数理解析研究所講究録
923 巻 1995 年 154-190

154

CONDITIONAL EXPECTATION IN
CLASSICAL AND QUANTUM WHITE NOISE CALCULINOBUAKI OBATAGRADUATE SCHOOL OF POLYMATHEMATICSNAGOYA UNIVERSITYNAGOYA, 464-01 JAPANIntroduction
The present paper continues the new approach to quantum stochastic processes on Fockspace developed in a series of papers [22], [23], [24], [25]. It is the noticeable feature ofthis approach that the quantum white noise, i.e., the time derivative of quantum Brownian
motion, is formulated as a-flow of operators on Fock space. More precisely, the role of
the annihilation process
and the creation process
in the works of Belavkin [1],Hudson-Parthasarathy [13], Meyer [19] and Parthasarathy [26] is played by their infinitesimal increments:,.
$C^{\infty}$$\{A_{t}\}$$\{A_{t}^{*}\}$$a_{t}= \frac{d}{dt}A_{t}$$a_{t}^{*}= \frac{d}{dt}A_{t}^{*}$

It is very common that these operators are understood as operator-valued distributions andhence are not defined pointwisely. On the other hand, it is also known (though not widelyused in practice) that the creation and annihilation operators are defined pointwisely using
a suitable Gelfand triple, see e.g., [3], [6], [14]. In particular, the special choice of Gelfand
triple of white noise functions$(E)\subset L^{2}(E^{*}, \mu)\cong\Gamma(L^{2}(\mathbb{R}))\subset(E)^{*}$

yields such situation; in fact, $a_{t}\in \mathcal{L}((E), (E))$ and. The above Gelfand
triple is referred to as the $Hida-Kubo^{-}Takenaka$ space [8], [15]. A similar structure called
Fock scale is introduced by Belavkin [1] in order to develop a non-adapted It\^o theory on
Fock space, though the pointwisely defined annihilation and creation operators are notformulated. A big advantage of the-Takenaka space is also found in [20] wherea general theory of operators in
is established systematically in terms ofpointwisely defined annihilation and creation operators, see also [21] for generalization tovector-valued white noise distributions.
There lives a canonical flow
called Brownian motion in.
$\{B_{S}s\leqt\}$Then the conditional expectation relative to the a-field generated bybecomes
one of the most fundamental concepts in both classical and quantum stochastic analyses.
$a_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$\mathrm{H}\mathrm{i}\mathrm{d}\mathrm{a}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}$$\mathcal{L}((E), (E)^{*})$$\{B_{t}\}_{t\in \mathbb{R}}$$E_{t}$$L^{2}(E*, \mu)\cong\Gamma(L^{2}(\mathbb{R}))$155

In fact, the conditional expectation
is an orthogonal projection acting on
and. In that sense it can be treated fully within our operator
therefore, belongs totheory; however, in various applications we need to discuss the conditional expectation ofa white noise distribution. Unfortunately, the conditional expectation is not defined on thewhole spaceof white noise distributions due to the fact that pointwise multiplication of
distributions is not defined in general. This would be one of the reasons why the conditional
expectation has not been discussed actively along with the-Takenaka space.
While, being based on a different framework of white noise distributions Hida [9] introduced).
the conditional expectation and suggested possibility of application to prediction
In this paper we propose an idea to overcome the above mentioned difficulty. Namely, we
introduce a certain space of test white noise functions, denoted by $(A)$ , which is bigger than$(E)$ and obtain by duality a space of white noise distributions, denoted by. Thereholds a simple inclusion relation among these spaces:$L^{2}(E^{*}, \mu)$$E_{t}$$\mathcal{L}((E), (E)^{*})$$(E)^{*}$$\mathrm{H}\mathrm{i}\mathrm{d}\mathrm{a}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}^{1}$$(A)^{*}$$(E)\subset(A)\subset L^{2}(E^{*}, \mu)\subset(A)^{*}\subset(E)^{*}$.

A white noise distribution belonging tois called admissible. It is shown that theconditional expectation
becomes a continuous operator from
into itself which keeps$(A)$ invariant. Accordingly, in both classical and quantum cases the notion of an admissible
process is naturally introduced and the conditional expectation of such a process becomesan interesting subject to study. In this paper we study the Hitsuada-Skorokhod integral ofan admissible process and observe how the conditional expectation acts on it. Moreover,
we derive prototypes of representation of a martingale in terms of stochastic integrals bothin classical and quantum cases. In particular, the result in classical case is thought of as avariant of the so-called Clark formula [4] which has been discussed with great interests invarious aspects, e.g., in connection with martingale representation, see also [28] for a white).
noise
The paper is organized as follows: Section 1 is devoted to assembling some technicalinstruments in the operator theory on white noise distributions. In Section 2 we introduce
admissible white noise distributions and the conditional expectation. In Section 3 we study
the Hitsuda-Skorokhod integral and derive a variant of the Clark formula. In Section 4
we introduce the conditional expectation for operators and the notion of an admissible
quantum stochastic process. In Section 5 we discuss quantum stochastic integrals in termsof white noise calculus. In particular, we obtain the conditional expectation of a quantum
Hitsuda-Skorokhod integral and discuss representation of a quantum martingale in termsof stochastic integrals.
$(A)^{*}$$(A)^{*}$$E_{t}$$\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{c}\mathrm{h}^{2}$

Preliminaries

1

In the recent development the basic framework of white noise calculus is constructed froman arbitrary topological space
keeping in mind applications to quantum and random
fields [15], [20]. This framework is called the standard
calculus [11].
of white
The present paper being devoted to a study of a stochastic “process,” we take$T$$\mathit{8}etup$$noi_{\mathit{8}}e$$T=\mathbb{R}$

$1)1$

$2)1$

thank Professor H.-H. Kuo for the information.
thank Professor H. Watanabe for interesting conversation on this topic.

156

and regard it as the time axis. Some of the results obtained below remain valid under the
standard setup after straightforward modification.

1.1

Triplet of white noise functionals

Let $H=L^{2}(\mathbb{R}, dt)$ be the Hilbert space of -valued -functions on
and inner product, and consider the Gelfand triple:$L^{2}$$\mathbb{R}$$\mathbb{R}$with norm$|\cdot|_{0}$$\langle\cdot, \cdot\rangle$$E=S(\mathbb{R})\subset H=L^{2}(\mathbb{R}, dt)\subset E^{*}=S’(\mathbb{R})$It is known that the topology of

$E$

.

(1.1)

is defined by the norms:$|\xi|_{p}=|A^{p}\xi|_{0}$,$\xi\in E$,$p\in \mathbb{R}$,$q\geq 0$,

where$A=1+t^{2}- \frac{d^{2}}{dt^{2}}$.

These norms are linearly ordered in the sense that$|\xi|_{p}\leq\rho^{q}|\xi|_{p+q}$,$p\in \mathbb{R}$

,

(1.2)

where
$\rho=\inf \mathrm{s}_{\mathrm{P}^{\mathrm{e}}}\mathrm{C}(A)=\frac{1}{2}$.

In fact, $E$ is a countable Hilbert nuclear space. The canonical bilinear form on,
being compatible with the real inner product of $H$ , is denoted also by.
The Gaussian measure associated with the Gelfand triple (1.1) is the unique probability
measure on
satisfying$E^{*}\cross E$$\langle\cdot, \cdot\rangle$$E^{*}$$\mu$$\exp(-\frac{1}{2}|\xi|_{0}2)=\int_{E^{*}}e^{i\langle x,\xi}\mu(dx)\rangle$

The probability space$(E^{*}, \mu)$,$\xi\in E$.

is called the Gaussian space. Let$(L^{2})\equiv L^{2}(E^{*}, \mu;\mathbb{C})$denote the Hilbert space of -valued -functions on the Gaussian spaceprobabilistic aspect is emphasized, we also use the symbol$\mathbb{C}$$L^{2}$$\mathrm{E}(\phi)=\int_{E^{*}}\phi(x)\mu(dX)$$(E^{*}, \mu)$. When a,which is the mean value (random average) of a random variable.
The canonical bilinear form on
is denoted byagain and its -bilinear
extension tois also denoted by the same symbol3) . For a non-negative integer
and $x\in E^{*}$ an element :is uniquely defined by$\phi\in L^{1}(E^{*}, \mu)$$(E^{\otimes n})^{*}\cross E^{\otimes n}$$\mathbb{C}$$\langle\cdot, \cdot\rangle$$(E_{\mathbb{C}}^{\otimes n})^{*}\cross E_{\mathbb{C}}^{\otimes n}$$n$$x^{\otimes n}:\in(E^{\otimes n})_{\mathrm{s}\mathrm{y}\mathrm{m}}^{*}$$\phi_{\xi}(x)\equiv\sum_{=n0}^{\infty}\langle:x^{\otimes n}:,$$3)\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}$$\frac{\xi^{\otimes n}}{n!}\rangle=\exp(\langle x, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)$,$\xi\in E_{\mathbb{C}}$,$x\in E^{*}$

the paper we do not use a specific symbol for the hermitian inner product.

,

(1.3)

157

is called the vacuum. As isis the so-called exponential vector. In particular,
wherewell known, each $\phi\in(L^{2})$ is expressed in the following form:$\phi_{0}$$\phi_{\xi}$$\phi(x)=\sum_{n=0}^{\infty}\langle:X^{\otimes n}:,$$f_{n}\rangle$,,$x\in E^{*}$$f_{n}\in H_{\mathbb{C}}^{\otimes n}\wedge$

(1.4)

,

and the convergence of the series are understood inwhere each functionof . In that case,the -sense. Expression (1.4) is called the Wiener-It\^o$x\vdasharrow\langle:x^{\otimes n}:, f_{n}\rangle$$L^{2}$$expan\mathit{8}i_{\mathit{0}}n$$\phi$$|| \phi||_{0}2\equiv\int E^{*}X|\phi()|2\mu(dx)=n\sum_{=0}n!|\infty fn|^{2}0^{\cdot}$, the Boson Fock space overandThus we have a unitary isomorphism between. This is the celebrated Wiener-It\^o-Segal isomorphism.
For $\phi\in(L^{2})$ with Wiener-It\^o expansion given as in (1.4) we put$(L^{2})$$\Gamma(H_{\mathbb{C}})$$H_{\mathbb{C}}$$\Gamma(A)\phi(_{X)}=n=\sum\langle\infty 0:X^{\otimes n}:,$$A^{\otimes n}f_{n}\rangle$

becomes a positive selfadjoint operator on
Then)is thereby obtained:
and a complex Gelfand$\Gamma(A)$$(L^{2})$.

with Hilbert-Schmidt inverse,$\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}^{4}$

(1.5)

.

$(E)\subset(L^{2})\equiv L^{2}(E*, \mu;\mathbb{C})\cong\Gamma(H_{\mathbb{C}})\subset(E)^{*}$) functional and a generalized
are called a test (white
Elements in $(E)$ andby
the canonical bilinear form on(white noise) functional, respectively. We denote$(E)^{*}\cross(E)$ and by, namely,
the norm induced from$(E)^{*}$$noi\mathit{8}e$$\langle\langle\cdot, \cdot\rangle\rangle$$\Gamma(A)$$||\cdot||_{p}$$|| \phi||_{p}^{2}=||\Gamma(A)^{p}\phi||_{0}^{2}=\sum_{n=0}^{\infty}n!|(A^{\otimes n})^{p}fn|_{0}^{2}=\sum_{n=0}^{\infty}n!|f_{n}|_{p}^{2}$,$p\in \mathbb{R}$

,

(1.6)

are related through the Wiener-It\^o expansion (1.4). It is obvious from
where andfor all and(1.6) that $\phi\in(L^{2})$ belongs to $(E)$ if and only if$p\geq0$.
for all
We use a similar (but formal) expression for a generalized white noise functional. For eachbe given and assume that
non-negative integer let$\Phi\in(E)^{*}$ such that$p\geq0$
uniquea. Then there exists
for some$(f_{n})_{n=0}^{\infty}$$\phi$$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$$n$$\phi$

and$(f_{n})_{n=0}^{\infty}$$\Sigma_{n=0}^{\infty}n!|f_{n}|_{p}^{2}<\infty$$\Sigma_{n=0}^{\infty}n!|F_{n}|_{-p}^{2}<\infty$$F_{n}\in(E_{\mathbb{C}}^{\otimes n})_{\mathrm{S}}*\mathrm{y}\mathrm{m}$$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle p_{n}, f_{n}\rangle$

where

$n$

,

$\phi\in(E)$are related as in (1.4). In that case$\Phi(x)=\sum_{n=0}^{\infty}\langle:X\otimes n:,$$F_{n}\rangle$$\Phi$.

,

is written in a formal series:(1.7)

notation (1.5) is commonly used in the standard setup of white noise calculus, see e.g., [20]. As
the white noise triplet discussed here is constructed from the special Gelfand triple (1.1), it is often denotedby $(S)\subset(L^{2})\subset(S)^{*}$ instead, see e.g., [10] or I. D\^oku’s paper in this volume. Note also the remark at the
beginning of this section.
$4)\mathrm{T}\mathrm{h}\mathrm{e}$158

Conversely, every $\Phi\in(E)^{*}$ is obtained in this way. Expression (1.7) is called the WienerIt\^o expansion of . Note that (1.6) is also true for . Moreover, for
we define theexponential vector
through its Wiener-It\^o expansion in a similar manner as in(1.3).
$\Phi$$\Phi$$f\in E_{\mathbb{C}}^{*}$$\phi_{f}\in(E)^{*}$

1.2

Brownian motion and white noise

Through the Wiener-It\^o-Segal isomorphism we define $B_{t}\in(L^{2})$ by$\langle x,$$1_{[0,t]}\rangle$,$t\geq 0$,$B_{t}(x)=\{$$-\langle x,$$1_{[t,0]}\rangle$, $t<0$ ,denotes the indicator function of. Note that :where, by construction
the delta function belongs to$J\subset \mathbb{R}$$1_{J}$$x^{\otimes 1}:=x$

by definition. Since$E^{*}=S’(\mathbb{R})$$\delta_{t}$$W_{t}(x)=\langle x, \delta_{t}\rangle$,$t\in \mathbb{R}$,is a white noise distribution, i.e., $W_{t}\in(E)^{*}$ . As is easily seen,$B_{0}=0$,$\mathrm{E}(B_{t})=0$,$\mathrm{E}(B_{s}B_{t})=s$

A $t \equiv\min\{s, t\}$ ,8, $t\geq 0$ ,is a Brownian motion. It is easily verified that the map
which means thatis continuous. An important consequence of our approach is illustrated in thefollowing$t\mapsto B_{t}\in$$\{B_{t}\}$$L^{2}(E^{*}, \mu)$Proposition 1.1 The map $t\mapsto B_{t}\in(E)^{*}$ is a$\underline{d}B_{t}=W_{t}$$dt$Hence $t\mapsto W_{t}\in(E)^{*}$ is also a$C^{\infty}$,$C^{\infty}-map^{5)}$$t\in \mathbb{R}$and it holds that

.

-map.

