数理解析研究所講究録982 巻 1997 年 196-208

196

Quantum Stochastic Process as

Continuous Flow of Fock Space OperatorsNOBUAKI OBATAGRADUATE SCHOOL OF POLYMATHEMATICSNAGOYA UNIVERSITYNAGOYA, 464-01 JAPANIntroduction
During the last decade quantum stochastic calculus on Fock space has developed into a
new field of mathematics with various applications to quantum physics by many authors,
see the excellent textbooks Meyer [9], Parthasarathy [16] and references cited therein. In
short, their discussion is in principle based on a quantum version of It\^o theory and a crucialrole has been played by three basic quantum stochastic processes
and,which are called respectively the annihilation process, the creation process and the number(gauge) process. In fact, quantum stochastic integrals of It\^o type are introduced by meansof Riemann-Lebesgue integrals against the infinitesimal increments
andjust asin the classical case where a stochastic integral of It\^o type is defined using the infinitesimalincrement of Brownian motion.
On the other hand, the (classical) It\^o theory has developed considerably together with
distribution theory on an infinite dimensional space. Among others Hida’s white noise calculus [2] allows to formulate the white noise as a smooth flow (with respect to the timeparameter) not as a generalized stochastic process in the sense of Gelfand and It\^o. To be
precise, let$\{A_{t}\},$$dA_{t},$$\{A_{t}^{*}\}$$dA_{t}^{*}$$\{\Lambda_{t}\}$$d\Lambda t$$dB_{t}$$(E)\subset L^{2}(E*, \mu)\subset(E)^{*}$

be the standard white noise triplet or-Takenaka space [6], where
Gaussian space associated with the Gelfand triple$\mathrm{H}\mathrm{i}\mathrm{d}\mathrm{a}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}$$E=S(\mathbb{R})\subset H=L2(\mathbb{R}, dt)\subset E^{*}=S’(\mathbb{R})$Then the Brownian motion, originally a continuous flow in
one inwith the derivative$\{B_{t}\}$.

$(E^{*}, \mu)$is the(0.1)$L^{2}(E^{*}, \mu)$, becomes a smooth$(E)^{*}$$W_{t}= \frac{d}{dt}B_{t}$or equivalently$dB_{t}=W_{t}dt$.

Then the smooth flow
is called the white noise process. It is the clue ofthe white noise approach to (classical) stochastic analysis (see e.g., [2], [3], [6], [7]) that the$t\vdash\Rightarrow W_{t}\in(E)^{*}$

197

infinitesimal increment of Brownian motion is replaced with the smooth flow of white noise
times the time increment .
the Boson
Through the celebrated Wiener-It\^o-Segal isomorphism between
we can discuss a quantum analogue of white noise calculus, that is, white
Fock space overandnoise approach to quantum stochastic calculus. In fact, the basic processes
are smooth flows of operators on white noise functions and we have$\delta$$dt$$L^{2}(E^{*}, \mu)\mathrm{a}\mathrm{n}\dot{\mathrm{d}}$$H_{\mathbb{C}}$$\{A_{t}\},$$\{A_{t}^{*}\}$$\{\Lambda_{t}\}$$dA_{t}=a_{t}dt$,$dA_{t}^{*}=a_{t}^{*}dt$,$d\Lambda_{t}=a_{t}^{*}a_{t}dt$,

where and are the pointwisely defined annihilation and creation operators, respectively.
by Hudson-Parthasarathy
Accordingly, quantum stochastic integrals of It\^o typeanalogy of the classicalonthetheorydistribution
noise
white
of
[5] can be discussed in termscase. This new approach was launched out by Huang [4] and Obata [11], [12], [13], [14], [15]with technical background [10].
The present short note is intended as a supplement to the above mentioned papers. We
re in detail and discuss
examine the canonical expression of a quantum stochastic process
the quantum It\^o formula. It is our hope that Hida’s idea of white noise works well also forquantum stochastic calculus.
$a_{t}^{*}$$a_{t}$$\dot{\mathrm{i}}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}$$\mathrm{m}o$be locally convex spaces.
General Notation Let $x,$
: the complexification of when it is a real space.
into ; equipped with the: the space of continuous linear operators fromtopology of bounded convergence.
with the strong dual
the space of continuous linear functionals on.
topology. This is a particular case of: the Hilbert space tensor product when both $x,$ are Hilbert spaces.
is: the completed -tensor product. When there is no danger of confusion,
simplicity.
denoted$\mathfrak{Y},$

$3$

$X$

$X_{\mathbb{C}}$$X$$\mathcal{L}(x, \mathfrak{Y})$$\mathfrak{Y}$$X;\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}_{\mathrm{P}\mathrm{p}\mathrm{e}\dot{\mathrm{d}}}$$X^{*}:$$\mathcal{L}(X, \mathfrak{Y})$$\mathfrak{Y}$$X\otimes \mathfrak{Y}$$\otimes_{\pi}$$\pi$$X\otimes_{\pi}\mathfrak{Y}$$\mathrm{b}\mathrm{y}\otimes \mathrm{f}\mathrm{o}\mathrm{r}$1Standard Triplet of White Noise Functions

