \mathrm{C}$$||\Phi_{\xi}||_{K,p}^{2}=||F_{\xi}\circ K^{p}||_{E^{2}(\nu)}^{2}=||(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq CG_{\alpha}(|\xi|_{q}^{2})$$\phi\in \mathcal{G}_{-p}$and define afunction$G_{\phi}$:$\mathcal{E}arrow \mathrm{C}$$G_{\phi}(\xi)=\langle\langle\phi, \Phi_{\xi}\rangle\rangle$

Then we can easily show that

$G_{\phi}$.

$S\Phi_{\xi}=F_{\xi}$$\Phi_{\xi}\in \mathcal{G}_{p}$Now, fix

be fixed and$F_{\xi}(K^{p}\cdot)\in E^{2}(\nu)$

.

by

,

$\xi\in \mathcal{E}$.

satisfies conditions (F1) and (F2). In fact,$|G_{\phi}(\xi)|^{2}\leq||\phi||_{K,-p}^{2}||\Phi_{\xi}||_{K,p}^{2}\leq C||\phi||_{K,-p}^{2}G_{\alpha}(|\xi|_{q}^{2})$Therefore, by Theorem 4.2, there exists$\Psi_{\phi}\in \mathcal{W}_{\alpha}^{*}$

such that,$S(\Psi_{\phi})(\xi)=G_{\phi}(\xi)=\langle\langle\phi, \Phi_{\xi}\rangle\rangle$$\xi\in \mathcal{E}$.

.

Moreover we have
(5.1)$||\Psi_{\phi}||_{-(q+q’),-}^{2}\leq C\tilde{G}_{\alpha}(||A^{-q’}||_{\mathrm{H}\mathrm{S}}^{2})||\phi||_{K,-p}^{2}$

by $—*\phi=$. Define alinear operator
for some $q’>1/2$ withby (5.1) and hence $$ is the symbol of,.
( is the adjoint of ).
During the above theorem we are convinced that the symbol is an operator-version of
the Segal-Bargmann transform$||A^{-q’}||_{\mathrm{H}\mathrm{S}}^{2}<R_{\alpha}$$\Psi_{\phi}$$\phi\in g_{-p}$$—*:\mathcal{G}_{-p}arrow \mathcal{W}_{\alpha}^{*}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}---*\in \mathcal{L}(\mathcal{G}_{-p}, \mathcal{W}_{\alpha}^{*})$$—\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})\mathrm{I}$

$-$

$—*$

70
5.3

Wick products

We first recall the following
Lemma 5.3 (Chung-Ji-Obata [5]) If the weight sequence $\alpha=\{\alpha(n)\}$ satisfies conditions $(Al)-(A\mathit{4})$ , then for two white noise operators
there exist
unique operator
such that$–1,–2–\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$$a$$—\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$,$-(\xi, \eta)=-1(\xi, \eta)_{-2}^{\underline{\underline{\wedge}}}(\xi, \eta)e^{-(\xi,\eta\rangle}\underline{\underline{\wedge}}\underline{\underline{\wedge}}$

$\xi$

,

$\eta\in \mathcal{E}$

.

(5.2)

The operatorin (5.2) is called the Wick product of –1- and –2-, and is denoted by$—=—10—2$ . Note that
equipped with the Wick product becomes acommutative-algebra. As for the annihilation and creation operators we have$—\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$*$$a_{s}\mathrm{o}$$a_{t}=a_{s}a_{t}$,$a_{s}^{*}\mathrm{o}$$a_{t}=a_{s}^{*}a_{t}$,$a_{s}\mathrm{o}$$a_{t}^{*}=a_{t}^{*}a_{s}$,$a_{s}^{*}\mathrm{o}$$a_{t}^{*}=a_{s}^{*}a_{t}^{*}$

.

(5.3)

More generally, it holds that,$a^{*}\cdots a--s_{1}s_{l^{-}}a_{t_{1}}\cdots a_{\mathrm{t}_{m}}=(*a_{s_{1}}^{*}\cdots a_{l}^{*},a_{t_{1}}\cdots a_{\mathrm{t}_{m}})0---$$—\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$.

In fact, the Wick product is aunique bilinear map from
into
which is (i) separately continuous; (ii) associative; and (iii) satisfying (5.3).
$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})\cross \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$and—1,$
Proposition 5.4 Let$C\geq0$$q\geq0$
andsuch that
if there exist$p\in \mathrm{R}$$—2\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$.

$Then—10—2\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$||\underline{\underline{\wedge}}-1(\xi, K^{p}\cdot)_{-2}^{\underline{\underline{\wedge}}}(\xi, K^{p}\cdot)e^{-\{\xi,K^{p}\cdot)}||_{E^{2}(\nu)}^{2}\leq CG_{\alpha}(|\xi|_{q}^{2})$,$\xi\in \mathcal{E}$

if and only.