Thus the one-parameter family of white noise distributions.
-flow incalled the white noise, is a

1.3$\{W_{t}\}$, which is justifiably$(E)^{*}$$C^{\infty}$

Integral kernel operators, symbols and Fock expansion, where andare locally convex spaces, denotes theThroughout the paper
carries thespace of continuous linear maps from into . Unless otherwise statedofboundedtopology(theeveryboundedsubsetofconvergenceon
topology of uniform
convergence).
We sketch briefly the essence of the operator theory on white noise distributions, see
there exists a unique operator[20] and [21] for the detailed account. For each
$D_{y}\in \mathcal{L}((E), (E))$ such that$X$$\mathcal{L}(X, \mathfrak{Y})$$X$$\mathfrak{Y}$$\mathcal{L}(X, \mathfrak{Y})$$\mathfrak{Y}$$X$$y\in E_{\mathbb{C}}^{*}$$D_{y}\phi_{\xi}=\langle y, \xi\rangle\phi_{\xi}$$5)\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}_{\mathrm{o}\mathrm{u}\mathrm{t}}$,$\xi\in E_{\mathbb{C}}$.

the paper the dual space of a locally convex space carries the strong dual topology.

159This is called the annihilation operator. In particular,$a_{t}=D_{s_{t}}$,$t\in \mathbb{R}$,is called the annihilation operator at a point or Hida’s differential $operator^{6)}$ . Thenand
. The latter is called the creation operator at a point.
It is emphasized that these operators are not operator-valued distributions but continuous
operators for themselves.
For eachtheer exists a unique operator
such that$a_{t}\in$$\mathcal{L}((E), (E))$$a_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$\kappa\in(E_{\mathbb{C}}m))^{*}\otimes(l+$$.-_{l,m}-(\kappa)\in \mathcal{L}((E), (E)^{*})$$\langle\langle_{-l,m}^{-}-(\kappa)\phi, \psi\rangle\rangle=\langle\kappa, \eta_{\phi,\psi}\rangle$,$\phi,$$\psi\in(E)$

,

where

.

$\eta_{\phi,\psi}(_{\mathit{8}_{1}}, \cdots, \mathit{8}_{l}, t_{1}, \cdots, t)m=\langle\langle a_{s_{1}}\cdots a_{s\iota^{a_{t}}}**1\ldots atm\phi,$ $\psi\rangle\rangle$

We use a formal (but descriptive) integral expression:
at 1$–l-,m( \kappa)=\int_{\mathbb{R}^{l+m}}\kappa(\mathit{8}_{1}, \cdots, \mathit{8}_{l1}, t, \cdots, t_{m})a^{*}\cdotsa^{*}$$\ldots atmd\mathit{8}1\ldots

d\mathit{8}ldtS1S_{l}1\ldots dt_{m}$

,

(1.8)

which is called an integral kernel operator with kernel distribution . It is known thatuniquely determined whenever it is taken from the subspace$\kappa$$(E_{\mathbb{C}}^{\otimes(l}m)^{*}+)=\mathrm{S}\mathrm{y}\mathrm{m}(l,m)\{\kappa\in(E_{\mathbb{C}}^{\otimes(l+}m))*;\mathit{8}_{l,m}(\kappa)=\kappa\}$

whereis the symmetrizing operator with respect to the firstindependently.
is a -valued function on
The symbol$l$$\mathit{8}_{l,m}$$\mathrm{o}\mathrm{f}---\in \mathcal{L}((E), (E)^{*})$$\mathbb{C}$

Since the exponential vectors
determines the operator uniquely.

$\{\phi_{\xi;}\xi\in E_{\mathbb{C}}\}$,$\xi,$$\eta\in E_{\mathbb{C}}$is,and the last $m$ variables$E_{\mathbb{C}}\cross E_{\mathbb{C}}$$—\wedge(\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$$\kappa$

defined by
(1.9)

.

span a dense subspace of $(E)$ , the symbol

For athere exists an $operator—\in \mathcal{L}((E), (E)^{*})$
such
if and only if the following two conditions are satisfied:, the function $(z, w)-\rangle\ominus(z\xi+\xi_{1}, w\eta+\eta_{1})$ is entire holo(O1) for any
morphic in(O2) there exist constant numbers $C\geq 0,$ $K\geq 0$ andsuch that$functi_{on}\ominus:E_{\mathbb{C}}\cross E_{\mathbb{C}}arrow \mathbb{C}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}_{\wedge,-}1.2$$that–=\ominus$$\xi,$$\xi_{1},$$z,$$\eta,$$\eta_{1}\in E_{\mathbb{C}}$$w\in \mathbb{C}_{i}$$p\in \mathbb{R}$$|\ominus(\xi, \eta)|\leq C\exp K(|\xi|_{p}^{2}+|\eta|^{2}p)$,$\xi,$$\eta\in E_{\mathbb{C}}$.

In that case,$||_{-}^{-}-\phi||-(p+q+1)\leq CM(K,p, q)||\phi||\mathrm{p}+q+1$,$\phi\in(E)$,where $M(K,p, q)\geq 0$ is anumber depending on $K\geq 0,$ $p\geq 0,$ $q>q_{0}(K,p)$ ; and$q_{0}(K,p)>0$ is also a constant number depending on $K\geq 0,$ $p\geq 0$ .
$con\mathit{8}tant$most literature of white noise calculus the annihilation operator at a point is denoted byrespecting the common notation among a wide audience, we use in this paper instead.
$6)\mathrm{I}\mathrm{n}$$a_{t}$$\partial_{t}$. However,

160

Theorem 1.3 For $any—\in \mathcal{L}((E), (E)^{*})$ there exists a unique family of kernel distributions
that$\kappa_{l,m}\in(E_{\mathrm{c}}^{\otimes}(\iota+m))\mathrm{s}*\mathrm{y}\mathrm{m}(\iota,m)\mathit{8}uch$$–=- \sum_{=l,,m0}^{\infty}---_{\iota,m}(\kappa\iota_{m},)$where the right hand side

$converge\mathit{8}$in$\mathcal{L}((E), (E)^{*})$,(1.10).

Expression (1.10) is called the expansion $of—$ in terms of integral kernel operators or
the Fock expansion. It seems that such expression of a Fock space operator in terms of normal ordered products of annihilation and creation operators is common among theoretical. The idea traces certainly back to Haag [7] and has been developed in various
contexts in quantum field theory, see e.g., [2], [3]. It is the strong point of our theory thata wide class of Fock space operators is determined to be discussed with mathematical rigor
using distribution theory. Thus our contribution here is purely mathematical.
Here are some of parallel results for an operator inwhich is a subspace of$\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{S}7)$$\mathcal{L}((E), (E))$$\mathcal{L}((E), (E)*)$.

Lemma 1.4 Let. In particular,$\kappa\in(E_{\mathbb{C}}^{\otimes(lm)}+)^{*}--\cdot Then--l,m-(\kappa)\in \mathcal{L}((E), (E))$$(E_{\mathbb{C}}^{\otimes m})^{*}$

for any$\cup 0_{m},(\kappa)\in \mathcal{L}((E), (E))$

if and only if$\kappa\in(E_{\mathbb{C}}^{\otimes m})^{*}$$\kappa\in(E_{\mathbb{C}}^{\otimes l})\otimes$.

Theorem 1.5 For a function :there exists an $operator—\in \mathcal{L}((E), (E))$
such $that—\wedge=\ominus$ if and only if (O1) in Theorem 1.2 and the next condition are satisfied:there exist $C\geq 0$ and $q\geq 0$ such thatfor any $p\geq 0$ and$\Theta$$E_{\mathbb{C}}\cross E_{\mathbb{C}}arrow \mathbb{C}$$\epsilon>0$$(\mathrm{O}2^{})$$|\ominus(\xi, \eta)|\leq C\exp\epsilon(|\xi|_{p+q}^{2}+|\eta|_{-p}^{2})$,$\xi,$$\eta\in E_{\mathbb{C}}$.

In that case,$||_{-}^{-}-\phi||_{p}-1\leq cM(\epsilon, q, r)|\phi|_{p+q}+r+1$

where$M(\epsilon, q, r)\geq 0i\mathit{8}$$r\geq r_{0}(q)j$and$r_{0}(q)\geq 0$,$\phi\in(E)$,a constant number depending on withis also a constant number depending on $q\geq 0$ .

$0<\epsilon<(2e^{3}\delta^{2})^{-1},$$\epsilon$$q\geq 0$,

Theorem 1.6 $For—\in \mathcal{L}((E), (E))$ let the Fock expansion be given as in (1.10). Then
all $m=0,1,2,$ , and the right hand side of (1.10) converges
in.
$\kappa_{l,m}\in(E_{\mathbb{C}}^{\otimes\iota})\otimes(E_{\mathbb{C}}^{\otimes m})^{*}for$$l,$$\cdots$$\mathcal{L}((E), (E))$

1.4

How to define an operator on white noise functions–An example

The operator symbol provides a useful criterion for checking whether or not an operator
formally defined in Fock space falls into a continuous operator on the white noise functions(Theorems 1.2 and 1.5). Here is a simple illustration.
Recall first thatis closed under the pointwise; in fact, it yields acontinuous bilinear map frominto
. Therefore multiplication ofand, denoted by, is defined as$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}_{\mathrm{P}^{\mathrm{l}\mathrm{i}\mathrm{a}}}\dot{\mathrm{c}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$E_{\mathbb{C}}$$E_{\mathbb{C}}\cross E_{\mathbb{C}}$$f\in E_{\mathbb{C}}^{*}$$f\xi=\xi f\in E_{\mathbb{C}}^{*}$$\langle f\xi, \eta\rangle=\langle f, \xi\eta\rangle$$7)1$$\xi\in E_{\mathbb{C}}$$E_{\mathbb{C}}$

thank Professor I. Ojima for this comment.

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

.

161

Proposition 1.7 For any.
$that-^{\phi=\phi}--\epsilon f\epsilon,$$f\in E_{\mathbb{C}}^{*}$

there exists a unique $operator—\in \mathcal{L}((E), (E)^{*})$ such$\xi\in E_{\mathbb{C}}$

Since the exponential vectors are linearly independent, the correspondence, is extended to a linear operator from the linear space spanned by the. We put
exponential vectors into
PROOF.

$\phi_{\xi}\mapsto\phi_{f\xi},$$\xi\in E_{\mathbb{C}}$$(E)^{*}$$\ominus(\xi, \eta)=\langle\langle\phi_{f\xi}, \phi_{\eta}\rangle\rangle=e^{\langle f\xi,\eta\rangle}$,$\xi,$$\eta\in E_{\mathbb{C}}$.

(1.11)It should be checked that satisfies conditions (O1) and (O2) in Theorem 1.2. Since (O1)
is obvious, we shall prove (O2). We choose $p\geq 0$ such that $|f|_{-p}<\infty$ . Then,$\ominus$$|\langle f\xi, \eta\rangle|=|\langle f, \xi\eta\rangle|\leq|f|_{-p}|\xi\eta|_{p}$By the continuity of pointwise multiplication of$E_{\mathbb{C}}$$|\xi\eta|p\leq c|\xi|p+q|\eta|p+q$ ’.

we choose $q\geq 0$ and $C\geq 0$ such that$\xi,$$\eta\in E_{\mathbb{C}}$,and hence$| \langle f\xi, \eta\rangle|\leq C|f|-p|\xi|p+q|\eta|p+q\leq\frac{C}{2}|f|_{-p}(|\xi|^{2}p+q|+\eta|^{2}p+q)$,

Thus (1.11) is estimated as$| \ominus(\xi, \eta)|\leq\exp\{\frac{C}{2}|f|_{-p}(|\xi|^{2}P+q+|\eta|_{p+q}2)\}$,

which proves (O2). It then follows from Theorem 1.2 that there exists $—\in \mathcal{L}((E), (E)^{*})$. In other words,
such that$—\wedge=\ominus$,$\langle\langle_{-}^{-}-\phi_{\zeta}, \phi_{\eta}\rangle\rangle=\langle\langle\phi_{f\xi}, \phi_{\eta}\rangle\rangle$

namely,$—\phi_{\xi}=\phi_{f\xi}$for any$\xi\in E_{\mathbb{C}}$$\xi,$$\eta\in E_{\mathbb{C}}$

,

qed

.

Proposition 1.7 is obtained easily. For $\phi\in(E)$
REMARK. (1) The explicit action
Wiener-It\^oexpansion is given asof which$\mathrm{o}\mathrm{f}_{-}^{-_{\mathrm{i}\mathrm{n}}}-$$\phi(x)=n\sum^{\infty}\langle:X^{\otimes n}:=0’ f_{n}\rangle$,$f_{n}\in E_{\mathbb{C}}^{\otimes n}$,

it holds that$— \phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes}:n,$$f^{\otimes n}\cdot f_{n\rangle}$(1.12),

identity
is pointwise multiplication. In fact, for an exponential vector
whereby the(1.12) is obvious. On the other hand, it is easy to see that the
Proposition 1.7 coincides.
right hand side of (1.12) belongs to$(E)$ .
spacewhole
, hence on thethe exponential vectors$\phi=\phi_{\xi}$$f^{\otimes n}\cdot f_{n}$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{0}\mathrm{r}--/\mathrm{d}-\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$$\mathcal{L}((E), (E)^{*})$$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}---/\mathrm{o}\mathrm{n}$$\{\phi_{\xi}\}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}---\mathrm{i}\mathrm{n}$162(2) Ifhave$f\in L^{\infty}(\mathbb{R})$, then$||_{-}^{-}- \phi||^{2}0=\sum_{n=}\infty 0n!|f^{\otimes}n$Choose $q\geq 0$ such thatfor some $q\geq 0$ . In fact, in view of (1.12) we$—\in \mathcal{L}((E)_{q}, (L^{2}))$$\rho^{q}||f||_{\infty}\leq 1$.

$f_{n}|^{2}0 \leq\sum_{n=0}^{\infty}n!||f^{\otimes}n||^{2}\infty|fn|^{2}0^{\cdot}$. Then$||_{-}^{-}- \phi||_{0}^{2}\leq\sum_{n}\infty=0n!||f||^{2n}\infty^{\rho^{2}}|nqf_{n}|_{q}2\leq\sum_{n=0}^{\infty}n!|fn|_{q}^{2}=||\phi||_{q}^{2}$, we see that
. In particular, if
Namely,typical example is the conditional expectation discussed below.
$—\in \mathcal{L}((E)_{q}, (L^{2}))$$||f||_{\infty}\leq 1$.

$—\in \mathcal{L}((L^{2}), (L^{2}))$

2

Conditional expectation for white noise distributions

2.1

Slowly increasing functions

For a$\mathbb{C}$-valued measurable function

$f$

on

$\mathbb{R}$. A

we put$|||f|||_{r}^{2}= \int_{-\infty}^{+\infty}|f(t)|^{2}(1+t^{2})^{r}dt$,$r\in \mathbb{R}$.

Note the obvious inequality:$r\in \mathbb{R}$$|||f|||_{r}\leq|||f|||r+r^{\prime_{2}}$,$r’\geq 0$.

Then $A_{\tau}=\{f;|||f|||_{r}<\infty\}$ becomes a Hilbert space with norm
functions) and forms an increasing chain of Hilbert spaces:...

(modulo null-$|||\cdot|||_{r}$$A_{2}\subset A_{1}\subset A_{0}=H_{\mathbb{C}}=L^{2}(\mathbb{R}, dt;\mathbb{C})\subset A_{-1}\subset A_{-2}\subset\cdots$

.