We adopt the standard notation following [10]. The Gelfand triple (0.1) is constructed fromthe differential operator$A=1+t^{2}- \frac{d^{2}}{dt^{2}}$-domain of equipped with the topology
is thein the standard manner. Namely,. The constant numbers
defined by the norms $|\xi|_{p}=|A^{p}\xi|_{H},$
$E$

$A$

$C^{\infty}$$p\in \mathbb{R}$’$\delta$$\rho$$=$

Hilbert-Schmidt norm of$=$operator norm of$A^{-1}<\infty$,$A^{-1}=( \inf \mathrm{S}_{\mathrm{P}^{\mathrm{e}}}\mathrm{c}(A))^{-1}=\frac{1}{2’}$and the obvious inequalities$0<\rho<1$ ;$|\xi|_{p}\leq\rho^{q}|\xi|_{pq}+$’$\xi\in E$,$p\in \mathbb{R}$,$q\geq 0$

,

(1.1)

198

are used throughout with no special notice. The probability space, where is thestandard Gaussian measure, is called the Gaussian space. For
we define an exponential vector by$(E^{*}, \mu)$$\mu$$\xi\in E_{\mathbb{C}}$$\phi_{\xi}(X)=\exp(\langle x, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)$,$x\in E^{*}$.

In particular,
is called the vacuum.
Letbe the Boson Fock space over
, the space of -valued -functions on .
An element in
is a sequencesuch that
and.
There is a unitary isomorphism between
anduniquely determined by thecorrespondence. This is the celebrated Wiener-It\^o-Segal isomorphism. For ageneralwe write the correspondence as. In fact, we can reconstruct afunction on
fromby means of Hermite polynomials, or more precisely, renormalized
tensor products :, for full details see [3] or [10].
The second quantized operator of is denoted by. For
its action is givenby. It is known thatis a positive selfadjoint operator on
ith Hilbert-Schmidt inverse. We thereby obtain a complex Gelfand triple:$\phi_{0}$$\Gamma(H_{\mathbb{C}})$$\Gamma(H_{\mathbb{C}})$$L^{2}$$\mathbb{C}$$H_{\mathbb{C}}$$f_{n}\in H_{\mathbb{C}}^{\otimes n}\wedge$$(f_{n})_{n=0}^{\infty}$$L^{2}(E^{*}, \mu)$$\mathbb{R}$$\sum_{n=0}^{\infty}n!|f_{n}|_{0}^{2}<\infty$$\Gamma(H_{\mathbb{C}})$$\phi_{\zeta}rightarrow(\xi^{\otimes n}/n!)$$\phi\in L^{2}(E^{*}, \mu)$$E^{*}$$\phi\sim(f_{n})$$(f_{n})$$x^{\otimes n}:$$A$$\Gamma(A)\phi\sim(A^{\otimes n}f_{n})$$\phi\sim(f_{n})$$\Gamma(A)$$\Gamma(A)$$L^{2}(E^{*}, \mu)$$.\mathrm{w}$$(E)\subset L^{2}(E*, \mu)\cong\Gamma(H_{\mathbb{C}})\subset(E)^{*}$,

which is called the standard white noise triplet or the $Hida-Kubo$ -Takenaka space [6]. Elements in $(E)$ andare called a test (white noise) function and a generalized (white noise)function, respectively. We denote by
the canonical bilinear form on $(E)^{*}\cross(E)$ . For
the defining normsof $(E)$ we have$(E)^{*}$$\langle\langle\cdot, \cdot\rangle\rangle$$||\cdot||_{p}$$|| \phi||_{p}^{2}=||\Gamma(A)^{p}\phi||_{0}^{2}=\sum_{=n0}^{\infty}n!|(A^{\otimes n})^{p}f_{n}|_{0}^{2}=\sum_{=n0}^{\infty}n!|f_{n}|_{p}^{2}$

where. Obviously, $\phi\in(E)$ if and only if
for all $p\geq 0$ . If
andunique $\Phi\in(E)^{*}$ such that$\phi\sim(f_{n})$$F_{n}\in(E_{\mathrm{c}}^{\otimes n})^{*}\mathrm{s}\mathrm{y}\mathrm{m}$$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$,$p\in \mathbb{R}$

,

(1.2)

for all andfor some $p\geq 0$ , then there exists a$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$$\sum_{n=0}^{\infty}n!|F_{n}|_{-p}^{2}<\infty$,$\sum_{n=0}^{\infty}n!|f_{n}|_{p}^{2}<\infty$$n$$\phi\sim(f_{n})\in(E)$.

Conversely, every $\Phi\in(E)^{*}$ is of this type. In that case we write $\Phi\sim(F_{n})$ and call it the
Wiener-It\^o expansion of .
For eachthere exists a unique operator $D_{y}\in \mathcal{L}((E), (E))$ such that$\Phi$$y\in E_{\mathbb{C}}^{*}$$D_{y}\phi_{\xi}=\langle y, \xi\rangle\phi_{\xi}$,$\xi\in E_{\mathbb{C}}$.