1

Proof. An immediate consequence from Theorem 5.2.

6Quantum Stochastic Processes
Acontinuous map

defined on an interval is called aquantum
stochastic process (in the sense of white noise theory), see [25]. The quantum white noise
process
is apair of quantum stochastic processes in this sense, see \S 5.1. In thissection we discuss some detailed properties of quantum stochastic processes in.
$t\vdash\star--t-\in \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\{a_{t}, a_{t}^{*}\}$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$6.1Continuity criterion

We first mention acriterion for the continuity ofsymbols.

$t$$|arrow---t\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$Theorem 6.1 Let
be a locally compact space and, $t\in T$ , the following conditions are equivalent:$T$$p\in \mathrm{R}$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$(i)$t\vdash\star--t-\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$is continuous;in terms of operator. Then for a map$t\vdasharrow--t-\in$

71(ii) for any $t_{0}\in T$ there exist $q\geq 0$ and an open neighborhood$\{_{-t}^{-}-;t\in U\}\subset \mathcal{L}(\mathcal{W}_{q}, \mathcal{G}_{p})$

$U$

of

$t_{0}$

such that;

and$\lim_{tarrow t_{0}}||_{-t-t_{\mathrm{O}}}^{--}---||_{\mathcal{L}(\mathcal{W}_{q\prime}\mathcal{G}_{p})}=0$(iii) for any $t_{0}\in T$ there exist an open neighborhoodof to, a set of positive numbers, constant number $q\geq 0$ such thatconverging to 0as$U$$tarrow t_{0}$$\{\epsilon_{t};t\in U\}$$||_{-t}^{-}-\wedge-\wedge(\xi, K^{p}\cdot)---_{t_{0}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq\epsilon_{t}G_{\alpha}(|\xi|_{q}^{2})$,$\xi\in \mathcal{E}$,(iv) for any $t_{0}\in T$ there exist $M\geq 0$ , $q\geq 0$ and an open neighborhood$||_{-t}^{\underline{\underline{\wedge}}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq MG_{\alpha}(|\xi|_{q}^{2})$and for each$\xi\in \mathcal{E}$,converges$-t\underline{\underline{\wedge}}(\xi, K^{p}\cdot)$,$\xi\in \mathcal{E}$$to-t_{0}(\xi, K^{p}\cdot)\underline{\underline{\wedge}}$in$t\in U$of$U$

$t_{0}$;

such that,,$t\in U$

$E^{2}(\nu)$.

The proof is asimple modification of that of [28, Theorem 1.8] and is omitted.
(1) The map

Proposition 6.2$t$$\vdasharrow a_{t}\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$(2) If there exists $p\geq 0$ such that$t\vdasharrow\delta_{t}\in D_{-p}$is continuous for all $p\geq 0$ .

is continuous, so is

Proof. (1) Note that there exists $q\geq 0$ such that the map
Moreover, for all $p\geq 0$ there exists $p’\geq p$ such that$t\mapsto\delta_{t}\in \mathcal{E}_{-q}$$||\hat{a}_{t}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}=|\langle\delta_{t}, \xi\rangle|^{2}||\phi_{\xi}||_{K,p}^{2}\leq|\delta_{t}|_{-q}^{2}|\xi|_{q}^{2}||\phi_{\xi}||_{\Psi,+}^{2}$

It then follows from Theorem 6.1 that
(2) By assumption we have$t$$\vdash*a_{t}\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$t|arrow a_{t}^{*}\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$,$\mathcal{E}\mathrm{c}arrow D_{-p}$.

is continuous.

,$\xi\in \mathcal{E}$.

is continuous, there exist $C\geq 0$ and $q\geq 0$ such that$||\hat{a_{t}^{*}}(\xi, K^{-p}\cdot)||_{E^{2}(\nu)}^{2}\leq C|\delta_{t}|_{K,-p}^{2}e^{|\xi|_{q}^{2}}$Then by Theorem 6.1 the map

is continuous.

$\xi\in \mathcal{E}$$||\hat{a_{t}^{*}}(\xi, K^{-p}\cdot)||_{E^{2}(\nu)}^{2}=||\langle\delta_{t}, K^{-p}\cdot\rangle e^{\langle\xi,K^{-\mathrm{p}}\cdot\rangle_{||_{E^{2}(\nu)}^{2}=\sum_{n=0}^{\infty}\frac{(n+1)!}{n!^{2}}|\delta_{t}|_{K,-p}^{2}|\xi|_{K,-p}^{2n}}}$Since the embedding.

$t\vdash ta_{t}^{*}\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$,$\xi\in \mathcal{E}$.

1is continuous.