(2.1)

Then
$A= \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim$$AT= \bigcap_{r}rarrow\infty\geq 0A_{r}$becomes a countable Hilbert space, and by general theory we have$A^{*}= \mathrm{i}\mathrm{n}\mathrm{d}rarrow\infty\lim A-r=r\bigcup_{0\geq}A_{-r}$,is equipped with the strong dual topology as we have agreed. We say that
whereconsists of slowly increasing functions.
$A^{*}$$A^{*}$

isLemma 2.1 For any $r\geq 0$ there exists $p\geq 0$ such that the natural injection
continuousandinjectionthenaturalwell defined and continuous. Therefore
has a dense image.
$E_{p}arrow A_{r}$$E_{\mathbb{C}}arrow Ai\mathit{8}$. Since the defining norms
is a continuous norm on
PROOF. It is obvious thatis linearly ordered (see (1.2)), given $r\geq 0$ there exist $p\geq 0$ and $C\geq 0$ such that
$E_{\mathbb{C}}$$|||\cdot|||_{r}$$|\cdot|_{p}$$|||\xi|||_{r}\leq C|\xi|_{p}$,$\xi\in E_{\mathbb{C}}$

(2.2)

.

is well defined and continuous. Therefore the naturalHence the natural injection
is proved with a
is a dense subspace ofis continuous. That
injectionqed
standard argument.
$E_{p}arrow A_{r}$$E_{\mathbb{C}}arrow A$$E_{\mathbb{C}}$$A_{r}$

163

Lemma 2.2 For any $r\geq 0$ there
such that the natural injection
is well defined and continuous. In particular,
is a continuous injection.
$exi\mathit{8}tsp\geq 0$$A_{-r}$-,$E_{-p}$$A^{*}arrow E_{\mathbb{C}}^{*}$PROOF.
we have

Given $r\geq 0$ we choose $p\geq 0$ and $C\geq 0$ satisfying (2.2). Then for $f\in A_{-r}$$|\langle f, \xi\rangle|\leq|||f|||-r|||\xi|||r\leq C|||f|||_{-}r|\xi|p$.

Therefore $f\in E_{-p}$ and$|f|_{-p}\leq C|||f|||_{-r}$,$f\in A_{-r}$.

This completes the proof.

qed

REMARK. We shall prove that $A$ is not a nuclear space. Let
mal basis of. Then$\{e_{n}\}$be a complete orthonor-$L^{2}(\mathbb{R}, dt)$$f_{n}(t)=e_{n}(t)(1+t^{2})^{-(}r+r^{l})/2$

forms a complete orthonormal basis of$A_{\tau+r^{t}}$. We note that$|||f_{n}|||_{r}^{2}= \int_{-\infty}^{+\infty}|f_{n}(t)|^{2}(1+t)2r_{d}t=\int_{-\infty}^{+\infty}|e(nt)|2(1+t)^{-}2r’dt$

Let

$T$

.

be the multiplication operator by $(1+t^{2})^{-r’/2}$ . Then$|||f_{n}|||^{2}r=\langle Te_{n}, \tau e\rangle n=|Te_{n}|_{0}^{2}$.

Thus the natural injection $A_{r+r’}arrow A_{r}$ is of Hilbert-Schmidt type if and only if so is theoperator on. If so should be compact. But this never occurs because there
is no non-zero multiplication operator onwhich is compact.
$T$

$T$

$L^{2}(\mathbb{R}, dt)$

$L^{2}(\mathbb{R}, dt)$

2.2
For

Cut-off operator$t\in \mathbb{R}$

we put$\chi_{t}(\mathit{8})=1_{(-\infty,t]}(_{\mathit{8})}=\{$The multiplication operator induced byhave

Lemma 2.3$\chi_{t}\in \mathcal{L}(A_{r}, A_{r})$$\chi_{t}$1$s\leq t$$0$$\mathit{8}>t$

is denoted by the same symbol. Obviously weand is an orthogonal projection for any$r\in \mathbb{R}$.

Lemma 2.4 For each $r\geq 0$ there exist $p\geq 0$ and $C\geq 0$ such that$|\chi_{t}f|_{-}p\leq C|||f|||_{-r}$,$|(\chi_{t}-\chi_{s})f|-p|\leq ct-\mathit{8}|1/2|||f|||_{-r}$where 8,$t\in \mathbb{R}$and $f\in A_{-r}$ . In particular,,$t\mapsto\chi_{t}\in \mathcal{L}(A_{-r}, E_{-})p$

is continuous.

164

Let $f\in A_{-r}$ . Then for

PROOF.

$\xi\in E_{\mathbb{C}}$we have by the Schwartz inequality$|\langle\chi_{t}f, \xi\rangle|\leq|||\chi_{t}f|||-r|||\xi|||r\leq|||f|||-r|||\xi|||r$In view of$\mathrm{L}_{\sim}\mathrm{m}\mathrm{m}\mathrm{a}2.1$

we take$p_{1}\geq 0$

andsuch that$C_{1}\geq 0$$|||\xi|||_{r}\leq C_{1}|\xi|_{p1}$,$\xi\in E_{\mathbb{C}}$

Then (2.3) becomes$|\langle\chi_{t}f, \xi\rangle|\leq C_{1}|||f|||_{-}r|\xi|_{p_{1}}$and therefore$|\chi_{t}f|_{-p1}\leq C_{1}|||f|||_{-r}$Suppose nex that$\mathit{8}\leq t$$\iota$

(2.3)

.

,

$t\in \mathbb{R}$,.
,$f\in A_{-r}$

(2.4)

.

. Since$|\langle(\chi_{t}-x_{s})f, \xi\rangle|^{2}$$=$$\leq$$| \int_{s}^{t}f(u)\xi(u)du|2$,$\int_{s}^{t}|f(u)|^{2}(1+u)^{-}2rdu\int_{s}^{t}|\xi(u)|2(1+u)2r_{du}$we have$| \langle(\chi\iota-x_{s})f, \xi\rangle|\leq|||f|||_{-}r(t-\mathit{8})1/2\max_{\in\tau l\mathbb{R}}|\xi(u)|(1+u)^{/2}2r$

Note that
and$\xi\mapsto\max_{u\in \mathbb{R}}|\xi(u)|(1+u^{2})^{r/2}$$C_{2}\geq 0$,$f\in A_{-r}$is a continuous norm on$E_{\mathbb{C}}$,$\xi\in E_{\mathbb{C}}$.

, one may find $p_{2}\geq 0$such that$\max_{u\in \mathbb{R}}|\xi(u)|(1+u)^{r/}22\leq C_{2}|\xi|_{p_{2}}$,$\xi\in E_{\mathbb{C}}$

Then we see that$|\langle(\chi_{t}-\chi s)f, \xi\rangle|\leq C_{2}(t-\mathit{8})1/2|||f|||_{-}r|\xi|_{p}2$.

,

and therefore$|(\chi_{t}-\chi s)f|-p2\leq C_{2}(t-\mathit{8})1/2|||f|||_{-r}$,$\mathit{8}\leq t$,$f\in A_{-r}$

.

(2.5)

Finally we take $p= \max\{p_{1},p_{2}\}$ and $C= \max\{C_{1}, C_{2}\}$ . Then in view of (2.4) we have$|\chi_{t}f|_{-}p=|x_{t}f|_{-}p_{1}-(p-p_{1})\leq\rho^{p-p}|1xtf|_{-p1}\leq\rho^{p-\mathrm{P}1}C1|||f|||_{-}r\leq C|||f|||-r$

which proves the first inequality. The second one follows similarly from (2.5).
Lemma 2.5 For each $r\geq 0$ there exist $p\geq 0$ and$|||\chi_{t}\xi|||_{r}\leq C|\xi|_{p}$$C\geq 0\mathit{8}uch$,$|||(\chi_{t}-\chi_{s})\xi|||_{r}\leq C|t-S|^{1}/2|\xi|_{p}$

where

$s,$

$t\in \mathbb{R}$and$\xi\in E_{p}$. In particular,

that

,

$t-t\chi_{t}\in \mathcal{L}(E_{p}, A_{r})i\mathit{8}continuou\mathit{8}$

.

,

qed

165

PROOF. This is the dual result of Lemma 2.4.

qed

REMARK. It follows from Lemma 2.3 and the chain (2.1) thatfor any$r’\geq
0$
. But
and
is not continuous whatever
and $r’\geq 0$ .
In fact, suppose thatis continuous atforand $r’\geq 0$ .
$t\geq0$$t\leq0$; the case ofis proved in a similar manner. Then we
We further assume that
have
.
(2.6)
$\chi_{t}\in \mathcal{L}(A_{r+r’}, A_{r})$$t-\rangle$$r\in \mathbb{R}$$r\in \mathbb{R}$$\chi_{t}\in \mathcal{L}(Ar+r^{l}’ Ar)$$t\mapsto\chi_{t}\in \mathcal{L}(A_{r+r’}, A_{r})$$r\in \mathbb{R}$$t\in \mathbb{R}$$\lim_{ts\downarrow|||f||}\sup|_{r+r},\leq 1|||(\chi_{S}-\chi t)f|||_{r}^{2}=0$

On the other hand, if $t<s$ there exists a measurable functionand$|||f|||_{r+r}^{2},$such that$f$$= \int_{t}^{s}|f(u)|^{2}(1+u^{2})^{r+r_{du}’}=1$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f\subset(t, \mathit{8})$.

Then$|||(\chi s-xt)f|||_{r}2$$=$$\int_{t}^{S}|f(u)|^{2}(1+u)2r_{du}$$=$$\int_{t}^{s}|f(u)|^{2}(1+u^{2})^{r+r’}(1+u^{2})^{-r_{du}’}$$\geq$$(1+\mathit{8}^{2})^{-r}$’

Therefore$|||f|||r\mathrm{s}\mathrm{u}\mathrm{p}||+r’\leq 1|(\chi_{S}-x_{t})f|||_{r}2\geq(1+\mathit{8})2-r’$,and hence$\lim_{s\downarrow t}\inf\sup||||||f|||_{r+r^{;\leq 1}}(x_{s}-\chi_{t})f|||r\geq 2(1+t2)^{-}r’>0$.

This contradicts (2.6).

2.3

Admissible white noise distributions

We introduce a new family of norms on white noise functions. For $\phi\in(E)$ with WienerIt\^o expansion$\phi(x)=\sum\langle n\infty=0:x:\otimes n,$$f_{n}\rangle$,$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$,

we put$||| \phi|||_{r,\beta}.2=\sum_{n=0}^{\infty}n!e^{2}|\beta n||fn|||_{r}^{2}$

Suppose $r\geq 0$ andsuch that$\beta\in \mathbb{R}$,$r,$ $\beta\in \mathbb{R}$

.

(2.7)

are fixed. According to Lemma 2.1 we choose $p\geq 0$ and $C\geq 0$$|||\xi|||_{r}\leq C|\xi|_{p}$,$\xi\in E_{\mathbb{C}}$.

Then we have$|||f_{n}|||_{r}\leq C^{n}|f_{n}|_{p}$,$f_{n}\in E_{\mathbb{C}}^{\otimes n}$

.

(2.8)

166

Combining (2.7) and (2.8), we obtain$|||\phi|||_{r,\beta}2$$\leq$$\sum_{n=0}^{\infty}n!e^{2\beta}n_{C}2n|fn|^{2}p$$\leq$$\sum_{n=0}^{\infty}n!ec2\beta n2nq|\rho^{2n}fn|_{p+q}2$$\leq$$\sum_{n=0}^{\infty}n!(Ce\rho^{q})^{2n}\beta|fn|_{p+q}2$

Take $q\geq 0$ sufficiently large to have

$Ce^{\beta}\rho^{q}\leq 1$. Then$||| \phi|||_{r,\beta}2\leq\sum_{n=0}^{\infty}n!|fn|_{p}^{2}+q=||\phi||_{p+q}^{2}$,$\phi\in(E)$Let
be the completion of $(E)$ with respect to the norm
above is summarized in the following$(A)_{r,\beta}$

Lemma 2.6 For any $r\geq 0$ and$\beta\in \mathbb{R}$.

$|||\cdot|||_{r,\beta}$.

. What we have proved

there exists $p\geq 0$ such that$|||\phi|||_{r,\beta}\leq||\phi||_{p}$,$\phi\in(E)$

.

(2.9)

In particular, the natural injection $(E)_{p}arrow(A)_{r,\beta}$ is well defined and continuous.

In an obvious manner$\{(A)_{r,\beta}\}_{r,\beta}\geq 0$

forms a projective system of Hilbert spaces. Then$(A)=\mathrm{p}\mathrm{r},\mathrm{o}\mathrm{i}^{\lim_{arrow\infty}(A}r\beta)_{r,\beta}$

becomes a countable Hilbert space. On the other hand,system of Hilbert spaces, we have$(A)^{*}= \mathrm{i}\mathrm{n}\mathrm{d}\lim_{\infty r,,\betaarrow}(A)_{-r},-\beta$$\{(A)_{-}r,-\beta\}r,\beta\geq 0$

being an inductive.

In view of Lemma 2.6 we obtain an inclusion relation:$(E)\subset(A)\subset(A)_{0,0}=(L2)\subset(A)^{*}\subset(E)^{*}$,

where the injections are all continuous. A white noise distribution belonging tocalled admissible. Suppose $\Phi\in(E)^{*}$ is given with Wiener-It\^o expansion$\Phi(x)=\sum_{n=0}\langle:x^{\otimes n}:\infty,$

Then$\Phi$$F_{n}\rangle$

is.

is admissible, i.e., $\Phi\in(A)^{*}$ if and only if there exist $r\geq 0$ andfor all and$F_{n}\in A_{-r}^{\otimes n}$$(A)^{*}$$n$$|| \Phi||_{-r,-}2\beta=n=\sum n!e-2\beta n||\infty 0|F|n||_{-}2<\infty r$.

$\beta\geq 0$such that

167

Conditional expectation on admissible white noise distributions

2.4

For an admissible white noise distribution $\Phi\in(E)^{*}$ with Wiener-It\^o expansion$\Phi(x)=n\sum^{\infty}\langle:=0X^{\otimes}:n,$$F_{n}\rangle$,

we put$E_{t} \Phi(x)=\sum_{n=0}^{\infty}\langle:X^{\otimes n}:,$Lemma 2.7

$E_{t}\in \mathcal{L}((A)_{r,\beta}, (A)_{r,\beta})$$\chi_{t}^{\otimes n}\cdot F_{n}\rangle$,$t\in \mathbb{R}$.

(2.10)and is an orthogonal projection for any$r,$ $\beta\in \mathbb{R}$.

. On the other hand,In particular, $E_{t}\in \mathcal{L}((A), (A))$ and hence
simplicity. The operator
for
,
of
we
write
being the unique continuous extension
is called the conditional expectation (on admissible white noise distribu,
belongs to any of the spaces:
tions). Thus the conditional expectation$\mathcal{L}((A), (A)),$ $c((A)^{*}, (A)^{*}),$ $\mathcal{L}((E), (A)),$.
, and$E_{t}^{*}\in \mathcal{L}((A)^{*}, (A)^{*})$$E_{t}^{*}$$E_{t}^{*}=E_{t}$$E_{t}$$E_{t}\in \mathcal{L}((A)^{*}, (A)^{*})$$\mathcal{L}((A)_{r,\beta}, (A)_{r,\beta})$$E_{t}$$\mathcal{L}((A)^{*}, (E)^{*})$$\mathcal{L}((E), (E)^{*})$

Theorem 2.8 Both $t\mapsto E_{t}\in \mathcal{L}((E), (A))$ and $t-*E_{t}\in \mathcal{L}((A)^{*}, (E)^{*})$ are continuous.
PROOF.