This is called the annihilation operator. In particular,$a_{t}=D_{\delta_{t}}$,is called the annihilation operator at a point
literature is used for ) The adjoint$\partial_{t}$$a_{t}.$$t\in \mathbb{R}$$t$,or Hidato$a_{t}^{*}\mathrm{b}\mathrm{e}1_{0}\mathrm{n}\mathrm{g}\mathrm{s}$$\mathrm{Z}s$differential operator.

$\mathcal{L}((E)^{*}, (E)^{*})$(In some
and is called the creation

199

operator at a point . These operators are not operator-valued distributions but continuousand }operators for themselves. Moreover, bothare smooth flows.
such that
there exists a unique
For each$t$$a_{t}\in \mathcal{L}((E), (E))$$t-\rangle$$\kappa\in(E_{\mathbb{C}}^{\otimes(+}\iota m))^{*}$$t$$arrow a_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\Gamma--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
$\eta_{\emptyset,\psi}(s_{1}, \cdots, Sl, t1, \cdots, t)m=\langle\langle a_{S_{1}}^{*}\cdots aa_{t_{1}t}\ldots a\phi m’\psi\rangle s\iota\rangle*$.

We use a formal (but descriptive) integral expression:$.-_{l,m}-( \kappa)=\int_{\mathbb{R}^{l+m}}\kappa(s1, \cdots , s_{l}, t_{1,m}\ldots, t)a\cdots a_{sl}a_{t_{1}}\cdots at_{m}d_{S_{1}\cdot d}S_{1}**.,s_{l1}dt\cdots dt_{m}$

.

(1.3)

an integral kernel operator with kernel distribution , see [10]. It is known
We callthat is uniquely determined whenever it is taken from the subspace$.-_{l,m}-(\kappa)$$\kappa$$\kappa$$s_{l,m}(\kappa)=\kappa\}$$(E_{\mathrm{c}}^{\otimes(})l+m)*\mathrm{s}\mathrm{y}\mathrm{m}(l,m)=\{\kappa\in(E_{\mathbb{C}}^{\otimes(m)*})+;l$,is the symmetrizing operator with respect to the first and the last $m$ variables
where, hence every
independently. It is known [10, Chapter 4] that every operator ina strongly
admits
as
well
space
Fock
Boson
bounded operator on theconvergent infinite series expansion in terms of integral kernel operators (Fock expansion).
$l$$s_{l,m}$$\mathcal{L}((E), (E)^{*})$$\Gamma(H_{\mathbb{C}})\cong L^{2}(E^{*}, \mu)$2

Brownian Motion and White Noise Process

Consider two white noise functions:$B_{t}\sim(0.’It is easy to see that

1_{[0},t],$$\mathrm{o},$$\cdots)$,$t\geq 0$,$\mathrm{E}(B_{t})=0$$W_{t}\sim..(0, \delta_{t}, 0, \cdots)$,$t\in \mathbb{R}$.

forms a Gaussian family satisfying$\{B_{t}\}\subset L^{2}(E^{*}, \mu)$$B_{0}=0$;,$\mathrm{E}(B_{s}B_{t})=\min\{s, t\}$,$\mathit{8},$$t\geq 0$,$t\geq 0$ .
is a Brownian motion. If necessary we put
namely,It is easily verified that $t\mapsto B_{t}\in L^{2}(E^{*}, \mu)$ is continuous but not differentiable; however, it. This is due to the fact that
becomes differentiable if regarded as a flow insummarize,-flow with derivative . To
is a$B_{-t}\sim(\mathrm{o}, -1_{1-t,0}],$ $0,$ $\cdots),$$\{B_{t}\}_{t\geq 0}$$(E)^{*}$$C^{\infty}$$t\mapsto 1_{[0,t]}\in E_{\mathbb{C}}^{*}$$\delta_{t}$Proposition 2.1 The map$trightarrow B_{t}\in(E)^{*}$

is a$\frac{d}{dt}B_{t}=W_{t}$with respect to the strong dual topology of,$(E)^{*}$$C^{\infty}$-flow and it holds that$t\in \mathbb{R}$,. Hence $trightarrow W_{t}\in(E)^{*}$ is also a$C^{\infty}$-flow.

-flow $\{W_{t}\}\subset(E)^{*}$ is called the white noise process. In other words, the whiteThe
noise is formulated as the time derivative of Brownian motion.
Remark There is another justification of white noise in terms of a generalized stochastic
called a
be a probabili.ty space. A map $X$ :process. In general, letifprocess.
generalized$C^{\infty}$$(\Omega, P)$$\mathit{8}tochaStiC$$\Omega\cross S(\mathbb{R})arrow \mathbb{R}\mathrm{i}.\mathrm{s}$200(i) for a fixed, a map
is a random variable;(ii) for a fixed, a map
is a continuous linear functional on.
This concept was introduced by Gelfand and by It\^o around 1955. Then $W$ :defined by $W(x, \xi)=\langle x, \xi\rangle$ satisfies the above properties, and hence is a generalized
stochastic process. This $W$ is also referred to as the white noise. Formal writing$\omegaarrow X(\omega, \xi)$$\xi\in S(\mathbb{R})$$\omega\in\Omega$$\xiarrow X(\omega, \xi)$$S(\mathbb{R})$$E^{*}\cross S(\mathbb{R})arrow$$\mathbb{R}$$W(x, \xi)=\langle x, \xi\rangle=I_{\mathbb{R}}^{\xi(}t)X(t)dt=\int_{\mathbb{R}}\xi(t)W_{t}(x)dt$

suggests the situation very well. The white noise
has no meaning (as a usual randomvariable) at each time point but is considered as a generalized stochastic process, i.e., a
distribution in . In ther words, the white noise receives a rigorous meaning only aftersmearing the time parameter and hence it is no longer a flow (time-parametrized stochastic
process). In contrast, once the spaceof white noise distributions is introduced, the$W_{t}(x)$white noise
is a distribution in and forms a-flow in.
$\{W_{t}\}$