By Theorem 6.1, the following result is immediate.

and letTheorem 6.3 Let$Then—n$ converges $to—in$$p\in \mathrm{R}$.
a sequence in$q\geq0$$M\geq0$such that
and
if and only if there exist$\{_{-n}^{-}-\}_{n=0}^{\infty}\backslash$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$be$$||_{-n}^{-}-\wedge(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq MG_{\alpha}(|\xi|_{q}^{2})$and for each$\xi\in \mathcal{E}$,$-n\underline{\underline{\wedge}}(\xi, K^{p}\cdot)$

converges$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$,$to-(\xi, K^{p}\cdot)\underline{\underline{\wedge}}$$\xi\in \mathcal{E}$in,$n=1,2$ ,$E^{2}(\nu)$.

$and—\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$\cdots$

,

72
6.2

Quantum stochastic integrals

Recall that the topology of$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$

is defined by the seminorms:$||^{-}--||_{B,B’}= \sup\{|\langle(_{-}^{-}-\phi, \psi\rangle)|;\phi\in B, \psi \in B’\}$

where

$B$

is the class of all bounded subsets ofis defined by$\mathcal{W}$,$B,B’\in B$ ,. Similarly, for fixed$p\in \mathrm{R}$, the topology of$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$||_{-}^{-}-||_{B,p}= \sup\{||_{-}^{-}-\phi||_{Kp} ; \grave{\phi}\in B\}$,

$B\in B$

Lemma 6.4 Let
be a quantum stochastic process in
andthere exists a unique operator$\{L_{t}\}$$f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathrm{R})$$–a,t-(f)\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$\langle\langle_{-a,t}^{-}-(f)\phi, \psi\rangle\rangle=\int_{a}^{t}f(s)\langle\langle L,\phi, \psi\rangle\rangle ds$

Moreover,Since$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$,$6$\mathcal{W}$. Then for any$a,t\in \mathrm{R}$such that,$\psi$$\in \mathcal{G}_{-p}$

.

(6.1)

is continuous.

$t|arrow--_{a,t}-(f)\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$Proof.
subset of$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$.

is continuous, the closed interval $[a, t]$ is mapped to acompact. Hence by applying (ii) in Theorem 6.1 there exists some $q\geq 0$ such that$s\vdasharrow L_{s}$.

$C \equiv\sup_{a\leq s\leq t}||L_{s}||_{\mathcal{L}(\mathcal{W}_{q\prime}\mathcal{G}_{\mathrm{P}})}<\infty$

Then for any $s\in[a, t]$ we have$|\langle\langle L_{s}\phi, \psi\rangle\rangle|\leq||L_{s}||_{\mathcal{L}(\mathcal{W}_{q},\mathcal{G}_{\mathrm{P}})}||\phi||_{q,+}||\psi$$||_{K,-p}\leq C||\phi||_{q,+}||\psi$ $||_{K,-p}$,and$| \int_{a}^{t}f(s)\langle\langle L_{s}\phi, \psi\rangle\rangle ds|\leq C||\phi||_{q,+}||\psi$$||_{K,-p} \int_{a}^{t}|f(s)|ds$,$\phi\in \mathcal{W}$,$\psi$$\in \mathcal{G}_{-p}$

.

(6.2)

Therefore, for each fixed
the right hand side of (6.1) is acontinuous linear functional. Hence, by the Riesz representation theorem there exists auniqueonsuch that$\phi\in \mathcal{W}$$g_{-p}$$\phi’\in \mathcal{G}_{p}$.

$\langle\langle\phi’, \psi\rangle\rangle=\int_{a}^{t}f(s)\langle\langle L_{s}\phi, \psi\rangle\rangle ds$

Define alinear operator $–a,t-(f)$ fromby $–a,t-(f)\phi=\phi’$ . Then by (6.2), $–a,t-(f)\in$
into
and (6.1) holds.
It is sufficient to prove the continuity on any finite interval. Taking constantnumbers $q\geq 0$ , $C\geq 0$ as above (considering the closed interval), we obtain that for
any $a_{1}<u<t<b_{1}$ and,$\mathcal{W}$$\mathcal{G}_{p}$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$(a_{1}, b_{1})$$[a_{1},$$\phi\in \mathcal{W}$$\psi$$b_{1}]$$\in g_{-p}$$|\langle\langle(_{-a,t}^{-}-(f)--_{a,\mathrm{u}}--(f))\phi, \psi\rangle\rangle|\leq C||\phi||_{q,+}||\psi$ $||_{K,-p} \int_{u}^{t}|f(s)|ds$

Then for bounded subset$B\subset \mathcal{W}$.

we have$||_{-a,t}^{-}-(f)--_{a,u}--(f)||_{Bp} \leq C||B||_{q,+}\int_{\mathrm{u}}^{t}|f(s)|ds$,$a_{1}<u<t<b_{1}$ ,73
where $||B||_{q,+}= \sup\{||\phi||_{q,+}$;$\phi\in B\}<\infty$ . It follows the continuity immediately.