For $\phi\in(E)$ with Wiener-It\^o expansion$\phi(x)=n\sum^{\infty}\langle=0:x^{\otimes}:n,$$f_{n}\rangle$,$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$,

we have by definition$|||(E_{s}-E_{t}) \phi|||_{r,\beta}^{2}=\sum_{n=0}^{\infty}n!e^{2}\beta n|||(\chi^{\bigotimes_{S}}-n\chi_{t})\otimes nfn|||_{r}^{2}$,8,$t\in \mathbb{R}$,$r,\beta\in \mathbb{R}$.

(2.11)

Since$\chi_{S}^{\otimes n}-x_{t}^{\otimes n}=\sum_{k=1}\chi^{\otimes k}s\otimes n-(xn\mathcal{E}^{-x)}t\otimes\chi_{t}\otimes k-1$,

we have$|||( \chi^{\bigotimes_{S}n}-\chi^{\bigotimes_{t}})nfn|||_{r}\leq\sum_{k=1}^{n}|||(\chi_{s}^{\otimes n-k}\otimes(\chi_{S}-\chi t)\otimes\chi t\otimes k-1)fn|||_{r}$,$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$

Take $p\geq 0$ and $C\geq 0$ exactly as in Lemma 2.5. Then (2.12) becomes$|||(x^{\bigotimes_{S}n}-xt)f_{n}\otimes n|||_{r}\leq nC^{n}|t-S|^{1}/2|f_{n}|_{p}$.

Inserting this into (2.11), we obtain$|||(E_{s}-E_{t})\phi|||_{r,\beta}2$$\leq$$\sum_{n=1}^{\infty}n!e^{2\beta n}n^{2}c^{2n}|S-t||f_{n}|_{p}^{2}$$\leq$$\sum_{n=1}^{\infty}n!n(2ce\rho)\beta q|2n\mathit{8}-t||fn|^{2}p+q$$\leq$$|_{\mathit{8}}-t| \sup n(2c_{e^{\beta}\rho)^{2n}||\phi}qn\geq 1||_{p+q}2$.

.

(2.12)

168

Take $q\geq 0$ large enough to have

$Ce^{\beta}\rho^{q}<1$. Then$M \equiv\sup_{1n\geq}n(Ce\rho\beta q)^{n}<\infty$and,$|||(E_{s}-E_{t})\phi|||r,\beta\leq M|\mathit{8}-t|1/2||\phi||p+q$$\phi\in(E)$.

$E_{t}\in \mathcal{L}((E), (A))$ is continuous.
This proves that
The second half of the statement
follows immediately by taking the adjoint.
qed$t-\rangle$

2.5

Fock expansion of the conditional expectation

It has been already noted that

$E_{t}\in \mathcal{L}((E), (E)^{*})$. Here we record the Fock expansion.

Lemma 2.9 The Fock expansion of the conditional expectation$E_{t}$

is given by.

$E_{t}= \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int^{+\infty}t\ldots\int_{t}+\infty\cdots daa_{Sn}a\cdots a\mathit{8}1\ldots dS_{1}**s_{1}Sns_{n}$PROOF.

By definition (2.10) we have$E_{t}\phi_{\xi}=\phi_{\chi_{2}\xi}$,$\xi\in E_{\mathbb{C}}$.

(In fact, the above relation characterizes the conditional expectation.) Then.

$\langle\langle E_{t}\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle\langle\phi xt\epsilon, \phi_{\eta}\rangle\rangle=\exp\langle x_{t}\xi, \eta\rangle=\exp\int_{-\infty}^{t}\xi(\mathit{8})\eta(_{S})d_{\mathit{8}}$Hence we have$e^{-\langle\xi,\eta\rangle} \langle\langle Et\phi\epsilon, \phi\eta\rangle\rangle=\exp(-\int_{t}^{+\infty}\xi(S)\eta(s)d\mathit{8})=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(\int_{t}^{+\infty}\xi(S)\eta(\mathit{8})d\mathit{8})^{n}$,which completes the proof.

qed

3

Adapted processes and the Hitsuda-Skorokhod integral

3.1

Adapted processes, admissible processes and martingales

The support of a distributionsubsetsuch thatvanishes in
Hida [9].

$F\in(E_{\mathbb{C}}^{\otimes n})^{*}$$K\subset \mathbb{R}^{n}$Definition 3.1 Let$F$$trightarrow\Phi_{t}\in(E)^{*}$, denoted by, is the smallest closed. The next definition is essentially due tobe a continuous map defined on an interval and$\Phi_{t}(x)=n\sum^{\infty}\langle=0:x^{\otimes}:n,$be the Wiener-It\^o expansion. Thenfor all $n\geq 1$ and .
$(-\infty, t]^{n}$$t$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}F$$\mathbb{R}^{n}-K$$\{\Phi_{t}\}$$F_{n}^{(t)}\rangle$

is called an adapted process if)$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}F_{n}^{(t}$$\subset$169

Definition 3.2 A continuous map
for all .
admissible for each , i.e.,$t\mapsto\Phi_{t}\in(E)^{*}$$\Phi_{t}\in(A\rangle^{*}$$t$, is called an admissible$proce\mathit{8}S$if$\Phi_{t}$

is

$t$

In the above definition we do not require thatis continuous with respect. Our condition above is weaker than this.
to the topology of$trightarrow\Phi_{t}\in(A)^{*}$$(A)^{*}$

be an admissible process. Then it is adapted if and only if
Proposition 3.3 Let,
only if
all
hence
and
.
for
if
for
$\{\Phi_{t}\}$$E_{t}\Phi_{t}=\Phi_{t}$$E_{s}\Phi_{t}=\Phi_{t}$$t$$\mathit{8}\geq t$

PROOF. The assertion follows immediately from the fact thatand only if
.

if
qed

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}F_{n}^{(t)}\subset(-\infty, t]^{n}$$x_{t}^{\otimes n}\cdot F_{n}^{()}t=F_{n}^{(t)}$Definition 3.4 An admissible process$\{\Phi_{t}\}$is called a martingale if$E_{s}\Phi_{t}=\Phi_{s}$for $s\leq t$ .

By definition a martingale is an adapted admissible process. The next assertion contains
a typical example.

Proposition 3.5 Let $\Phi\in(A)^{*}$ be an admissible whiteis a martingale.

$noi\mathit{8}edi\mathit{8}tributi_{\mathit{0}}n$. Then$\{E_{t}\Phi\}_{t\in \mathbb{R}}$It follows from Theorem 2.8 that $t\mapsto E_{t}\Phi\in(E)^{*}$ is continuous. Obviously
any , which means thatis an admissible process. Since $E_{s}(E_{t}\Phi)=$for $s\leq t$ obviously,
is a martingale.
qed

PROOF.

$t$$E_{t}\Phi\in(A)^{*}\mathrm{f}\mathrm{o}\Gamma$$E_{s}\Phi$$\{E_{t}\Phi\}$$\{E_{t}\Phi\}$The Brownian motion
is expressed as $B_{t}=E_{t}\Phi$ , where $\Phi(x)=\langle x,$particular, the Brownian motion is a martingale.
$\{B_{t}\}_{t\geq 0}$

Proposition 3.6 Any martingale$\{\Phi_{t}\}$admits an$expres\mathit{8}ion$$\chi_{t}F\infty n\otimes n.n\rangle$$\Phi_{t}(x)=\sum_{n=0}\langle:x^{\otimes}:,$where$F_{n}$

is a

PROOF.

$\mathbb{C}$-valued measurable function on$\mathbb{R}^{n}$of the form:,(3. 1)

Let

be the Wiener-It\^o expansion of , whereby assumption, we havefor

$F_{n}^{(t)}$$\Phi_{t}$$F_{n}^{(t)}\rangle$

is a slowly increasing function on$\mathbb{R}^{n}$. Since$\mathit{8}\leq t$$\chi_{s}^{\otimes n}\cdot F_{n}(t)=F_{n}^{(s)}$,$\mathit{8}\leq t$

Therefore, we can define a measurable funtion$F_{n}(u_{1,n}\ldots, u)=F_{n}^{(t)}$

Then

. In

.

$\Phi_{t}(x)=\sum_{n=0}\langle:x^{\otimes n}:\infty,$$E_{s}\Phi_{t}=\Phi_{s}$$1_{[0,+\infty})\rangle$$F_{n}^{(t)}=\chi_{t}^{\otimes n}\cdot F_{n}$( $u1,$$F_{n}$$\cdots,$on,$n\geq 1$.

by$\mathbb{R}^{n}$u)n’$t\geq u_{1},$ $\cdots,$ $u_{n}$and we obtain (3.1).

.
qed

REMARK. In Proposition 3.6 one might consider a formal series:$\Phi(x)=\sum_{n=0}^{\infty}\langle:X^{\otimes n}:,$$F_{n}\rangle$.

However, is not necessarily a white noise distribution because there is no guarantee thatis slowly increasing. Neverthless, one can write
at a formal level.
$\Phi$$F_{n}$$E_{t}\Phi=\Phi_{t}$

170

3.2

Hitsuda-Skorokhod integral

We first introduce the integral of an$(E)^{*}$-valued function.

Lemma 3.7 [10, Proposition 8.1] Let $t\mapsto\Phi_{t}\in(E)^{*}$ be a map defined on a (finite or
infinite) interval I. Assume that for any $\phi\in(E)$ the function
belongs to$L^{1}(I, dt)$ . Then there exists a unique $\Psi\in(E)^{*}$ such that$t-\rangle$$\langle\langle\Psi, \phi\rangle\rangle=\int_{I}\langle\langle\Phi_{t}, \phi\rangle\rangle dt$,$\phi\in(E)$$\langle\langle\Phi_{t}, \phi\rangle\rangle$.

In that case we write$\Psi=\int_{I}\Phi_{t}dt$.

For example, if is a finite closed interval and $trightarrow\Phi_{t}\in(E)^{*}$ is continuous, the above
integral exists.
Again suppose we are given a map $t-,$, where runs over an interval . Notethatis defined because. If in addition
belongs to $L^{1}(I, dt)$ for any $\phi\in(E)$ , then$I$$\Phi_{t}\in(E)^{*}$$\alpha_{t}^{*}\Phi_{t}\in(E)^{*}$

$I$

$t$

$a_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$t-\mathrm{k}\langle\langle a_{t}^{*}\Phi_{t}, \phi\rangle\rangle=$$\langle\langle\Phi_{t}, a_{t}\phi\rangle\rangle$$\Psi=\int_{I}a_{t}^{*}\Phi_{t}dt\in(E)^{*}$

is defined according to Lemma 3.7. This is called the Hitsuda-Skorokhod integral of.
If is a finite closed interval andis continuous, the Hitsuda-Skorokhod
integral exists. In fact,
is a continuous function on the interval for any$\{\Phi_{t}\}$$I$$t-\rangle$$\Phi_{t}\in(E)^{*}$$I$$t\mapsto\langle\langle a_{t}^{*}\Phi_{t}, \phi\rangle\rangle$$\phi\in(E)$3.3.

Conditional expectation of Hitsuda-Skorokhod integral

Lemma 3.8 Let$t\mapsto\Phi_{t}\in(A)^{*}$

be a map defined on a$clo\mathit{8}ed$

finite interval $[a, b]$ such that
(3.2)$\sup_{a\leq t\leq b}|||\Phi_{t}|||_{-r,-\beta}<\infty$

for somethat the$r,$ $\beta\geq 0$$Hit\mathit{8}uda$. Assume that$trightarrow\langle\langle a_{t}^{*}\Phi_{t}, \phi\rangle\rangle$-Skorokhod integral

belongs to $L^{1}(a, b)$ for any $\phi\in(E)$ and$\int_{a}^{b}a_{s}\Phi_{S}*d_{\mathit{8}}$belongs to$(A)^{*},$$i.e.$, is an admissible white noise distribution. Then’$\int_{a}^{t\wedge b}a_{s}^{*}E_{t}\Phi_{s}d_{\mathit{8}}$,$\alpha\leq t$

,

(3.3)

$E_{t}( \int_{a}^{b}\alpha_{s}*\Phi_{S}ds)=$$0$$t\leq a$.

171

PROOF. Suppose
is fixed throughout. First note that the map
belongs to
is well defined. We shall show that
In fact,$t\in \mathbb{R}$$s\mapsto a_{s}^{*}E_{t}\Phi_{s}\in(E)^{*}$$L^{1}(\alpha, b)$$\mathit{8}\mapsto\langle\langle\alpha_{s}E_{t}*\Phi_{s}, \phi\rangle\rangle$

for any $\phi\in(E)$ .

$|\langle\langle\alpha_{s}^{*}E_{ts}\Phi, \phi\rangle\rangle|=|\langle\langle E_{tS}\Phi, a_{s}\phi\rangle\rangle|\leq|||E_{t}\Phi_{S}|||_{-}r,-\beta|||\alpha\phi s|||_{r,\beta}\leq|||\Phi_{S}|||_{-}r,-\beta|||as\phi|||r,\beta$.

In view of Lemma 2.6 we may find $p\geq 0$ such that$|||\alpha_{S}\phi|||_{r,\beta}\leq||\alpha_{s}\phi||_{p}$.

On the other hand, by [20, Theorem 4.1.1] there exist $q>0$ and $C\geq 0$ such that$||\alpha_{s}\phi||pC\leq|\delta S|_{-(+}pq)||\phi||_{pq}+\cdot$

Thus we obtain.

$|\langle\langle\alpha_{s}^{*}E_{ts}\Phi, \phi\rangle\rangle|\leq C|||\Phi_{S}|||_{-}r,-\beta|\delta|_{-}s(p+q)||\phi||_{p}+q$

Since 8$[]arrow\delta_{s}$is continuous, taking $q>0$ large enough we see that$\sup_{a\leq s\leq b}|\delta t|-(p+q)<\infty$.

is bounded on
Combining this with (3.2).’ we see that
integrable. Then by Lemma 3.7 the Hitsuda-Skorokhod integral exists:$s-\succ|\langle\langle\alpha_{s}E_{t}*\Phi_{s}, \phi\rangle\rangle|$$\int_{a}^{t\Lambda b}\alpha_{s}E*\Phi tsd_{\mathit{8}}\in(E)^{*}$$[\alpha, b]$

and hence.

For simplicity we put$\Psi=\int_{a}^{b}\alpha_{s}^{*}\Phi_{s}dS$.

For the assertion it is sufficient to prove that
$E_{t} \Psi=\int_{a}^{t\wedge b}\alpha_{s}^{*}Et\Phi d\mathit{8}s$’$t>\alpha$;$E_{t}\Psi=0$,$t\leq\alpha$.

Since both sides in the above identities are white noise distributions, it is sufficient to provefor any
Suppose,$\langle\langle E_{t}\Psi, \phi_{\eta}\rangle\rangle$$=$$\int_{a}^{t\wedge b}\langle\langle\alpha_{S}^{*}E_{t}\Phi_{S}, \phi_{\eta}\rangle\rangle ds$$\langle\langle E_{t}\Psi, \phi_{\eta}\rangle\rangle$

$=$

$0$

,

$t>\alpha$

(3.4)

,

$t\leq\alpha$

,

(3.5)

. We shall prove (3.4) for (3.5) is verified in a similar manner.
is fixed as $t>a$ . Note first that$\eta\in E_{\mathbb{C}}$$t$$\langle\langle E_{t}\Psi, \phi_{\eta}\rangle\rangle=\langle\langle\Psi, E_{t}\phi_{\eta}\rangle\rangle=\langle\langle\Psi, \phi_{x}\mathrm{t}\eta\rangle\rangle$

We take an approximate sequence$A$ . Since $\Psi\in(A)^{*}$ by assumption,$\eta_{n}\in E_{\mathbb{C}}$

with

.