$t$

$t$

$.0$

$(E)^{*}$

3$(E)^{*}$$C^{\infty}$$x$

Quantum Stochastic Processes

Definition ([12]) A family ofis called a quantum stochastic
process (on Fock space) if
is continuous, where runs over an interval of . A
continuous linear map:
is called a generalized quantum
process. A generalized quantum stochastic process
called regular if it admits a continuous
extenstion from
into $c((E), (E)^{*})$ . The extension will be denoted by the same symbol.
$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}--\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\{_{-t}^{-}-\}\subset \mathcal{L}((E), (E)^{*})$$t\mapsto\cup t$

$—$

$t$

$\mathbb{R}$$stocha\mathit{8}ti_{C}$$E_{\mathbb{C}}arrow \mathcal{L}((E), (E)^{*})$$—\mathrm{i}\mathrm{s}$$E_{\mathbb{C}}^{*}$

The continuity condition is not very restrictive as is seen below; also recall that the white
noise process is a-flow within our formulation. If a generalized quantum stochastic
regular,is a quantum stochastic process. However, not every
quantum stochastic process is of this form. From now on a regular generalized quantum
stochastic process is also called a regular quantum stochastic process.
It is known that bothandare regular quantum stochastic processes. In fact,and, both of whish are-flows. The integral
kernel operators:$At=—0,1(11^{0},t])$ ,,(3.1)$C^{\infty}$$\{_{-t}^{-}-=---(\delta_{t})\}t\in \mathbb{R}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}---\mathrm{i}\mathrm{s}$$\{a_{t}\}$$t\vdasharrow a_{t}\in \mathcal{L}((E), (E))$$\{a_{t}^{*}\}$$t\mapsto a_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$C^{\infty}$$A_{t}^{*}=---1,\mathrm{o}(1_{[}0,t])$form quantum stochastic processes. These are called the annihilation and creation processes,
respectively.
Using the natural inclusion:by multiplication, we regard a continuous flow
as a quantum stochastic process. In that sense the white noise
processand the Brownian motion
give rise to the quantum white noise and thequantum Brownian motion, respectively. It is known that$(E)^{*}\llcorner_{arrow}\mathcal{L}((E), (E)^{*})$$t\mapsto\Phi_{t}\in(E)^{*}$$\{W_{t}\}$$\{B_{t}\}$$W_{t}=a_{t}+a_{t}^{*}$,$t\in \mathbb{R}$;$B_{t}=A_{t}+A_{t}^{*}$,$t\geq 0$.

Obviously, the quantum white noise is regular.
There are many examples arising from quantum stochastic differential equations, see e.g.,[5], [9], [16]. A systemtic study of such examples from our viewpoint would be interesting,see also [13].

201

4

Integral Kernel Expansion of a Quantum Stochastic Process

Recall first the canonical isomorphism:$\mathcal{L}((E), (E)^{*})\cong((E)\otimes(E))^{*}\cong \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty}((E)\otimes(E))_{-P}$,

where $((E)\otimes(E))_{-p}$ is the Hilbert space obtained by completing $((E)\otimes(E))$ with respectdenote the subspace of all $—\in$. Then, for $p\geq 0$ let
to the norm
which correspond to elements in $((E)\otimes(E))_{-p}$ . In particular,. The topology ofis identified with the space of all Hilbert-Schmidt operators on$((E)\otimes(E))_{-P}$
which is denoted also by
is naturally induced from the norm of. With this notation,$\mathcal{L}_{p}((E), (E)^{*})$$||\cdot||_{-p}$$\mathcal{L}_{0}((E), (E)^{*})$$\mathcal{L}((E), (E)^{*})$$L^{2}(E^{*}, \mu)$$\mathcal{L}_{p}((E), (E)^{*})$$||\cdot||_{-\mathrm{P}}$$||_{-}^{-}-\phi||_{-P}\leq||_{-}^{-}-||-p||\phi||p$’$\phi\in(E)$

.