1

The white noise operator $–a,t-(f)$ defined in (6.1) is denoted by$–a,t-(f)= \int_{a}^{t}f(s)L_{s}ds$

Theorem 6.5 Let two quantum stochastic processes.
and$\{L_{t}\}$$\{_{-t}^{-}-\}$

in$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$be related

as$—t= \int_{a}^{t}L_{s}ds$

Then the map$t\vdasharrow---t\in \mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$,$t\in \mathrm{R}$.

is differentiate and$\frac{d}{dt}-_{t}--=L_{t}$holds in$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$.

The proof is straightforward by modifying the argument in [25].

7White Noise Differential Equations
In this section, we study the following white noise differential equation:$\frac{d_{-}^{-}-}{dt}=F(t,---)$,$–|_{t=0=}----0$

,

,

(7.1)

$0\leq t\leq T$

is awhite noiseis acontinuous function and
where
$[0,
T]$
.
with values inoperator. Asolution of (7.1) must be a -map defined on.
value
Obviously, the solution depends on the “regularity property” of the initial$–0-$$F:[0, T]\cross \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})arrow \mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$C^{1}$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$–0-$

Unique existence

7.1

and $=\{\omega(n)\}$ satisfying conditions
We now consider two weight sequences, the generating functions of which are related in such away that$\alpha=\{\alpha(n)\}$$\omega$$(\mathrm{A}1)-(\mathrm{A}4)$$G_{\alpha}(t)=\exp\gamma\{G_{\omega}(t)-1\}$,

where $\gamma>0$ is acertain constant. In that case, we have continuous inclusions:$\mathcal{W}_{\alpha}\subset \mathcal{W}_{\omega}\subset L^{2}(\mathcal{E}_{\mathrm{R}}^{*}, \mu)\subset \mathcal{W}_{\omega}^{*}\subset \mathcal{W}_{\alpha}^{*}$

and$\mathcal{L}(\mathcal{W}_{\omega}, \mathcal{W}_{\omega}^{*})\subset \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$.

The relation given as in (7.2) is abstracted from the case of Bell numbers, see [5], [6]

(7.2)

74
Theorem 7.1 (Ji-Obata [16]) Let $\alpha=\{\alpha(n)\}$ and $=\{\omega(n)\}$ be trno weight sequences
satisfying conditions $(Al)-(A\mathit{4})$ , and assume that their generating functions are related as
be a continuous function and assumein (7.2). Let F : [0, T]$\inL^{1}[0,T]$
such thatand a nonnegative function M
there nist p$\omega$$\cross \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})arrow \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$$\geq 0$(i) for all ,$\xi$$\eta\in \mathcal{E},---1,$, and $s\in[0,T]$$–2-\in \mathcal{L}(\mathcal{W}_{\alpha},\mathcal{W}_{\alpha}^{*})$$|\hat{F}(s,---1)(\xi,\eta)-\hat{F}(s,---2)(\xi, \eta)|^{2}\leq M(s)G_{\omega}(|\xi|_{p}^{2})G_{\omega}(|\eta|_{p}^{2})|_{-1}^{\underline{\underline{\wedge}}}(\xi,\eta)--2\underline{\underline{\wedge}}(\xi,\eta)|^{2}$(ii) for all ,$\xi$$\eta\in \mathcal{E},---\in \mathrm{C}(\mathrm{W}\mathrm{a}, \mathcal{W}_{\omega}^{*})$, and $s\in[0,T]$$|\hat{F}(s,---)(\xi,\eta)|^{2}\leq M(s)G_{\omega}(|\xi|_{p}^{2})G_{\omega}(|\eta|_{p}^{2})(1+|_{-}^{\underline{\underline{\wedge}}}(\xi,\eta)|^{2})$Then, for$any—0\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{W}_{\omega}^{*})$$\mathcal{L}(\mathcal{W}_{\alpha},\mathcal{W}_{\alpha}^{*})$, $t\in[0,T]$ .

;.

the initial value problem (7.1) has a unique solution $–t-\in$,
Example Let
be two quantum stochastic processes, where runs
over $[0, T]$ . Then the initial value problem$\{L_{t}\}$$t$$\{M_{t}\}\subset \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{W}_{\omega}^{*})$$\frac{d}{dt}-_{tt-t}--=L\mathrm{o}--+M_{t}$

,

,

(7.3)

$—|_{t=0=}---0\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{W}_{\omega}^{*})$. Note that equation (7.3) is already beyond atraditional
has aunique solution inquantum stochastic differential equation.
$\mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$