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta_{n}\subset(-\infty, t]$$\langle\langle\Psi, \phi_{\chi \mathrm{r}\eta}\rangle\rangle=\lim_{narrow\infty}\langle\langle\Psi, \phi\eta_{n}\rangle\rangle$

.

(3.6)
such that$\eta_{n}arrow\chi_{t}\eta$

in

(3.7)

172

Now we see that$\langle\langle\Psi, \phi_{\eta_{n}}\rangle\rangle$$=$$\int_{a}^{b}\langle\langle\alpha_{s}\Phi_{S}*, \phi\eta n\rangle\rangle d_{\mathit{8}}$$=$$\int_{a}^{b}\langle\langle\Phi_{SS}, a\phi\eta_{n}\rangle\rangle d\mathit{8}$$=$$\int_{a}^{b}\langle\langle\Phi_{s}, \phi_{\eta_{7l}}\rangle\rangle\eta_{n}(\mathit{8})d_{\mathit{8}}$$=$$\int_{a}^{t\wedge b}\langle\langle\Phi_{s}, \phi\eta n\rangle\rangle\eta n(_{\mathit{8}})d_{\mathit{8}}$.

Then in view of (3.6) and (3.7) we obtain.

$\langle\langle E_{t}\Psi, \phi_{\eta}\rangle\rangle=\mathrm{l}\mathrm{i}\mathrm{m}narrow\infty\langle\langle\Psi, \phi_{\eta_{n}}\rangle\rangle=\lim_{narrow\infty}\int_{a}^{t\wedge b}\langle\langle\Phi_{s}, \phi_{\eta_{n}}\rangle\rangle\eta n(S)d\mathit{8}=\int_{a}^{t\Lambda b}\langle\langle\Phi_{s}, \phi_{\chi\iota\eta}\rangle\rangle\eta(_{\mathit{8})d}\mathit{8}$Therefore, viewing$\langle\langle\Phi_{s}, \phi_{\chi_{t}\eta}\rangle\rangle\eta(S)=\langle\langle\alpha_{S}*Et\Phi s’\phi_{\eta}\rangle\rangle$

we come to (3.4).

qed

Proposition 3.9 Let
be an adapted admissible process, where. Assume that
finite interval$\{\Phi_{t}\}$$t$runs over a closed$[\alpha, b]$$\sup_{a\leq t\leq b}|||\Phi t|||-r,-\beta<\infty$for$\mathit{8}omer,\beta\geq 0$, that$t-*\langle\langle a_{t}^{*}\Phi_{t}, \phi\rangle\rangle$

belongs to$L^{1}(\alpha, b)$

and that the$Hit\mathit{8}uda$-Skorokhod

integral$\int_{a}^{b}\alpha_{s}^{*}\Phi_{S}d\mathit{8}$$belong_{\mathit{8}}$to$(A)^{*},$$i.e.$, is an admissible white noise distribution. Then$\int_{a}^{t\wedge b}a^{*}\Phi sSdS$,$a\leq t$,$t<\alpha$.

$E_{t}( \int_{a}^{b}\alpha^{*}\Phi sd\mathit{8}\mathrm{I}S=\{$$0$

PROOF. By the assumption of adaptedness we havefrom Lemma 3.8 that

$E_{t}\Phi_{s}=\Phi_{s}$$E_{t}( \int_{a}^{b}\alpha_{s)}^{*}\Phi sd\mathit{8}=\int_{a}^{t\wedge b}a_{S}E_{ts}*\Phi d\mathit{8}=\int_{a}^{t\wedge b}\alpha^{*}\Phi sd\mathit{8}s’$

which completes the proof.

for$t\geq \mathit{8}$. It then follows$a\leq t$,

qed

Theorem 3.10 Let $\Phi\in(A)^{*}$ be an admissible white
expansion$noi_{\mathit{8}}e$$F_{n}\otimes n\rangle$$\Phi(x)=\sum_{n=0}^{\infty}\langle:x:,$.

distribution with Wiener-It\^o173is a continuous function. Then there exists an adapted admissible
Assume that every
thatprocess$F_{n}$$\{\Psi_{t}\}\mathit{8}uch$$E_{t} \Phi=\mathrm{E}(\Phi)\phi 0+\int_{-\infty}^{t}\alpha_{s}^{*}\Psi_{s}d_{S}$where$\mathrm{E}(\Phi)=\langle\langle\Phi, \phi_{0}\rangle\rangle=F_{0}$

,

(3.8)

is the vacuum expectation of .
$\Phi$PROOF. Since
it is sufficient to prove (3.8) under the assumption that
, i.e., $F_{0}=0$ . By assumption there exists $r\geq 0$ such thatfor all $n\geq 1$ .
Now for $n\geq 0$ we put$E_{t}\phi_{0}=\phi_{0}$$\mathrm{E}(\Phi)=0$$F_{n}\in A_{-r}^{\otimes n}$$G_{n}^{(s)}(u1, \cdots, u)n=(n+1)F_{n+1}(s, u1, \cdots, u_{n})x_{s}(u_{1})\cdots x_{s}(u_{n})$Obviously,$G_{n}^{(_{S})}\in A_{-r}^{\otimes n}$$\{\Psi_{s}\}$8,$u_{1},$$\cdots,$$u_{n}\in \mathbb{R}$.

. We put$\Psi_{s}(x)=\sum\langle n\infty=0:X^{\otimes n}:,$

Then

,

$G_{n}^{(s)}\rangle$.

is an adapted admissible process. To prove (3.8) it is sufficient to see that$\int_{-\infty}^{t}\langle\langle\alpha_{s}\Psi_{s}*, \phi_{\xi}\rangle\rangle d_{\mathit{8}}=\langle\langle E_{t}\Phi, \phi_{\xi}\rangle\rangle$,$\xi\in E_{\mathbb{C}}$.

We first observe that.

$\int_{-\infty}^{t}\langle\langle a^{*}s\Psi_{S}, \phi_{\xi}\rangle\rangle d_{S}=\int_{-\infty}^{t}\langle\langle\Psi_{s}, a_{s}\phi_{\xi}\rangle\rangle d\mathit{8}=\int_{-\infty}^{t}\xi(S)\langle\langle\Psi_{S}, \phi_{\xi}\rangle\rangle d_{\mathit{8}}$Since by definition$\langle\langle\Psi_{S}, \phi_{\xi}\rangle\rangle$$=$$\sum_{n=0}^{\infty}n!\langle G_{n}^{(_{S})},$$=$$\sum_{n=0}^{\infty}\int-\infty+\infty\cdots\int_{-\infty}+\infty)G^{(}ns)(u_{1}, \cdots, u_{n})\xi(u_{1})\cdots\xi(u_{n}du1\ldots du_{n}$$=$$\sum_{n=0}^{\infty}(n+1)\int^{S}-\infty\cdots\int_{-\infty}^{s}F_{n}+1(\mathit{8}, u_{1}, \cdots, u_{n})\xi(u_{1})\cdots\xi(u)nu_{1}d\cdots du_{n}$$\frac{\xi^{\otimes n}}{n!}\rangle$,

we obtain$\int_{-\infty}^{t}\langle\langle a_{s}^{*}\Psi_{S}, \phi_{\zeta}\rangle\rangle d_{\mathit{8}}=$$= \sum_{n=0}^{\infty}(n+1)\int-\infty)t\xi(_{\mathit{8}}d\mathit{8}\int_{-\infty}s\ldots\int_{-\infty}s(F\mathit{8}, u_{1}n+1, \cdots , u_{n})\xi(u_{1})\cdots\xi(u_{n})du_{1}\cdots du_{n}$On the other hand, by symmetry we have$\int_{-\infty}^{s}\cdots\int_{-\infty}^{s}F+1(nus,1, \cdots, un)\xi(u1)\cdots\xi(u_{n})du1\ldots du_{n}=$.

$=n! \int_{-\infty}^{s}du_{1}\int_{-\infty}^{u_{1}}du_{2}\cdots\int_{-\infty}^{u_{n-1}}du_{n}Fn+1(S, u_{1}, \cdots, u_{n})\xi(.u_{1})\cdots\xi(u_{n})$

.

174

Finally we come to$\int_{-\infty}^{t}\langle\langle\alpha_{s}\Psi_{\mathrm{s}}*, \phi_{\xi}\rangle\rangle d_{\mathit{8}}=$$=$$\sum_{n=0}^{\infty}(n+1)!\int_{-}^{t}\infty)\xi(Sd\mathit{8}\int_{-\infty}sdu1\ldots\int-\infty)u_{n}-1du_{n}F+1(nS,u_{1},\cdots,u_{n})\xi(u_{1})\cdots\xi(u_{n}$$=$$\sum_{n=0}^{\infty}\int-\infty t\ldots\int^{t}-\infty)F_{n+1}(_{\mathit{8}}, u1, \cdots, u)n\xi(_{\mathit{8})\xi(u)\cdots\xi}1(u_{n}d\mathit{8}du_{1}\cdots du_{n}$$=$$\sum_{n=0}^{\infty}\langle x_{t}^{\otimes(1}Fn+)\xi^{\otimes}n+1,(n+1)\rangle$$=$$\sum_{n=1}^{\infty}\langle x^{\otimes n}tF_{n},$$=$$\langle\langle E_{t}\Phi, \phi_{\xi}\rangle\rangle$$\xi^{\otimes}n\rangle$.
qedThis completes the proof.

is a martingale for $\Phi\in(A)^{*}$ . The above
We have shown in Proposition 3.5 that
result is a prototype of representation of a martingale by means of the Hitsuda-Skorokhod
integral.
$\{E_{t}\Phi\}_{t\in \mathbb{R}}$3.4

Clark formulaSince
consider$E_{t}\in \mathcal{L}((E), (E)^{*})$, the composition$M_{t} \equiv\int_{-\infty}^{t}\alpha_{s}*E\alpha d\mathit{8}ss$

In fact,$M_{t}$’$\alpha_{t}^{*}E_{t}\alpha_{t}\in \mathcal{L}((E), (E)^{*})$$-\infty<t\leq+\infty$

is defined. We shall.

is defined in the following

Lemma 3.11 There exists a unique$-\infty<t\leq+\infty$$M_{t}\in \mathcal{L}((E), (E)^{*}),$,$\langle\langle M_{t}\phi_{\xi}, \phi_{\eta}\rangle\rangle=\int_{-\infty}^{t}\langle\langle\alpha_{s}^{*}Ess\phi_{\xi}\alpha, \phi_{\eta}\rangle\rangle d_{\mathit{8}}$$\xi,$$\eta\in E_{\mathbb{C}}$.

, such that
(3.9)

Moreover, $M_{t}^{*}=M_{t}$ .

PROOF.

Note first that$\langle\langle\alpha_{s}^{*}ES\alpha\phi s\xi, \phi_{\eta}\rangle\rangle$$=$$\xi(s)\eta(_{\mathit{8}})\langle\langle E_{s}\phi\xi, \phi_{\eta}\rangle\rangle$$=$$\xi(\mathit{8})\eta(\mathit{8})\exp\int_{-\infty}^{s}\xi(u)\eta(u)du$$=$$\frac{d}{ds}\exp\int_{-\infty}^{s}\xi(u)\eta(u)du$.

Therefore$\int_{-\infty}^{t}\langle\langle\alpha_{s}^{*}E_{s}a\phi_{\xi}s’\phi_{\eta}\rangle\rangle d_{\mathit{8}}$$=$$\exp\int_{-\infty}^{s}\xi(u)\eta(u)du|s=-\infty S=t$$=$$\exp\int_{-\infty}^{t}\xi(u)\eta(u)$du–l.

(3.10)

175

Consequently,.

$| \int_{-\infty}^{t}\langle\langle\alpha^{*}E_{S}\alpha_{s}\phi_{\xi}S’\phi_{\eta}\rangle\rangle ds|\leq\exp\int_{-\infty}^{t}|\xi(u)\eta(u)|du+1\leq\exp(|\xi|_{0}|\eta|_{0})+1$

Hence

.

$| \int_{-\infty}^{t}\langle\langle\alpha^{*}E_{S}\alpha_{s}\phi_{\xi}s’\phi_{\eta}\rangle\rangle d_{\mathit{8}}|\leq\exp\frac{1}{2}(|\xi|_{0^{+}}^{2}|\eta|^{2}0)+1\leq 2\exp\frac{1}{2}(|\xi|_{0}2+|\eta|_{0}2)$

It follows from Theorem 1.2 that the right hand side of (3.9) is the symbol of an operator
qed. That $M_{t}^{*}=M_{t}$ is obvious by definition.
, which we denote byin. Hence
One may prove by a slightly modified argument that.
and is the unique continuous extension of$M_{t}$$\mathcal{L}((E), (E)^{*})$$M_{\infty}\in \mathcal{L}((E), (E))$$M_{\infty}^{*}\in$$M_{\infty}$$\mathcal{L}((E)^{*}, (E)^{*})$

Lemma 3.12 It holds that,

$E_{t}\phi=\mathrm{E}(\phi)\phi 0+M_{t}\phi$$\phi\in(E)$.

(3.11)PROOF. In (3.10) we have already established$\langle\langle M_{t}\phi_{\xi}, \phi_{\eta}\rangle\rangle=\exp\int_{-\infty}^{t}\xi(u)\eta(u)$$du– l=\exp\langle\chi_{t}\xi, \eta\rangle-1$,$\xi,$$\eta\in E_{\mathbb{C}}$.

In other words,$\langle\langle M_{t}\phi\epsilon, \phi\eta\rangle\rangle=\langle\langle\phi x\iota\xi, \phi_{\eta}\rangle\rangle-\langle\langle\phi_{0}, \phi_{\eta}\rangle\rangle$,$\xi,$$\eta\in E_{\mathbb{C}}$

.

Hence
$M_{t}\phi_{\xi}=\phi_{\chi\epsilon}\iota-\phi 0=\phi_{x\mathrm{t}}\xi-\langle\langle\phi\xi, \phi 0\rangle\rangle\phi_{\mathrm{o}t}=E\phi\xi-\mathrm{E}(\phi_{\xi})\phi 0$.

qedThen by continuity we obtain (3.11).
is called the vacuum projection and, obviously, is extended to a
The map. It is known
into $(E)$ by putting
continuous linear operator from, while from the above consideration so does thebelongs to(\S 2.4) thatis extended to a continuous
vacuum projection. Therefore from Lemma 3.12 we see that
. In that sense we obtain a variant of the Clark formula.
operator in$\phi\mapsto \mathrm{E}(\phi)\phi_{0}$$\mathrm{E}(\Phi)=\langle\langle\Phi, \phi_{0}\rangle\rangle$$(E)^{*}$$\mathcal{L}((A)^{*}, (A)^{*})$$E_{t}$$M_{t}$$\mathcal{L}((A)^{*}, (A)^{*})$

Theorem 3.13 $For-\infty<t<+\infty$ it holds that
$E_{t} \Phi=\mathrm{E}(\Phi)\phi 0+(\int_{-\infty}^{t}\alpha_{s}^{*}E\alpha ssd\mathit{8})\Phi$,$\Phi\in(A)^{*}$.