(4.1)

It is noted thatas vector$\mathcal{L}((E), (E)^{*})\cong \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty}\mathcal{L}_{p}((E), (E)^{*})(=\cup \mathcal{L}_{P}((p\geq 0E), (E)^{*})$On the other hand, it is also known that$L^{2}(E^{*}, \mu)$$\mathcal{L}((E), (E)^{*})$$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}\mathrm{s})$.

contains all bounded operator on.

be the expansion $of—\in \mathcal{L}_{p}((E), (E)^{*}),$ $p\geq 0$ , inLemma 4.1
terms of integral kernel operators. Then
$Let—=\Sigma_{l,m=}^{\infty}0^{-}--_{\iota},m(\kappa_{l,m})$$|\kappa_{t},m|-(p+1)\leq Gl,m;p||_{-}^{-}-||_{-p}$

(4.2)

,

where

(4.3)

$G_{l,m;p}=(l^{l}m^{m})-1/2(e^{3}\delta^{2}(1+\rho)2P)^{(m)}\iota+/2$

PROOF. Since by definition$—\wedge(\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\}=\langle\langle^{-}--, \phi_{\zeta}\otimes\phi_{\eta}\rangle\rangle$,$\xi,\eta\in E_{\mathbb{C}}$,we have$|_{-}^{-}- \wedge(\xi, \eta)|\leq||_{-}^{-}-||-\mathrm{P}||\phi\epsilon^{\otimes}\phi\eta||\leq p||_{-}^{-}-||-_{\mathrm{P}}\exp\frac{1}{2}(|\xi|_{p}^{2}+|\eta|_{\mathrm{p}}2)$.

The assertion then follows by applying the result in [10, Theorem 4.4.6].
We now prepare a general lemma. Let be a countable Hilbert space overthere exists a sequence of Hilbert spaces$X$

qed$\mathbb{R}$or$\cdots\subset H_{2}\subset H_{1}\subset H_{0}\subset H_{-1}\subset H_{-2}\subset\cdots$$X \cong \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim Hparrow\infty p$’$\mathfrak{X}^{*}\cong \mathrm{i}\mathrm{n}\mathrm{d}\lim_{\infty parrow}H_{-P}$$\mathbb{C}$. Thensuch that.

is a nuclear space, we may assume without loss of generality that the natural injection.
the norm of$H_{p+1}arrow H_{p}$ is of Hilbert-Schmidt type for any $p\geq 0$ . We denote by

If

$X$

$|\cdot|_{p}$$H_{p}$

202

Lemma 4.2 Let be a countable Hilbert nuclear space andthe same as above. Let
be a locally compact space. Then for a map :the following two conditions areequivalent:(i) is continuous;(ii) for each
there exists $p\geq 0$ such that
and$X$$\{H_{p}\}$$\Omega$$f$$\Omegaarrow X^{*}$$f$$\omega_{0}\in\Omega$$f(\omega_{0})\in H_{-P}$$\omegaarrow")\lim_{0}|f(\omega)-f(\omega 0)|_{-p}=0$

In that case for any compact subset
continuous.

$\Omega_{0}\subset\Omega$

there

.

$exi_{S}t\mathit{8}p\geq 0$such that$f$:$\Omega_{0}arrow H_{-p}$is

PROOF.
Let
be an open neighborhood ofwith compact closure.
Since is cotinuous,is compact and hence bounded. Thenisbounded for some . In other words, there exists $M\geq 0$ such that$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$V\subset\Omega$$\omega_{0}$$f(\overline{V})\subset X^{*}$$f$$f(\overline{V})\subset H_{-p}$$p$$|f(\omega)|_{-p}\leq M$

Let$\{e_{j}\}_{j=1}^{\infty}$.

,

be a complete orthonormal basis of$\omega\in V$$H_{p+1}$. Then by definition,$|f( \omega)-f(\omega_{0})|_{-}^{2}(P+1)=j1\sum_{=}^{\infty}\langle f(\omega)-f(\omega_{0}), e_{j}\rangle^{2}$

We note that$\langle f(\omega)-f(\omega 0), e_{j}\rangle^{2}\leq|f(\omega)-f(\omega_{0})|^{2}-p|ej|p2\leq 4M2|ej|_{p}^{2}$Given$\epsilon>0$,.

$\omega\in V$

we choose $N$ such that
$4M^{2} \sum_{Nj>}|e_{j}|_{P}2<\frac{\epsilon}{2}$which is possible since
is of Hilbert-Schmidt type and hence
On the other hand,is continuous by assumption. Then for each $j=1,$
one may find an open neighborhoodofsuch that$H_{\mathrm{P}+1}arrow H_{p}$$\sum_{j=1}^{\infty}|e_{j}|_{p}^{2}<\infty$$\omega\mapsto\langle f(\omega), e_{j}\rangle$$U_{j}\subset\Omega$$\omega_{0}$$|\langle f(\omega), e_{j}\rangle-\langle f(\omega_{0}), e_{j}\rangle|<\sqrt{\frac{\epsilon}{2N}}$,$\omega\in U_{j}$.

Put $U=V\cap U_{1}\cap\cdots\cap U_{N}$ . Then$|f(\omega)-f(\omega_{0})|^{2}-(p+1)$

This is the assertion of (ii).

$=$$\sum_{i=1}^{N}\langle f(\omega)-f(\omega_{0}), e_{j}\rangle^{2}+\sum\langle j>Nf(\omega)-f(\omega_{0}), e_{j}\rangle^{2}$$\leq$$\sum_{j=1}^{N}\frac{\epsilon}{2N}+4M^{2}\sum_{>jN}|e_{j}|_{p}^{2}$$<$$N \cross\frac{\epsilon}{2N}+\frac{\epsilon}{2}=\epsilon$,.