7.2

Regularity of solutions
In this section we study regularity properties of solution of (7.1) as usual operators in. Let $\alpha=\{\alpha(n)\}$ and $=\{\omega(n)\}$ be two weight sequences satisfying conditions, and assume that their generating functions are related as in (7.2).
$–t-$$\mathcal{L}(\mathcal{W}, \mathcal{G}_{p})$$\omega$$(\mathrm{A}1)-(\mathrm{A}4)$

be a continuous function andTheorem 7.2 Let $F$ : $[0, T]$$M\inL^{1}[0, T]$ , and a nonnegative,$q\geq0$
and a nonnegative function
assume that there exist
satisfying
locally bounded function on$\cross \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})arrow \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$$g$$\mathcal{E}\cross D_{-p}$$||g(\xi, K^{p}\cdot)^{n}||_{E^{2}(\nu)}^{2}\leq n!(RG_{\omega}(|\xi|_{q}^{2}))^{n}$for sorne$R\geq 0$(i) for all,$n=1,2$ ,(7.4)$\cdots$such that, and $s\in[0,T]$$\xi,\eta\in \mathcal{E},---1,$ $–2-\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{W}_{\alpha}^{*})$$|\hat{F}(s,---1)(\xi, \eta)-\hat{F}(s,---2)\cdot(\xi, \eta)|^{2}\leq M(s)g(\xi, \eta)^{2}|_{-1}^{\wedge\underline{\underline{\wedge}}}--(\xi,\eta)--2(\xi, \eta)|^{2}$(ii) for all 4,$\eta\in \mathcal{E},---\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$, and $s\in[0,T]$$|\hat{F}(s,---)(\xi,\eta)|^{2}\leq M(s)g(\xi, \eta)^{2}(1+|_{-}^{\underline{\underline{\wedge}}}(\xi,\eta)|^{2})$

.

;

75
Then, for$any—0\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$satisfying$||g(\xi, K^{p}\cdot)_{-0}^{n^{\underline{\underline{\wedge}}}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq n!(R’G_{\omega}(|\xi|_{q}^{2},))^{n}$$n=1,2$ ,,$\cdots$,

for some $R’\geq 0$ and $q’\geq 0$ , the initial value problem (7.1) has a unique solution$\mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$(7.5)$–t-\in$, $t\in[0, T]$ .

PROOF. In principle, the proof is based on the standard Picard-Lindel\"of method of
successive approximations (see e.g., [11]) applied to the operator symbols. We define$–t-(0)=_{-0}--$,$–t-(n)=—0+ \int_{0}^{t}F(s,---_{s}(n-1))ds$

Then by applying Theorem 5.2 we see that
fact, by (ii)$| \int_{0}^{t}\hat{F}(s,---0)(\xi, \eta)ds|^{2}$,$n\geq 1$$–t-(n)\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$.

for all$n$

and $0\leq t\leq T$ . In$\leq$$T \int_{0}^{t}|\hat{F}(s,---0)(\xi, \eta)|^{2}ds$$\leq$$T( \int_{0}^{t}M(s)ds)g(\xi, \eta)^{2}(1+|_{-0}^{\underline{\underline{\wedge}}}(\xi, \eta)|^{2})$and hence by assumption we have$|| \int_{0}^{t}\hat{F}(s,---0)(\xi, K^{p}\cdot)ds||_{E^{2}(\nu)}^{2}\leq T(\int_{0}^{t}M(s)ds)\{R_{1}G_{\omega}(|\xi|_{q_{1}}^{2})+R_{2}G_{\omega}(|\xi|_{q_{2}}^{2})\}$

for some
5.2 that$R_{1}$, $R_{2}\geq 0$ and$q_{1}$,$q_{2}\geq 0$. Moreover, since$–0-\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$, we see from Theorem$||_{-0}^{\underline{\underline{\wedge}}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq R_{3}G_{\omega}(|\xi|_{q3}^{2})$for some$R_{3}\geq 0$

and$q_{3}\geq 0$

(7.6)

(7.7)

. Put

$R= \max\{2T(\int_{0}^{t}M(s)ds)R_{1},2T(\int_{0}^{t}M(s)ds)R_{2},2R_{3}\}$and $q= \max\{q_{1}, q_{2}, q_{3}\}$ . Then by (7.6), (7.7) we have$||_{-t}^{\overline{-(1)}}-( \xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq 2(||\int_{0}^{t}\hat{F}(s,---0)(\xi, K^{p}\cdot)ds||_{E^{2}(\nu)}^{2}+||_{-0}^{\underline{\underline{\wedge}}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2})$$\leq 3RG_{\omega}(|\xi|_{q}^{2})$Hence by Theorem 5.2 we see thatcan be repeated to conclude that
Theorem 6.1 we see that.
for all $0\leq t\leq T$ . The above argument
for all and $0\leq t\leq T$ . Moreover, fromis aquantum stochastic process$–t-(1)\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$$–t-(n)\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$$\{_{-t}^{-(n)}-\}\subset \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$

$n$

76
We shall prove step by step that
simplicity we put$\lim_{narrow\infty-t}^{-(n)}-$is the desired solution to our equation. For$\Theta_{n}(t;\xi,\eta)=---t(\overline{(n)}\xi,\eta)=\langle \mathrm{t}_{-t}^{-(n)}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$,$\xi,\eta\in \mathcal{E}$,$0\leq t\leq T$.