(3.12)For $t=+\infty$ we have$\Phi=\mathrm{E}(\Phi)\phi 0+(\int_{-\infty}^{+\infty}\alpha^{*}sEa_{s}d_{\mathit{8})\Phi}S’$$\Phi\in(E)^{*}$.

The above result is closely related to representation of a martingale (Theorem 3.10). For
instance we have the following

176

Theorem 3.14 Let $\Phi\in(A)^{*}$ be an admissible white noise distribution with Wiener-It\^oexpansion$\Phi(x)=\sum\langle n\infty=0:X^{\otimes n}:,$Assume that every$F_{n}$

is a continuous function. Then,

$E_{t} \Phi=\mathrm{E}(\Phi)\phi_{0}+\int_{-\infty}^{t}a_{s}^{*}ESas\Phi d_{\mathit{8}}$REMARK. Since$F_{n}$.

$F_{n}\rangle$

is a continuous function,$\alpha_{s}\Phi$$\alpha_{s}\Phi(X)=\sum_{n=0}\langle:\infty X^{\otimes n}:,$PROOF. It is easily verified that$\Psi_{s}$$-\infty<t\leq+\infty$.

(3.13)

is defined by$\delta_{s}\otimes_{1n}F\rangle$.

defined in Theorem 3.10 coincides with

$E_{s}a_{s}\Phi$.
qed

REMARK. Note the difference between (3.12) and (3.13). The latter is a more direct
generalization of the so-called Clark formula, see [28] for a white noise approach.

4

Quantum Stochastic Processes

4.1

Definition and basic processes

Definition 4.1 [24] A one-parameter family of operators
is called aquantum
process if $t\mapsto--t-\in \mathcal{L}((E), (E)^{*})$ is continuous, where runs over aninterval. A continous linear map
isa generalized quantum
stochastic process. A generalized quantum stochastic process
is called regular if it isextended to a continuous linear map from.
$\{_{-t}^{-}-\}\subset \mathcal{L}((E), (E)^{*})$$\mathit{8}tochaStiC$$t$$—:E_{\mathbb{C}}arrow \mathcal{L}((E), (E)^{*})$$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d}--$$\cup$$E_{\mathrm{c}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{o}}}^{*}c((E), (E)^{*})$

Since
continuous, for a regular generalized quantum stochastic
one obtains a quantum stochastic process by putting$t\mapsto\delta_{t}\in E_{\mathrm{c}^{\mathrm{i}_{\mathrm{S}}}}^{*}$$–t–=–(\delta_{t})$,$t\in \mathbb{R}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}---$.

A quantum stochastic process obtained in this way is also called regular. Note that not
every quantum stochastic process is regular.

Proposition 4.2 The families of annihilation operatorsand creation operators
are both regular quantum stochastic processes. Moreover, bothand-maps.
are$\{\alpha_{t}\}_{t\in \mathbb{R}}$$t-\rangle$$\{\alpha_{t}^{*}\}t\in \mathbb{R}$$t\mapsto\alpha_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$PROOF.

$C^{\infty}$

Consider an integral kernel operator:$–0,1-(f)= \int_{\mathbb{R}}f(t)\alpha tdt$,

$f\in E_{\mathbb{C}}^{*}$.

$a_{t}\in \mathcal{L}((E), (E))$

177

is
It is proved [20, Theorem 4.1.1 and Proposition 4.3.10] that $–0,1-$ :. Since the natural injection $\mathcal{L}((E), (E))arrow \mathcal{L}((E), (E)^{*})$ is continua
forms a regular quantum stochastic process. It follows by a directous,. In fact,
is infinitely many times differentiable inverification that
$E_{\mathbb{C}}^{*}arrow \mathcal{L}((E), (E))$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{m}--\mathrm{a}\mathrm{p}$$\{\alpha_{t}=-0,1(\delta_{t})\}$$\mathcal{L}((E), (E))$$t\mapsto a_{t}$$\frac{d^{n}}{dt^{n}}a_{t}=(-1)n---(0,1\delta_{t}(n))$

By taking adjoint one may prove the assertion for
In a similar manner one obtains$\alpha_{t}^{*}$.
qed

easily.

Proposition 4.3 Put$A_{t-0}=--,1(1_{[0,t]})$

Then$\{A_{t}\}_{t\geq 0}$

and$\{A_{t}^{*}\}_{t\geq 0}$,$t\geq 0$

.

(4.1)

are quantum stochastic processes. Moreover, it holds that$\alpha_{t}=\underline{d}A_{t}$$dt$with respect to the topologies ofand$t\mapsto A_{t}\in \mathcal{L}((E), (E))$,$A_{t}^{*}=---1,0(1[0,t])$,$\mathcal{L}((E), (E))$$a_{t}^{*}=\underline{d}A_{t}^{*}$

$dt$

and

,
, respectively. In particular,-maps.

$\mathcal{L}((E)^{*}, (E)^{*})$$t\mapsto A_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$are$C^{\infty}$

defined in (4.1) areandDefinition 4.4 The quantum stochastic processes
respectively.
process,
creationtheandcalled the annihilation
The correspondence between classical and quantum stochastic processes is stated in the$\{A_{t}\}$$\{A_{t}^{*}\}$$proces\mathit{8}$following

Proposition 4.5 If $trightarrow\Phi_{t}\in(E)^{*}$ is continuous, regarded as multiplication operators.
becomes a quantum stochastic$\{\Phi_{t}\}$$proce\mathit{8}\mathit{8}$PROOF. Since the pointwise multiplication of white noise functions yields a continuous
bilinear map $(E)\cross(E)arrow(E)$ , multiplication of $\phi\in(E)$ and $\Phi\in(E)^{*}$ , denoted by, is defined by$\Phi\phi=\phi\Phi$$\langle\langle\Phi\phi, \psi\rangle\rangle=\langle\langle\Phi, \phi\psi\rangle\rangle$,$\phi,$$\psi\in(E)$,$\Phi\in(E)^{*}$.

It is then easily verified that $\phi-t\Phi\phi,$ $\phi\in(E)$ , is continuous and linear; namely, each
. Moreover, as is easily seen, thus obtained natural
gives rise to an operator in$(E)^{*}arrow\mathcal{L}((E),(E)^{*})$
qed
This completes the proof.
iscontinuous.
injection
The quantum Brownian motion and the quantum white noise are quantum stochastic
and the classicalprocesses respectively corresponding to the classical Brownian motion, for the definitions see \S 1.2, in such a way as described in Proposition
white noise
4.5. The quantum Brownian motion, again denoted by , is decomposed into the sum of
the annihilation and creation processes:$\Phi$$\mathcal{L}((E), (E)^{*})$$\{B_{t}\}$$\{W_{t}\}$$B_{t}$$B_{t}=A_{t}+A^{*}t$ ’$t\geq 0$.

$t\in \mathbb{R}$.

Similarly, for the quantum white noise we have$W_{t}=\alpha_{t}+a_{t}*)$

It is also noteworthy that the conditional expectations
process (see Theorem 2.8).

$\{E_{t}\}_{t\in \mathbb{R}}$form a quantum stochastic

178

4.2

Conditional expectation for admissible operators

Definition 4.6 An operator $.–\in \mathcal{L}((E), (E)^{*})$ is called admissible if there exists a continuous operator inof which restriction to $(E)$ coincides. For an admissible
the conditional expectation is defined as.
$\mathcal{L}((A), (A)^{*})$$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}---$$E_{t-}--E_{t}\in \mathcal{L}((A), (A)^{*})$$\mathrm{o}_{\mathrm{P}^{\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}-}}\mathrm{o}\mathrm{r}--\in \mathcal{L}((A), (A)^{*})$

Lemma 4.7 Let
function with$i.e.,$
be a slowly increasing function on$r\geq 0$ . Then$\beta>0$
some
any
for
for
$\mathbb{R}^{l+m},$$\kappa$$a\mathbb{C}$-valued measurable$|||\kappa|||_{-r}<\infty$$|||_{-l,m}^{-}-(\kappa)\phi|||_{-r,-\beta}\leq C|||\kappa|||-r|||\phi|||_{r},\beta$$\phi\in(A)$

’

,

where
.

(4.2)

$C= \sup_{n\geq 0}\{\frac{(l+n)!}{n!}\frac{(m+n)!}{n!}\}^{1/2}e^{-(n}2+m+l)\beta$

In particular, $–l,m-(\kappa)\in \mathcal{L}((A), (A)^{*})$ .
PROOF. The action of an integral kernel
Let $\phi\in(E)$ be given with Wiener-It\^o expansion$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{0}\mathrm{r}---_{l,m}(\kappa)$$\phi(x)=n\sum^{\infty}\langle:=0X^{\otimes}:n,$$f_{n}\rangle$

is given explicitly as follows:.

Then$–l-,m( \kappa)\phi(_{X)}=n=\sum\frac{(m+n)!}{n!}\infty 0\langle:x^{\otimes}(l+n):,$$\kappa\otimes_{m}f_{m+n}\rangle$.

By definition$|||_{-l,m}^{-}-(\kappa)\phi|||_{-r,-\beta}2$$=$$\sum_{n=0}^{\infty}(l+n)!e-2(l+n)\beta\{\frac{(m+n)!}{n!}\}^{2}|||\kappa\otimes_{m}fm+n|||_{-T}^{2}$$=$$\sum_{n=0}^{\infty}\frac{(l+n)!}{n!}\frac{(m+n)!}{n!}e-2(l+n)\beta(m+n)!|||\kappa\otimes_{m}f_{m}+n|||_{-r}2$.

Using the inequality$|||\kappa\otimes_{m}f_{m}+n|||_{-r}\leq|||\kappa|||-r|||f_{m}+n|||r$,$r\geq 0$

,

(4.3)

which is verified easily with the Schwartz inequality, we come to$|||_{-l,m}^{-}-(\kappa)\phi|||_{-r,-\beta}2$

where

$C$

!

$|||\kappa|||2-r|||fm+n|||_{r}^{2}$$\leq$$\sum_{n=0}^{\infty}\frac{(l+n)!}{n!}\frac{(m+n)!}{n!}e^{-2(}l+n)\beta(m+n)$$\leq$$\sum_{n=0}^{\infty}c2(n+m)\beta e^{2}(m+n)!|||\kappa|||2-r|||fm+n|||_{r}^{2}$$\leq$$C^{2}|||\kappa|||2-r|||\phi|||_{r,\beta}2$

is defined as in (4.2).

,
qed

179

Moreover, we can give a sufficient condition for an integral kernel operator to belong to$\mathcal{L}((A), (A))$ . For a measurable function
we puton$\mathbb{R}^{l+m}$$\kappa$$=$$|||\kappa|||^{2}\iota,m;r,S$$\int_{\mathbb{R}^{1+}}n\mathrm{t})^{r_{\mathrm{X}}}|\kappa(\mathit{8}1, \cdots, \mathit{8}l, t1, \cdots, tm)|2(1+\mathit{8}_{1}2)^{r}\cdots(1+s_{l}^{2}$$(1+t_{1}^{2})^{s}\cdots(1+t_{m}^{2})^{s_{d}}\mathit{8}1\ldots d\mathit{8}_{l}dt_{1}$Obviously,$|||\kappa|||_{\iota,m;}r,r=|||\kappa|||_{r}$Lemma 4.8 Let
that$\kappa$$|||\kappa|||_{l,mr};,-r<\infty$$dt_{m}$.

.

-valued measurable function on, then for any $\beta>0$ andfor allbe a...

$\mathbb{R}^{l+m}$$\mathbb{C}$$r\geq r_{0}$$|||_{-l,m}^{-}-(\kappa)\phi|||r,\beta\leq C|||\kappa|||_{\iota_{m;}r},r,-|||\phi|||r,\beta+\epsilon$. If there exists
we have$\phi\in(A)$’$r_{0}\geq 0$

such$\epsilon>0$

,

where
$C= \sup_{n\geq 0}\{\frac{(l+n)!}{n!}\frac{(m+n)!}{n!}\}^{1/2}e^{-\epsilon n}-(\beta+\epsilon)m+\beta\iota$.

In particular, $–l-,m(\kappa)\in \mathcal{L}((A), (A))$ .
PROOF.

We need only to modify the proof of Lemma 4.7 using$|||\kappa\otimes_{m}fm+n|||_{r}\leq|||\kappa|||\iota,m;r,-r|||f_{m}+n|||_{r}$,$r\geq 0$,

qed

instead of (4.3).

REMARK. The converse assertions of Lemma 4.7 and 4.8 are not true. In fact, there exists
an admissible integral kernel operator of which kernel distribution is not slowly increasing,see e.g., Proposition 4.12. On the other hand, we have a partial result for characterizing
an admissible operator in terms of Fock expansion. Let $—\in \mathcal{L}((A), (A)^{*})$ be given with
the Fock expansion$—= \sum_{l,m=0}^{\infty}---_{l,m}(\kappa_{l,m})$.

By general theory of countable Hilbert spaces (see e.g., [5], [20]) there exist $r\geq 0$ , $\beta>0$
and $C\geq 0$ such that$\phi\in(A)$ .
,$|||_{-}^{-}-\phi|||_{-r},-\beta\leq C|||\phi|||_{r,\beta}$Then$| \langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle|\leq C|||\phi_{\xi}|||r,\beta|||\phi\eta|||_{r,\beta r}=c\exp\frac{e^{2\beta}}{2}(|||\xi|||_{r}^{2}+|||\eta|||2)$and hence$| \langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle e^{-\langle}|\epsilon,\eta\rangle\leq C\exp\frac{e^{2\beta}+1}{2}(|||\xi|||_{r}2+|||\eta|||_{r}2)$

Then, applying the Cauchy estimate to$\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle e^{-}\langle\xi,\eta\rangle=\sum_{l,m=0}^{\infty}\langle\kappa l,m’\eta^{\otimes l_{\otimes\xi^{\otimes m}}}\rangle$

,

.

,

180

we obtain$|\langle\kappa\iota_{m},,$$\eta\otimes\xi\otimes l\otimes m\rangle|\leq C\{e(e^{2}+1\beta)\}(l+m)/2l-^{\iota/-}2mm/2|||\eta|||lr|||\xi|||^{m}r$

(4.4)

’

where the calculation is modelled after [20, Lemma 4.4.8]. However, it does not follow fromandis slowly increasing. This is a typical difference between(4.4) that; the former is based on $A$ which is not nuclear, while the latter is based on.
the nuclear space$\mathcal{L}((A), (A)^{*})$$\kappa_{l,m}$$\mathcal{L}((E), (E)^{*})$$E_{\mathbb{C}}$

Lemma 4.9 Let$\kappa_{l,m}\in A_{-r}^{\otimes(+m)}\iota,$$r\geq 0$. Then$Et—l,m(\kappa l,m)E_{t}=$$= \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int^{t\ldots ttt+}-\infty d\mathit{8}1\int-\infty d\mathit{8}l\int-\infty dt_{1}\cdots\int-\infty t_{m}d\int tu_{1}d\cdots\int\infty t+\infty du_{n}\cross$$\kappa\iota_{m},(\mathit{8}1, \cdots, sl, t1, \cdots, tm)\alpha_{u_{1}}\cdots a\alpha_{s}**u_{n}*\ldots*1a\alpha t_{1}\ldots a_{t_{m}}a_{u_{1}}\cdots\alpha\delta\iota u_{n}$PROOF.

4.3.