$\omega\in U$.

$N$$\cdots,$

203$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$The topology of$X^{*}$

is defined by the seminorms$||f||_{B}= \sup_{\omega\in B}|\langle f, \omega\rangle|$

where $B$ runs over the bounded subsets of$||f(\omega)-f(\omega 0)||_{B}$$\leq$$X$,$f\in \mathrm{X}^{*}$. Then for any$f$$–t-=\Sigma^{\infty}l,m=0^{-}--l,m(\kappa\iota_{m},(t))$

we have$|B|_{p}|f(\omega)-f(\omega_{0})|-parrow 0$

by assumption, which shows that is continuous at
The rest of the statement is already clear.
$\{_{-t}^{-}-\}$$B$$\sup_{\omega\in B}|f(\omega)-f(\omega 0)|_{-}p|\omega|_{p}$$=$Theorem 4.3 Let,

be a quantum. Then$\mathit{8}tochaStiC$$\omega_{0}$,$\omegaarrow\omega_{0}$

,

.
qed

process wit the integral kernel expansion:is continuous.

$t\mapsto\kappa_{l,m}(t)\in(E_{\mathbb{C}}^{\otimes(m)}\iota+)_{\mathrm{s}}^{*}\mathrm{y}\mathrm{m}(\iota,m)$

We shall prove the continuity at a fixed . Since$((E)\otimes(E))^{*}$ is continuous, there exists $p\geq 0$ such that
PROOF.

$t$$s\vdasharrow--s-\in \mathcal{L}((E), (E)^{*})\cong$$||_{-t}^{-}-||_{-p}<\infty$

and$\lim_{sarrow t}||^{-}--_{S}---_{t}-||-p=0$

by Lemma 4.2. On the other hand, by Lemma 4.1 we have$|\kappa_{l,m}(S)-\kappa_{l,m}(t)|-(p+1)\leq G_{l,m;p}||--_{S}-----|t|-p$ ’and therefore$\lim_{Sarrow t}|\kappa_{l,m}(\mathit{8})-\kappa l,m(t)|-(p+1)=0$

Again from Lemma 4.2 we see that$t\mapsto\kappa_{l,m}(t)$.
qedin continuous.

Thus a question about a quantum stochastic process can be discussed in terms of continin [12]uous flows in the space of kernel distributions. A similar result as above wasunder a superfluous assumption which is now eliminated.
$\mathrm{p}.\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d}$5Generalized Integral Kernel Operators and Quantum Stochastic Integrals

In [13] we introduced an operator of which formal expression is giveIi as$\int_{\mathbb{R}^{l+m}}a_{S_{1}}^{*}\cdots aL(S_{1,l,1,m}\ldots, st\cdots, t)s\iota a*t_{1}\ldots a_{tm}ds_{1}\cdots ds_{f}dt1\ldots dtm$

where
space

$L$

is an

$\mathcal{L}((E), (E)^{*})$-valued distribution on$\mathbb{R}^{l+m}$, more precisely,

$L$

’

(5.1)

is taken from the.

$\mathcal{L}(l(E), (E)^{*}))\cong(E_{\mathbb{C}}\otimes(l+m))*c(\otimes(E), (E)*)$$\mathcal{L}(E_{\mathrm{c}^{(}}^{\otimes}m+),$Such an operator is called a (generalized) integral kernel operator. In fact, by the characterization theorem of operator symbols [10, Chapter 4.4] an operator $—\in \mathcal{L}((E), (E)^{*})$ is
uniquely determined by$\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle\langle L(\eta\otimes\otimes \mathrm{t}\xi\otimes m)\phi\epsilon,$$\phi\eta\rangle\rangle$,$\xi,$$\eta\in E_{\mathbb{C}}$

.

(5.2)

204

Then (5.1) is a formal integral expression. An integral kernel
as in(1.3) is obtained from (5.1) by taking to be a-valued distribution.
Letbe a quantum stochastic process, being, and fix $a\in J$ as a time
origin. Then for any $t\in J$ there exists a uniquesuch that$\mathrm{f}\mathrm{o}\mathrm{r}---$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}--l,m-(\kappa)$$L$$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}- \mathrm{o}_{\mathrm{P}}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$$J$$\{L_{t}\}_{t\in J}$$\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{l}--$$\mathrm{o}_{\mathrm{P}^{\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}}-t}\mathrm{o}\mathrm{r}\in \mathcal{L}((E), (E)^{*})$$\langle\langle_{-t}^{-}-\phi, \psi\rangle\rangle=\int_{a}^{t}\langle\langle L_{S}\phi, \psi\rangle\rangle dS$,$\psi\in(E)$$\phi,$.

(5.3)

We write$—t= \int_{a}^{t}L_{s}ds$

It is proved by astand.ard argument that$\{_{-t}^{-}-\}_{tJ}\in$

.

(5.4)

is a quantum stochastic process.