By (i) we see that$| \Theta_{n}(t;\xi,\eta)-\Theta_{n-1}(t;\xi,\eta)|=|\int_{0}^{t}\mathrm{t}\hat{F}(s,---(.n-1))(\xi,\eta)-\hat{F}(s,---_{s}(n-2))(\xi,\eta)\}ds|$

,

(7.8)

$\leq g(\xi,\eta)\int_{0}^{t}\sqrt{M(s)}|\Theta_{n-1}(s;\xi,\eta)-\Theta_{n-2}(s;\xi,\eta)|ds$

for ,
$\xi$

$\eta\in \mathcal{E}$and $0\leq t\leq T$ . Then, repeating this argument we come to$|\Theta_{n}(t;\xi, \eta)-\Theta_{n-1}(t;\xi, \eta)|$$\leq\{g(\xi, \eta)\}^{n-1}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{\mathfrak{n}-2}}dt_{n-1}$$\cross\sqrt{M(t_{1})}\sqrt{M(t_{2})}\cdots\sqrt{M(t_{n-1})}|\Theta_{1}(t_{n-1};\xi, \eta)-\Theta_{0}(t_{n-1};\xi, \eta)|$

.

(7.9)

As for the last quantity it follows from (ii) that$| \Theta_{1}(t;\xi, \eta)-\Theta_{0}(t;\xi, \eta)|=|\int_{0}^{t}\hat{F}(s,---0)(\xi, \eta)ds|\leq\int_{0}^{t}|\hat{F}(s,---0)(\xi,\eta)|ds$.

(7.10)For simplicity we put,$H(\xi, \eta)=\overline{M}g(\xi, \eta)\sqrt{1+|_{-0}^{\underline{\underline{\wedge}}}(\xi,\eta)|^{2}}$$\overline{M}=\int_{0}^{T}\sqrt{M(s)}ds$.

Then (7.10) becomes$|\Theta_{1}(t;\xi,\eta)-\Theta_{0}(t;\xi, \eta)|\leq H(\xi,\eta)$.

Similarly, (7.9) becomes$|\Theta_{n}(t;\xi, \eta)-\Theta_{n-1}(t;\xi, \eta)|\leq\{g(\xi, \eta)\}^{n-1}H(\xi, \eta)\cross$$\cross\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots$$\int_{0}^{t_{n-2}}dt_{n-1}\sqrt{M(t_{1})}\sqrt{M(t_{2})}\cdots\sqrt{M(t_{n-1})}$$\leq\frac{1}{(n-1)!}\{\overline{M}g(\xi,\eta)\}^{n-1}H(\xi,\eta)$

It then follows that for each ,$\xi$$\eta\in \mathcal{E}$.

, the series$\sum_{n=1}^{\infty}\{\Theta_{n}(t;\xi, \eta)-\Theta_{n-1}(t;\xi, \eta)\}$(7.11)

77
converges absolutely and uniformly in $t\in[0, T]$ . In fact,$\sum_{n=1}^{\infty}|\Theta_{n}(t;\xi, \eta)-_{n-1}(t;\xi, \eta)|$$\leq$$H(\xi, \eta)\exp\{\overline{M}g(\xi, \eta)\}$.

(7.12)$\exp\{2\overline{M}g(\xi, \eta)\}\sqrt{1+|_{-0}^{\underline{\underline{\wedge}}}(\xi,\eta)|^{2}}$$\leq$

We now put.

$\Theta_{t}(\xi, \eta)=\lim_{narrow\infty}_{n}(t;\xi, \eta)=--\wedge-\mathrm{o}(\xi, \eta)+\sum_{n=1}^{\infty}\{_{n}(t;\xi, \eta)-_{n-1}(t;\xi, \eta)\}$Since is bounded on every bounded subset of, the series
any 4,and ,$g$$\xi’\in \mathcal{E}$$\eta$$\mathcal{E}\cross D_{-p}$, by (7.12) we can easily see that for$\eta’\in D_{-p}$$\sum_{n=1}^{\infty}\{_{n}(t;\lambda\xi+\xi’, \gamma\eta+\eta’)-_{n-1}(t;\lambda\xi+\xi’, \gamma\eta+\eta’)\}$

converges uniformly on every compact subset ofmap$(\lambda, \gamma)\vdash*_{t}(\lambda\xi+\xi’, \gamma\eta+\eta’)$$\mathrm{C}\cross \mathrm{C}$is holomorphic on. Therefore for each $0\leq t\leq T$ the$\mathrm{C}\cross \mathrm{C}$. Also, by (7.12)$|| \sum_{n=1}^{\infty}\{\Theta_{n}(t;\xi, K^{p}\cdot)-_{n-1}(t;\xi, K^{p}\cdot)\}||_{E^{2}(\nu)}^{2}$$\leq||\exp\{2\overline{M}g(\xi, K^{p}\cdot)\}\sqrt{1+|_{-0}^{\underline{\underline{\wedge}}}(\xi,K^{p}\cdot)|^{2}}||_{E^{2}(\nu)}^{2}$(7.13)On the other hand, by the Schwartz inequality, for any $0<r<1$$\exp\{2\overline{M}g(\xi, \eta)\}\leq\frac{1}{\sqrt{1-r}}(\sum_{n=0}^{\infty}\frac{(2\overline{M})^{2n}}{r^{n}n!^{2}}g(\xi, \eta)^{2n})^{1/2}$