By a direct computation modelled after Lemma 2.9.

qed

Admissible processes

Definition 4.10 A quantum stochastic process
for each .

$\{_{-t}^{-}-\}\subset \mathcal{L}((E), (E)^{*})$

is called admissible$t$$\mathrm{i}\mathrm{f}---t\in \mathcal{L}((A), (A)^{*})$

Here are typical examples.

Proposition 4.11 The annihilation process
andadmissible. Moreover,$A_{t}\in \mathcal{L}((A), (A))$$\{A_{t}\}$and the creation process$\{A_{t}^{*}\}$

are both$A_{t}^{*}\in \mathcal{L}((A)^{*}, (A)^{*})$PROOF. It is proved in Proposition 4.3 thattic process. Since$\{A_{t}=---_{0,1}(1_{[0,t]})\}_{t\geq 0}$$|||1_{[]}0,t|||^{2}0,1;r,-r= \int_{0}^{t}(1+\mathit{8}^{2})^{-}rd_{\mathit{8}<\infty}$,$r\geq 0$

is a quantum stochas-,qed

the assertion follows immediately from Lemma 4.8.

The number$proce\mathit{8}S$(gauge process) is defined as$\Lambda_{t}=\int_{0}^{t}a_{s}a_{S}d_{\mathit{8}}*$Proposition 4.12 The number
PROOF.

$proce\mathit{8}\mathit{8}$is,$t\geq 0$$admi_{\mathit{8}}sible$. Moreover,

For $\phi\in(E)$ with Wiener-It\^o expansion$\phi(x)=\sum n=0\infty\langle:x^{\otimes n}:,$.

$f_{n}\rangle$

,

(4.5)
$\Lambda_{t}\in \mathcal{L}((A), (A))$

.

181

we put$gn\rangle\infty n$$\Lambda_{t}\phi(X)=\sum_{n=0}\langle:x^{\otimes}:,$.

Then by a direct computation we obtain$g_{n}(u_{1}, \cdots, u_{n})=nf_{n}(u_{1}, \cdots, u_{n})1[0,t](u_{1})$Hence for an arbitrary$\epsilon>0$

we have$|||\Lambda_{t}\phi|||^{2}r,\beta$$=$$\sum_{n=0}^{\infty}n!e^{2}|\beta n||gn|||^{2}r$$\leq$$\sum_{n=0}^{\infty}n!e^{2\beta}n^{2}|n||f_{n}|||_{r}^{2}$$=$$\sum_{n=0}^{\infty}n^{2}e-2\epsilon nn!e^{2}|(\beta+\epsilon)n||fn|||_{r}^{2}$$\leq$

which proves that.

$\Lambda_{t}\in \mathcal{L}((A), (A))$$(_{n\geq} \sup ne-\epsilon n)^{2}0\epsilon|||\phi|||_{r}^{2},\beta+$

,

.

qed

rise to
REMARK. As is stated in Proposition 4.5, any continuous map
a quantum stochastic process by multiplication. It can be proved with a similar argument
as in [20, \S 3.5] that the pointwise multiplication yields a continuous bilinear map from$(A)\cross(A)$ into $(A)$ . Therefore a (classical) admissible process is always considered as anadmissible quantum stochastic process by multiplication.
$t\mapsto\Phi_{t}\in(E)^{*}\mathrm{g}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{s}$5Qunatum stochastic integrals

5.1

Integrals of quantum stochastic processes

be a quantum stochastic process defined on an interval andLet. Then by general theory of topological vector spaces there exists
fix
as aa unique operator $-t\in \mathcal{L}((E), (E)^{*})$ such that
$I$$\{L_{t}\}\subset \mathcal{L}((E), (E)^{*})$$\alpha\in I$$\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}--\mathrm{n}$$\langle\langle_{-t}^{-}-\phi, \psi\rangle\rangle=\int_{a}^{t}\langle\langle L_{s}\phi, \psi\rangle\rangle d\mathit{8}$Moreover, it is proved that$\{_{-t}^{-}-\}$,$\phi,$$\psi\in(E)$,$t\in I$.

is again a quantum stochastic process. We write$–t-= \int_{a}^{t}Ld_{\mathit{8}}S$

against time. It is also proved thatand call it an integral ofandwith respect to the topology of$\{L_{s}\}$$\{_{-t}^{-}-\}$

is differentiable$\mathcal{L}((E), (E)^{*})$$\frac{d}{dt}---_{t}=L_{t}$

.

(5.1)

182

is a regular quantum stochastic process, by definition there exists
If. It then holds thatsuch thata continuous linear map$\{L_{t}\}\subset \mathcal{L}((E), (E)^{*})$$L_{t}=L(\delta_{t})$$L:E_{\mathbb{C}}^{*}arrow \mathcal{L}((E), (E)^{*})$$L(1_{[]}a,t)= \int_{a}^{t}L_{s}d_{\mathit{8}}$.

In particular, it holds that
$A_{t}= \int_{0}^{t}a_{s}d\mathit{8}$,$A_{t}^{*}= \int_{0}^{t}a_{s}d_{\mathit{8}}*$,$t\geq 0$.

Again we obtain$\underline{d}A_{t}=a_{t}$

$dt$

,

$\underline{d}A_{t}^{*}=a_{t}^{*}$

$dt$

(5.2)

,

, respectively.
andwith respect to the topologies ofbe a regular quantum stochastic process. Then we have
REMARK. Let$\mathcal{L}((E), (E))$$\mathcal{L}((E)^{*}, (E)^{*})$$\{L_{t}\}$$\lim_{aarrow-\infty}L(1[a,t])=L(1_{(t]}-\infty,)$in$\mathcal{L}((E), (E)^{*})$, since$\lim_{aarrow-\infty}1_{[}a,t$] $=1_{(-\infty,t]}$ in$E_{\mathbb{C}}^{*}$. Hence$\int_{-\infty}^{t}Ld\mathit{8}s\equiv\lim_{aarrow-\infty}\int_{a}^{t}L_{S}d\mathit{8}=L(1_{(-\infty,\iota]})$is well defined.

5.2

Quantum stochastic integrals

is a quantum stochastic process, so are
It is known [24] that if. Then one may define quantum stochastic processes by
and$\{L_{t}\}\subset \mathcal{L}((E), (E)^{*})$$\{L_{t}\alpha_{t}\}$$\{a_{t}^{*}L_{t}\}$$\int_{a}^{t}L_{s}\alpha_{s}d_{\mathit{8}}$,$\int_{a}^{t}a_{S}^{*}L_{s}d_{\mathit{8}}$

(5.3)

,

as in the previous section. The number process (4.5) is understood in this sense as well.
Definition 5.1 The former in (5.3) is called the quantum stochastic integral against the
integral against the creation
annihilation process, and the latter the quantumor the quantum Hitsuda-Skorokhod integral.
$stocha\mathit{8}tiC$$proces\mathit{8}$These generalize “quantum stochastic integrals” so far discussed by many authors invarious contexts. For example, in view of (5.2) we may write$\int_{a}^{t}L_{s}\alpha_{S}d\mathit{8}=\int_{a}^{t}L_{s}dA_{s}$If$a_{s}^{*}$

and$L_{s}$.

(5.4)

commute8) , then$\int_{a}^{t}\alpha_{S}^{*}LsdS=\int_{a}^{t}L_{s}a^{*}d\mathit{8}S=\int_{a}^{t}L_{s}dA_{s}^{*}$.

we need further assumption, e.g.,
the well-definedness ofhand, this condition relates deeply to the formulation of the adaptedness of$8)\mathrm{p}\mathrm{o}\mathrm{r}$$[a_{S}^{*}, L_{s}]$(5.5)$L_{s}\in \mathcal{L}((E)^{*}, (E)^{*})$$\{L_{s}\}$.

. On the other183

The right hand sides of (5.4) and (5.5) are introduced by Hudson-Parthasarathy [13] using
Riemann-Stieltjes integral of It\^o type. To develop a non-adapted quantum stochastic
integration a quantum Hitsuda-Skorokhod integral has been discussed in a different context
by Belavkin [1], Lindsay [18], and others.
. It is then easily verified thatis a quantum stochastic process, so is
If$\{L_{t}^{*}\}$$\{L_{t}\}$$( \int_{a}^{t}L_{ss}\alpha d\mathit{8})=*\int_{a}^{t}\alpha_{ss}^{*}L*d_{S}$

.

(5.6)

Thus in many cases it is sufficient to discuss only quantum Hitsuda-Skorokhod integrals.
We shall observe how the above quantum Hitsuda-Skorokhod integral generalizes a clasbe a continuous map defined on an interval . Fixsical one (\S 3.2). Let$a\in I$ and consider the (classical) Hitsuda-Skorokhod integral:$t-\rangle$$I$$\Phi_{t}\in(E)^{*}$$\Psi_{t}=\int_{a}^{t}a^{*}\Phi_{s}dSs\in(E)^{*}$.

becomes a quantum stochastic process which
On the other hand, by multiplication
for clarity. Then we have the quantum Hitsuda-Skorokhod integral:
we denote by$\{\Phi_{t}\}$$\{\tilde{\Phi}_{t}\}$$–t-= \int_{a}^{t}\alpha_{s}^{*}\tilde{\Phi}_{s}dS\in \mathcal{L}((E), (E)*)$.

Proposition 5.2 Notations and $as\mathit{8}umptions$ being as above, it holds that$\Psi t=--t-\phi 0$

where$\phi_{0}$,$t\geq a$,is the vacuum.

The proof is straightforward. Note that $–t-$ is no longer a multiplication operator. Let
is the quantum stochastic process
denote the multiplication operator by , i.e.,. Then we havecorresponding to$\tilde{\Psi}_{t}$$\Psi_{t}$$\{\tilde{\Psi}_{t}\}$$\{\Psi_{t}\}$$\tilde{\Psi}_{t}=\int_{a}^{t}(a_{s}\tilde{\Phi}_{S}*+\overline{\Phi}_{s}\alpha)sd_{S}$

.

(5.7)

In classical case, in addition to the Hitsuda-Skorokhod integral another approach to astochastic It\^o integral with non-adapted integrand has been discussed by Kuo-Russek [17],see also Kuo [16]. Our observation (5.7) would be a key to study a quantum analogue of
their results.
5.3

Generalized integral kernel operators

It is possible to replace the kernel distribution in an integral kernel operator (1.8)
with an operator-valued distribution, for generalities for such distributions see [21]. With$\mathcal{L}l((E), (E)^{*}))$ we may associate anby theeachformula:(5.8).
,$\kappa$$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes(+}m),$$\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle\langle L(\eta^{\otimes\otimes}\xi\iota_{\otimes}m)\phi_{\xi},$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\in \mathcal{L}((E), (E)^{*})$$\phi_{\eta}\rangle\rangle$$\xi,$$\eta\in E_{\mathbb{C}}$184

well defined is due to the characterization theorem of symbols (see Theorem
That1.2). It is reasonable to write$—\mathrm{i}\mathrm{s}$$—= \int_{\mathbb{R}^{1+m}}a^{*}\cdots a^{*}L(\mathit{8}1, \cdots, \mathit{8}l, t1, \cdots, tS1slm)\alpha t_{1}\ldots a_{tm}d\mathit{8}1\ldots ds\iota dt1^{\cdot}$,$.dtm$.

(5.9)

In fact, if is a-valued distribution (5.9) is reduced to an integral kernel
operator as in (1.8).
We observe that quantum stochastic integrals (5.3) are special cases of (5.9). Let
be a quantum stochastic process and consider$L$$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}-_{\mathrm{O}}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$$\{L_{s}\}$$\tilde{L}_{t}(\eta)=\tilde{L}(t, \eta)=\int_{a}^{t}\eta(\mathit{8})L_{s}dS$

where is fixed. It is then easily verified thatabove consideration we can define$a$,$\eta\in E_{\mathbb{C}}$,$\tilde{L}_{t}\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)^{*}))$. Hence by the$–t-= \int_{\mathbb{R}}\alpha_{s}^{*}\tilde{L}t(S)dS$

as a generalized integral kernel operator, the symbol of which is$\langle\langle_{-t}^{-}-\phi\epsilon’\phi\eta\rangle\rangle=\langle\langle\tilde{L}_{t}(\eta)\phi_{\xi}, \phi_{\eta}\rangle\rangle=\int_{a}^{t}\eta(_{S})\langle\langle L\phi_{\xi}S’\phi_{\eta}\rangle\rangle d_{\mathit{8}}=\int_{a}^{t}\langle\langle a^{*}L_{s}\phi_{\zeta}S’\phi_{\eta}\rangle\rangle ds$.

Therefore$\int_{\mathbb{R}}a_{s}^{*}\overline{L}_{t}(\mathit{8})d_{\mathit{8}=}\int at\alpha_{S}^{*}Ld_{\mathit{8}}s$.

In other words, the Hitsuda-Skorokhod integral of a quantum stochastic process is a special
case of a generalized integral kernel operator. The situation for a quantum stochastic
integral against the annihilation process is similar.

5.4

Adapted processes

Consider the annihilation and creation operators associated with
$D_{\eta}=--0,1-( \eta)=\int_{\mathbb{R}}\eta(t)atdt$,$D_{\eta}^{*}=---1,0( \eta)=\int_{\mathbb{R}}\eta(t)ad*tt$

It follows from general theory [20] thatto an operator from
into itself, i.e.,the commutators
andare meaningful.

$D_{\eta}^{*}\in \mathcal{L}((E), (E))$$(E)^{*}$$[D_{\eta}, ---]$$\eta\in E\mathrm{c}$and$D_{\eta}\in \mathcal{L}((E)^{*}, (E)^{*})$$D_{\eta}$:.

is continuously extended. Thus for$—\in \mathcal{L}((E), (E)^{*})$$[D_{\eta}^{*-}, --]$

Definition 5.3 A generalized quantum stochastic process
called adapted if$—\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)^{*}))$

is$[D_{\eta}, ---(\xi)]=[D_{\eta}^{*-}, --(\xi)]=0$

for any choice of$t\in \mathbb{R},$$\xi,$$\eta\in E_{\mathbb{C}}$such that$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\xi\subset(-\infty, t)$andLemma 5.4 [24] Assume that a generalized quantum stochastic
it is adaptedand only if $[D_{\eta’-t}--]=[D_{\eta}^{*-}, --t]=0$ for any$?f$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta\subset(t, +\infty)$.

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta\in(t, +\infty)$$process^{-}--i\mathit{8}$$t\in \mathbb{R}$and.

regular. Thenwith$\eta\in E_{\mathbb{C}}$185

Definition 5.5 A quantum stochastic process$\{_{-t}^{-}-\}$

is called adapted if$[D_{\eta’-t}--]=[D_{\eta}^{*}, ---_{t}]=0$

for any$t$and$\eta\in E_{\mathbb{C}}$with$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta\subset(t, +\infty)$.