Theorem 5.1 If two quantum stochastic processes
andthenis differentiable with respect to the topology of$\{L_{t}\}$$t-\rangle$$\{_{-t}^{-}-\}$

are related as in (5.4),
and itthat$\mathcal{L}((E), (E)^{*})$$\cup t$$\frac{d}{dt}---_{t}=L_{t}$Corollary 5.2 For the annihilation process(3.1) it holds that
$A_{t}= \int_{0}^{t}a_{s}ds$,$\frac{d}{dt}A_{t}=a_{t}$,$\{A_{t}\}$and$hold_{\mathit{8}}$.
and the creation process$A_{t}^{*}= \int_{0}^{t}a_{s}^{*}ds$,$\{A_{t}^{*}\}$$\frac{d}{dt}A_{t}^{*}=a_{t}^{*}$

defined in.

If
is a quantum stochastic process, so areand. The proof being
given in [12], we only note that the bilinear map $\mathcal{L}((E), (E)^{*})\chi \mathcal{L}((E), (E))arrow \mathcal{L}((E), (E)^{*})$
defined by composition is separately continuous but is not (jointly) continuous. We are now
in a position to introduce quantum stochastic integrals in terms of white noise distribution
theory.
$\{L_{t}\}$$\{L_{t}a_{t}\}$Definition Let
processes defined as$\{L_{t}\}$$\{a_{t}^{*}L_{t}\}$be a quantum stochastic process. Then the quantum stochastic$\int_{a}^{t}L_{s}a_{s}dS$

,

(5.5)

$\int_{a}^{t}a_{s}^{*}Ldss$

are called the quantum stochastic integral ofagainst the annihilation and creation processes, respectively. The latter is also called a quantum Hitsuda-Skorokhod integral.
$L_{t}$

Obviously the quantum stochastic integrals in (5.5) are dual to each other. The number
process
is defined as$\{\Lambda_{t}\}$$\Lambda_{t}=\int_{0}^{t}a_{s}^{*}a_{S}d_{S}$,$\frac{d}{dt}\Lambda_{t}=a_{t}^{*}a_{t}$

.

(5.6)

This is the quantum stochastic integral ofagainst the annihilation process as well as
the quantum stochastic integral ofagainst the creation process.
There are slightly different definitions of a quantum Hitsuda-Skorokhod integral, see
Lindsay [8] where the gradient operator plays a role; see also [13] where the integral isdefined only for a regular quantum stochastic process.
$\{a_{t}^{*}\}$$\{a_{t}\}$$\{L_{t}\}$

205

6

Stochastic Integral Representation of a Quantum Stochastic Process
we devide its integral kernel expansion into three partial sums:$\mathrm{G}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}---\in \mathcal{L}((E), (E)^{*})$

(6.1)

.

$–=- \sum_{\geq\iota\geq 0,m1}--l-,(m\kappa\iota_{m},)\dagger\sum_{\iota\geq 1}---_{\iota,0}(\kappa l,0)+---_{0,0}(\kappa_{0,0})’$. Note that each integral kernel
The third term is a scalar operator:
operator in the first term of (6.1) contains at least one annihilation operator, and that eachin the second term consists only creation operators. Then one may expect an expression of
the form:$–0,0-(\kappa 0,0)=\langle\langle_{-}^{-}-\phi 0, \phi_{0}\rangle\rangle I$$\iota\geq 0,\sum_{m\geq 1}--l,m(\kappa-l,m)=\int_{\mathbb{R}}L(t)atdt$,$\sum_{l\geq 1}--l-,(0\kappa_{l},0)=\int_{\mathbb{R}}a_{s}*M^{*}(s)dS$

.

(6.2)

a
In fact, by iterated integration (guaranteed by the Fubini type theorem, see [13]) and byroutine examination of convergence by means of norm estimates we obtain the following

Theorem 6.1 $Let—\in \mathcal{L}_{p}((E), (E)^{*})_{J}p\geq 0$ . Then it admits an$–=- \int_{\mathbb{R}}L(t)atdt+\int_{\mathbb{R}}a_{s}^{*}M^{*}(s)d_{\mathit{8}}+cI$

where (i)$L\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)^{*}))$$||L(\xi)\phi||-\langle p+q+1$

)

(ii)

(6.3)

,

satisfies$\leq C_{1}||_{-}^{-}-||-p|\xi|_{p}+q+1||\phi||p+q+1$

for $q>q_{1}=q_{1}(p)\geq 0$ andof the form:$expres\mathit{8}i_{on}$,$\xi\in E_{\mathbb{C}}$

,

(6.4)

,

$\phi\in(E)$$C_{1}=C_{1}(p, q)\geq 0_{i}$$M\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)))$satisfies $[M(\xi), a_{t}]=0$ for all$||M(\xi)\phi||p|\leq c_{2}|_{-}^{-}-||-p|\xi|_{p}+q+1||\phi||p+q\dagger 1$ ’$\xi\in E_{\mathbb{C}}$$\xi\in E_{\mathbb{C}}$,and$t\in \mathbb{R}$$\phi\in(E)$

and

(6.5)

,

for $q>q_{2}=q_{2}(p)\geq 0$ and $C_{2}=C_{2}(p, q)\geq 0$ ;(iii)

$c\in \mathbb{C}$is given by$c=\langle\langle_{-}^{-}-\phi_{0}, \phi 0\rangle\rangle$.