Therefore, by (7.4) and (7.5) there exist$R_{1}$,$R_{2}\geq 0$

and$q_{1}$,$q_{2}\geq 0$such that$||\exp\{2\overline{M}g(\xi, K^{p}\cdot)\}\sqrt{1+|_{-0}^{\underline{\underline{\wedge}}}(\xi,K^{p}\cdot)|^{2}}||_{E^{2}(\nu)}^{2}$$\leq\frac{1}{1-r}\sum_{n=0}^{\infty}\frac{(2\overline{M})^{2n}}{r^{n}n!^{2}}\{n!(R_{1}G_{\omega}(|\xi|_{q_{1}}^{2}))^{n}+n!(R_{2}G_{\omega}(|\xi|_{q2}^{2}))^{n}\}$$\leq\frac{2}{1-r}\sum_{n=0}^{\infty}\frac{(2\overline{M})^{2n}}{r^{n}n!^{2}}n!(RG_{\omega}(|\xi|_{q}^{2}))^{n}$$= \frac{2}{1-r}\sum_{n=0}^{\infty}\frac{1}{n!}(\frac{4\overline{M}^{2}R}{r}G_{\omega}(|\xi|_{q}^{2}))^{n}$(7.14),

where $R=R_{1}\vee R_{2}$ and $q=q_{1}\vee q_{2}$ . We fix $0<r<1$ in such away that
$M_{0}( \gamma)\equiv\frac{4\overline{M}^{2}R}{r\gamma}\geq 1$,

78
where 7is the constant defined in (7.2). Then by (3.2) we have$\frac{4\overline{M}^{2}R}{r}G_{\omega}(|\xi|_{q}^{2})$Choosing$r_{0}\geq 0$$=$$\gamma M_{0}(\gamma)G_{\omega}(|\xi|_{q}^{2})$$=$$\gamma\{M_{0}(\gamma)\{$$G_{\omega}(|\xi|_{q}^{2})-1]\}+\gamma M_{0}(\gamma)$$\leq$$\gamma\{G_{\omega}(M_{0}(\gamma)|\xi|_{q}^{2})-1\}+\gamma M_{0}(\gamma)$.

such that$M_{0}(\gamma)\rho^{2r_{0}}\leq 1$,

we obtain$\frac{4\overline{M}^{2}R}{r}G_{\omega}(|\xi|_{q}^{2})\leq\gamma\{G_{\omega}(|\xi|_{q+r_{0}}^{2})-1\}+\gamma M_{0}(\gamma)$.

(7.15)

Hence, by (7.2), (7.13), (7.14) and (7.15) we have$|| \sum_{n=1}^{\infty}\{\Theta_{n}(t;\xi, K^{p}\cdot)-\Theta_{n-1}(t;\xi, K^{p}\cdot)\}||_{E^{2}(\nu)}^{2}\leq\frac{2}{1-r}e^{(4\overline{M}^{2}R)/r}G_{\alpha}(|\xi|_{q+r_{0}}^{2})$.

Therefore, for the function
condition (ii) in Theorem 5.2 is satisfied since
Hence by Theorem 5.2 there exists aunique operator
such that$\Theta_{t}$$–0-\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$.

$–t-\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$\Theta_{t}(\xi, \eta)=-t(\xi, \eta)\underline{\underline{\wedge}}$

,

$\xi$

,

$\eta\in \mathcal{E}$,$t\in[0, T]$.

By applying Theorem 6.3 with (7.11) we see that$–t-= \lim_{narrow\infty}---t(n)$and hence the map
We now prove that
integraluniformly in ,$t$

is continuous.
is asolution of (7.1). We first note that, by Lemma 6.4, the$t|arrow---t\in \mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$$\{_{-t}^{-}-\}$$\int_{0}^{t}F(s,---s)ds$

is well-defined as an operator in$\mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{p})$. On the other hand, by assumption (i) we have$|| \int_{0}^{t}(\hat{F}(s,---s)(\xi, K^{p}\cdot)-\hat{F}(s,-_{s}--(n))(\xi, K^{p}\cdot))ds||_{E^{2}(\nu)}^{2}$$\leq||\int_{0}^{t}\sqrt{M(s)}g(\xi, K^{p}\cdot)|\underline{\underline{\wedge}}-_{C}(\xi, K^{\mathrm{P}}\cdot)--_{l}(\underline{\underline{\wedge}}n)(\xi, K^{\mathrm{P}}\cdot)|ds||_{E^{2}(\nu)}^{2}$$\leq||\int_{0}^{t}\sqrt{M(s)}g(\xi, K^{p}\cdot)\sum_{k=n+1}^{\infty}|\Theta_{k}(s;\xi, K^{p}\cdot)-\Theta_{k-1}(s;\xi, K^{p}\cdot)|ds||_{E^{2}(\nu)}^{2}$