Adaptedness of a quantum stochastic process was introduced by Hudson-Parthasarathy[13] and has been discussed by Huang [12] along with white noise calculus. Thier definitions
are compatible with ours.
is adapted and differen. Moreover, if
is adapted, so is
By definition, if
is also adapted.
, thentiable in$\{_{-t}^{-}-\}$$\{_{-t}^{-*}-\}$$\{_{-t}^{-}-\}$$\{d_{-t}^{-}-/dt\}$$\mathcal{L}((E), (E)^{*})$

Theorem 5.6 [24] Let$\{_{-t}^{-}-\}$

be a quantum$\mathit{8}tochastiC$$–t-= \sum_{l+m\geq 1}\cup--\iota,m(\kappa_{l},m(t))+c_{t}I$

be the Fock expansion. Then
and $l+m\geq 1$ .
any$\{_{-t}^{-}-\}i_{\mathit{8}}$process and let,$t\in \mathbb{R}$adapted if and only if,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\kappa_{l,m}(t)\subset(-\infty, t]^{l+m}$for$t\in \mathbb{R}$and the number, the creation process
For example, the annihilation processadapted.
areandare adapted. It is also obvious thatprocess
The notion of adaptedness for both classical and quantum stochastic processes (Definitions 3.1 and 5.5) are compatible.
$\{A_{t}^{*}\}$$\{A_{t}\}$$\{\alpha_{t}\},$$\{\Lambda_{t}\}$$\{a_{t}^{*}a_{t}\}$$\{\alpha_{t}^{*}\}$

be a $continuou\mathit{8}$ map defined on an interval. Then it
Proposition 5.7 Let
is adapted (in the classical sense) if and only if it is an adapted quantum stochastic
as multiplication operators.
$t-\rangle$$\Phi_{t}\in(E)^{*}$$proce\mathit{8}S$PROOF.

Let$\Phi_{t}=\sum_{n=0}^{\infty}\langle:x:\otimes n,$$F_{n}^{(t)}\rangle$

be the Wiener-It\^o expansion. Then, as multiplication operator the Fock expansion ofis given as$\Phi_{t}=\sum_{=l,,m0}^{\infty}\frac{(l+m)!}{l!m!}---l,m(F_{l+m}^{(})t)$$\Phi_{t}$,see [20, Proposition 4.6.4]. Hence, in view of Theorem 5.6 it is adapted (in the quantum
is adapted.
for all and $n\geq 1$ , i.e.,
sense) if and only if
qed
5.5$\{\Phi_{t}\}\subset(E)^{*}$$t$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}F_{n}^{(t)}\subset(-\infty, t]^{n}$

Quantum stochastic integrals of an adapted process

Proposition 5.8 Let
runs over an interval I. Then both$\{L_{t}\}\subset \mathcal{L}((E), (E)^{*})$

be an adapted quantum$st_{oC}ha\mathit{8}ticproce\mathit{8}S$$t$$\int_{a}^{t}L_{s}\alpha_{s}d\mathit{8}$are adapted.

,$\int_{a}^{t}\alpha_{S}^{*}L_{s}d\mathit{8}$,$t\geq\alpha$,$t\in I$

,

where

186

It is sufficient to prove the assertion for the qunatum Hitsuda-Skorokhod

PROOF.
integral$–t-= \int_{a}^{t}\alpha_{s}^{*}L_{s}d\mathit{8}$,$t\geq\alpha$,$t\in I$.

For that purpose we shall show that$[D_{\eta’-t}--]=[D_{\eta}^{*-}, --_{t}]=0$

whenever $t\geq a$ and$\eta\in E_{\mathbb{C}}$with$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta\subset(t, \infty)$. For$\phi,$$\psi\in(E)$$\langle\langle_{-t\eta}^{-}-D\phi, \psi\rangle\rangle=\int_{a}^{t}\langle\langle a^{*}L_{S}D\eta s\phi, \psi\rangle\rangle ds$

Since both$\{a_{s}^{*}\}$

and$\{L_{s}\}$we have by definition.

are adapted, we have$\langle\langle\alpha_{s}^{*}LD_{\eta}s\phi, \psi\rangle\rangle=\langle\langle D_{\eta s}\alpha^{*}L_{s}\phi, \psi\rangle\rangle=\langle\langle\alpha_{s}^{*}L_{s}\phi, D_{\eta}^{*}\psi\rangle\rangle$.

Therefore$\langle\langle_{-t\eta}^{-}-D\phi, \psi\rangle\rangle=\int_{a}^{t}\langle\langle a_{s}L*\phi s’ D_{\eta}^{*}\psi\rangle\rangle d_{\mathit{8}}=\langle\langle_{\cup t}^{-}-\phi, D_{\eta}^{*}\psi\rangle\rangle=\langle\langle D_{\eta^{-}t}--_{\phi}, \psi\rangle\rangle$

which proves $[D_{\eta}, ---t]=0$ . Similarly,$[D_{\eta}^{*-}, --_{t}]=0$,

is proved.

qed

Conditional expectation of a quantum Hitsuda-Skorokhod integral

5.6

Lemma 5.9 Let$\{L_{t}\}$be an admissible process such that$M \equiv\sup_{a\leq t\leq b|||\phi|}\sup||||_{r},\beta\leq 1|Lt\phi|||-r,-\beta<\infty$

for some$r,$ $\beta\geq 0$. Assume that the quantum Hitsuda-Skorokhod integral$–=- \int_{a}^{b}\alpha_{s}^{*}L_{s}d\mathit{8}$$i\mathit{8}admi_{\mathit{8}}sible,$$i.e.$, belongs to$\mathcal{L}((A), (A)^{*})$. Then’$\int_{a}^{t\wedge b}a_{s}^{*}EtLsEtdS$,$t\geq a$,(5.10)$E_{t}---E_{t}=$$0$$t<a$ .

,PROOF. Note that the composition
for $\psi\in(E)$ we have$\alpha_{s}^{*}E_{ts}LE_{t}\in \mathcal{L}((E), (E)^{*})$

is well defined. Moreover,$\phi,$$|\langle\langle a_{s}^{*}EtLEt\phi S’\psi\rangle\rangle|\leq|\langle\langle E_{tt}L_{s}E\phi, a_{s}\psi\rangle\rangle|\leq|||E_{ts}LE_{t}\phi|||_{-r,-\beta}|||\alpha_{s}\psi|||_{r,\beta}$

According to Lemma 2.6 we take $p\geq 0$ such that$|||\alpha_{S}\psi|||r,\beta\leq||a_{s}\psi||_{p}$

.

.

187

Noting also that

$E_{t}$

is an orthogonal projection, we obtain.

$|\langle\langle\alpha_{s}^{*}E_{ts}LEt\phi, \psi\rangle\rangle|\leq|||L_{st}E\phi|||-r,-\beta||\alpha\psi S||_{p}$By assumption we have$|\langle\langle\alpha_{s}^{*}E_{tt}L_{S}E\phi, \psi\rangle\rangle|\leq M|||E_{t}\phi|||r,\beta||\alpha_{s}\psi||_{p}\leq M|||\phi|||_{r},\beta||\alpha_{s}\psi||_{p}\leq M||\phi||p||\alpha_{S}\psi||_{p}$.

it follows from a similar argument as in the proof of Lemma 3.8 there exist$M’\geq 0$ and $q>0$ such that

As for$||\alpha_{S}\psi||_{p}$$||\alpha_{s}\phi||_{p}\leq M’||\phi||p+q$$\phi\in(E)$’,$a\leq s\leq b$.

Thus we come to$|\langle\langle\alpha_{s}^{*}E_{t}L_{st}E\phi, \psi\rangle\rangle|\leq MM’||\phi||_{p}||\psi||_{p+q}$,$\alpha\leq \mathit{8}\leq b$,$\phi,$$\psi\in(E)$,and hence the integral$\int_{a}^{t\wedge b}\alpha^{*}EtssLE_{t}d_{\mathit{8}}$,$t\geq a$,.
is well defined in
To see (5.10) we compute the operator symbol.
$\mathcal{L}((E), (E)^{*})$$\langle\langle Et---Et\phi\epsilon, \phi_{\eta}\rangle\rangle$$=$$\langle\langle_{\cup}^{-}-\phi xt\xi, \phi_{\chi\iota\eta}\rangle\rangle$$=$$\int_{a}^{b}\langle\langle\alpha_{s}^{*}L_{s}\phi_{\chi t}\epsilon, \phi_{\chi\iota\eta}\rangle\rangle d_{\mathit{8}}$$=$$\int_{a}^{b}\langle\langle L_{s}\phi\chi\iota\zeta, \alpha_{S}\phi_{\chi t}\eta\rangle\rangle ds$$=$$\int_{a}^{b}\langle\langle Ls\phi x\mathrm{z}\xi, \phi_{\chi_{t}\eta}\rangle\rangle\chi_{t}(_{S)}\eta(\mathit{8})d_{S}$.

In fact, we need a suitable argument of approximation which is similar to the proof of
Lemma 3.8. Hence for $t\leq a$ we have $E_{t\cup}--E_{t}=0$ . Suppose that $t>a$ . Then, the last
A ], we come tointegral being taken over the interval [$\alpha,$

$t$

$b$

$\langle\langle Et---Et\phi_{\xi}, \phi\eta\rangle\rangle=\int_{a}^{t\wedge b}\langle\langle L_{s}\phi_{\chi_{t}\xi}, \phi_{\chi_{t}\eta}\rangle\rangle\eta(s)d\mathit{8}$.

Finally we compute the integrand.
$\langle\langle L_{s}\phi_{\chi\xi}t’\phi_{\chi\iota\eta}\rangle\rangle\eta(\mathit{8})$$=$$\langle\langle L_{s}E_{t}\phi_{\xi}, E_{t}\phi_{\eta}\rangle\rangle\eta(\mathit{8})=\langle\langle EtL_{S}E_{t}\phi_{\xi}, \phi_{\eta}\rangle\rangle\eta(s)$$=$$\langle\langle E_{t}L_{st}E\phi_{\xi}, a_{s}\phi_{\eta}\rangle\rangle=\langle\langle a_{S}^{*}E_{t}LE_{t}s\phi_{\xi}, \phi_{\eta}\rangle\rangle$

qedwhich completes the proof.

Proposition 5.10 Let$\{L_{t}\}$,be an admissible process such that$\sup$$\sup$$a\leq t\leq b|||\emptyset|||r,\beta\leq 1$$|||L_{t}\phi|||-r,-\beta<\infty$

188

for some$r,$ $\beta\geq 0$. Assume that the quantum Hitsuda-Skorokhod integral$–t-= \int_{a}^{t}a_{s}^{*}L_{S}d\mathit{8}$is admissible, $i.e.,$to$belong_{\mathit{8}}$$\mathcal{L}((A), (A)^{*})$. If$E_{s-}-_{ts}-E=_{-s}--$PROOF.

,$t\geq\alpha$,

for all , then we have$E_{t}L_{t}E_{\ell}=L_{t}$,$t$$a\leq s\leq t$.

Suppose $a\leq s\leq t$ . It follows from Lemma 5.9 that
$E_{s-}-_{ts}-E= \int_{a}^{s}\alpha_{u}^{*}E_{s}L_{u}E_{s}du$By assumption we have $E_{s}L_{u}E_{S}=E_{u}L_{u}E_{u}=L_{u}$ for.

$\alpha\leq u\leq \mathit{8}$$E_{s-tS}--E= \int_{a}^{s}\alpha_{u}^{*}L_{u}du=---s$. Hence

.
qed

5.7

Quantum martingales

Definition 5.11 An adapted processand$\{_{-t}^{-}-\}$

is called a quantum martingale if it is admissible$E_{s-ts}--E=E_{s}---_{s}ES$$\mathit{8}\leq t$’.

(5.11)

This is a straightforward extension of Parthasarathy-Sinha’s one [27]. By definition theadjoint process of a quantum martingale is again a quantum martingale.
Theorem 5.12 Let$\{L_{t}\}$be an adapted$\sup$$admi_{\mathit{8}}sible$$\sup$

process such that$|||L_{t}\phi|||-r,-\beta<\infty$$a\leq t\leq b|||\phi|||r,\beta\leq 1$

for$\mathit{8}omer,\beta\geq 0$. Assume that the quantum Hitsuda-Skorokhod integral$–t-= \int_{a}^{t}\alpha_{S}^{*}L_{s}d\mathit{8}$$i\mathit{8}$admissible. Then$\{_{-t}^{-}-\}t\geq a$,$t\geq a$,

is a quantum martingale.

PROOF. Thatis adapted follows from Proposition 5.8. Condition (5.11) is
qedeasily verified with Lemma 5.9.
$\{_{-\ell}^{-}-\}t\geq a$

Proposition 5.13 The annihilation process$ber$ process
are quantum martingales.

$\{A_{t}\}$, the creation process$\{A_{t}^{*}\}$and the num-$\{\Lambda_{t}\}$is a quantum martingale; hence so is
PROOF. It follows from Theroem 5.12 that
. It has been already shown that the number process is an adapted admissble process.
qed
Condition (5.11) is checked easily.
$\{A_{t}\}$$\{A_{t}^{*}\}$

Finally we prove a quantum analogue of Proposition 3.6.

189

. If
be one-parameter family of slowly increasing
Proposition 5.14 Let$–t-=–l,m-(\kappa,)l(t)m$ is a quantum martingale, then there exists a-valued measurable function
such thaton$\{\kappa_{l,m}^{(t)}\}$$function\mathit{8}$$a$$\mathbb{C}$$\mathbb{R}^{l+m}$$–t=—-_{l},m(\chi_{t}\otimes(l+m).

\kappa_{l,m})$.

is adapted we see from Theorem 5.6 that
PROOF. Since
Now, in view of Lemma 4.9 we see that $E_{s^{-}}-E-_{ts}=E_{s-SS}--_{E}$ for$\{_{-t}^{-}-\}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\kappa_{l,m}^{(t)}\subset(-\infty, t]l+m$$\mathit{8}\leq t$.

if and only if$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int^{s}-\infty d\mathit{8}_{1}\cdots\int_{-\infty}^{\delta}d\mathit{8}\iota\int_{-\infty}^{s}dt_{1}\cdots\int^{S}-\infty dt_{m}\int sd+\infty u_{1}\cdots\int_{S}+\infty du_{n}\cross$$\kappa_{\iota^{t}}^{()},(m1,lt\mathit{8}\cdots, \mathit{8},1, \cdots, tm)a_{u1}^{*}\cdots a_{un}\alpha_{s_{1}}\cdots a_{sl}a_{t_{1}t_{m}}\ldots\alpha\alpha\cdots\alpha_{u_{n}}***u1$$= \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int^{s}-\infty dS_{1}\cdots\int_{-\infty}SldS\int_{-}^{s}\infty dt_{1}\cdots\int-\infty sdt_{m}\int_{s}^{+}\infty ud1\ldots\int^{+\infty}Su_{n}d\mathrm{x}$$\kappa_{\iota,m}^{(s})(_{\mathit{8}_{1,\cdots,\mathit{8}}}1, t_{1}, \cdots, t_{m})\alpha\alpha^{*}a_{s_{1}}\cdots\alpha_{s}**\ldots*a_{t_{m}}a_{t_{1}}\cdots a_{u_{1}}\cdots au_{1}un\iota u_{n}$.

Since the Fock expansion is unique, we obtain$\kappa_{l,m}^{(t})(\mathit{8}1, \cdots, \mathit{8}l, t_{1}, \cdots, t_{m})=\kappa_{l,m}^{(s})(s_{1}, \cdots, sl, t_{1}, \cdots, t_{m})$$t1,$
for 81,such that$\cdots,$$\mathit{8}\iota,$$\cdots,$$tm\leq \mathit{8}$$\mathbb{R}^{l+m}$. Therefore there exists a -valued measurable. function.
$\mathbb{C}$$\kappa_{l,m}^{(t)}$$\kappa_{l,m}$on$=\chi t\kappa_{l}\otimes(l+m).,m$’qed

which completes the proof.

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