79
Therefore, by (7.11), for any $0<r<1$ we have$|| \int_{0}^{t}(\hat{F}(s,---s)(\xi, K^{p}\cdot)-\hat{F}(s,---_{S}(n))(\xi, K^{p}\cdot))ds||_{E^{2}(\nu)}^{2}$$\leq||\overline{M}g(\xi, K^{p}\cdot)\sum_{k=n}^{\infty}\frac{1}{k!}\{\overline{M}g(\xi, K^{p}\cdot)\}^{k}H(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}$$\leq\frac{r^{n}}{1-r}\sum_{k=n}^{\infty}\frac{\overline M^{2(k+2)}}{r^{k}k!^{2}}||g(\xi, K^{p}\cdot)^{k+2}\sqrt{1+|_{-0}^{\underline{\underline{\wedge}}}(\xi,K^{p}\cdot)|^{2}}||_{E^{2}(\nu)}^{2}$$\leq\frac{2r^{n}}{1-r}\sum_{k=n}^{\infty}\frac{\overline{M}^{2(k+2)}}{r^{k}k!^{2}}(k+2)!(RG_{\omega}(|\xi|_{q}^{2}))^{k+2}$,

where $R\geq 0$ and $p\geq 0$ are pointed out in (7.14). It follows from Theorem 6.3 that$\lim_{narrow\infty}\int_{0}^{t}F(s,--_{s}-(n))ds=\int_{0}^{t}F(S,---s)ds$.

Hence we see that

三 t=nl\rightarrow im\infty ---t(n)$=—0+ \int_{0}^{t}F(s,---s)ds$

is asolution.
which shows from Theorem 6.5 that,
andFinally we prove the uniqueness. Assume that we have two solutions
which satisfies the same integral equation. Modeled after the derivation of (7.8), we come to$\{_{-t}^{-}-\}$$\{_{-t}^{-}-\}$$\{X_{t}\}$.

$|_{-\iota}^{\wedge}--( \xi, \eta)-\hat{X}_{t}(\xi, \eta)|\leq g(\xi, \eta)\int_{0}^{t}\sqrt{M(s)}|_{-s}^{\wedge}--(\xi, \eta)-\hat{X}_{s}(\xi, \eta)|ds$Ifollows by astandard argument with the Gronwall inequality.
Then
The following result for regular solution of normal-0rdered white noise differential equation is immediate from Theorem 7.2. For more relevant study of regularity of solutions ofnormal-0rdered white noise differential equations, we refer to [5].
$–_{t}-\wedge=\hat{X}_{t}$be two quantum stochastic prO,
and letCorollary 7.3 Let$[0,T]$exist $q\geq 0$ and a nonnegative
there, satisfying the conditions:cesses, where runs oversatisfying
function $H\in L^{1}[0, T]$ , and a nonnegative, locally bounded function on$\{L_{t}\}$$p\in \mathrm{R}$$\{M_{t}\}\subset \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$$t$$\mathcal{E}\cross D_{-p}$$g$$||g(\xi, K^{p}\cdot)^{n}||_{E^{2}(\nu)}^{2}\leq n!(RG_{\omega}(|\xi|_{q}^{2}))^{n}$for some$R\geq 0$

such that for all ,$\xi$$\eta\in \mathcal{E}$$any—0\in \mathcal{L}(\mathcal{W}_{\omega}, \mathcal{G}_{p})$$\cdots$

and $s\in[0, T]$$\max\{|\hat{L}_{s}(\xi, \eta)e^{-\langle\xi,\eta\rangle}|^{2}$

Then for$n=1,2$ ,,,$|\overline{M}_{s}(\xi, \eta)|^{2}\}\leq H(s)g(\xi, \eta)^{2}$.

satisfying$||g(\xi, K^{p}\cdot)_{-0}^{n^{\underline{\underline{\wedge}}}}(\xi, K^{p}\cdot)||_{E^{2}(\nu)}^{2}\leq n!(RG_{\omega}(|\xi|_{q}^{2}))^{n}$,$n=1,2$ ,$\cdots$,

and $q\geq 0$ , the (nomal-Ordered) white noise differential equation (7.3) has, $t\in[0, T]$ .
a unique solution infor some$R\geq 0$$–\iota-$$\mathcal{L}(\mathcal{W}_{\alpha}, \mathcal{G}_{\mathrm{P}})$

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