With the help of Theorem 6.1 one can derive stochastic integral representation (in abroad sense) of a quantum stochastic process.
Theorem 6.2 Let$\{_{-t}^{-}-\}$

be a quantum stochastic process. Then there exist continuous maps
such that
and$trightarrow L_{t}\in \mathcal{L}(E\mathrm{c}, \mathcal{L}((E), (E)^{*})),$$t\mapsto M_{t}\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)))$$—t= \int_{\mathbb{R}}L_{t}(s)a_{S}dS+\int_{\mathbb{R}}a_{s}^{*}M_{t}^{*}(S)dS+c_{t}I$PROOF. The topology of$\mathcal{L}(E_{\mathbb{C}}, \mathcal{L}(\langle E), (E)^{*}))$$t[]arrow c_{t}\in \mathbb{C}$.

is defined by the seminorms$||L||_{K,B_{1},B_{2}}= \sup\{|\langle\langle L(\xi)\phi, \psi\rangle\rangle|;\xi\in K, \phi\in B_{1},\psi\in B2\}$

where$K\subset E_{\mathbb{C}},$$B_{1},$$B_{2}\subset(E)$,are bounded subsets. Then by (6.4) we obtain$||L_{s}-L_{t}||_{K,B_{1},B_{2}}\leq C_{1}||_{-s}^{--}----t||_{-p}|K|_{\mathrm{P}+q+1}||B_{1}||_{\mathrm{P}+q+1}||B_{2}||_{p+q+1}$

.

(6.6)

206

For a fixed$t$there exists $p\geq 0$ such thatand$||_{-t}^{-}-||_{-p}<\infty$$\lim_{sarrow t}||^{-}-_{S}----_{t}-||-p=0$,

see Lemma 4.2. Then from (6.6) we see thatis continuous at . Thatiscontinuous is proved similarly. The continuity ofis obvious.
qed
A similar result was proved in [12] under a superfluous condition which is now eliminated.
There are similar results for a generalized quantum stochastic process and a regular quantum
stochastic process, see [12].
$s\mapsto L_{s}$$t$$\mathrm{t}\mapsto M_{t}$$trightarrow c_{t}=\langle\langle_{-t}^{-}-\phi_{0}, \phi_{0}\rangle\rangle$

7

Admissible White Noise Distributions and Quantum It\^o Formula

The quantum It\^o formula is related to the rule of composition of two quantum stochastic
integrals. The original discussion due to Hudson-Parthasarathy [5] was restricted to adapted
processes and the recent work of Belavkin [1] discusses without assuming adaptedness. A
white noise approach was discussed by Huang [4] at somehow formal level and needs more
careful study. As the first step we here discuss the simplest but an instructive case ofand. Before going into the discussion we note first the following$\{A_{t}\}$$\{A_{t}^{*}\}$

Proposition 7.1 Let$\{L_{t}\}$and$\{M_{t}\}$be two quantum stochastic processes and put$—t= \int_{a}^{t}L_{s}ds$

If,$\Omega_{t}=\int_{a}^{t}M_{S}ds$.

is continuous, or if $t\mapsto L_{t}\in \mathcal{L}((E)^{*}, (E)^{*})$ is continuou..s, thenis a quantum stochastic
and it holds that$trightarrow M_{t}\in \mathcal{L}((E), (E))$$\{_{-t}^{-}-\Omega t\}\subset \mathcal{L}((E), (E)^{*})$$proces\mathit{8}$$d(_{-}^{-}-_{t}\Omega t)=d_{-}^{-}-t$.

.

$d\Omega$$\Omega t+--_{tt}-$,

or equivalently$—t \Omega t=\int_{a}^{t}L_{s}\Omega_{s}ds+\int_{a}^{t}---_{s}M_{s}d_{S}$.

The proof is obvious. Modelled after the discussion in [12, \S 3.3] we can prove that$\int_{a}^{t}---ss\lim\sum Mds=-\Delta--_{s_{i}}(\Omega Si+1-\Omega_{s_{i}})\equiv\int_{a}^{t}---Sd\Omega s$$\int_{a}^{t}L\Omega d_{S}=\lim\sum SS(_{-+1}^{-}-si-\Delta---_{s:})\Omega\equiv S:\int_{a}^{t}d_{-\Omega}^{-}-_{s}s$

.

’

(7.1)
(7.2)

The It\^o type Riemmanian approximations in (7.1) and (7.2) are not essential just for the; but play an important role when we consider adapted processes. Namely, if$[_{-t}^{-}-+h-\cup t, \Omega t]=0$ whenever $h>0$ , then (7.2) becomes$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}--\mathrm{C}\mathrm{e}$$\int_{a}^{t}L_{S}\Omega_{S}d_{S=}\lim_{\Delta}\sum\Omega_{s}i(----s_{i+1}---_{s}i)\equiv\int_{a}^{t}\Omega_{s}d_{-}-_{S}-$

.

207

As a direct consequence of Proposition 7.1 we have
(7.3)
(7.4)
(7.5)

,

$d(A_{t}A_{t})$$=$$dA_{t}\cdot A_{t}+A_{t}\cdot dA_{t}=2A_{t}a_{t}dt$$d(A_{t}^{*}A_{t})$$=$$dA_{t}^{*}\cdot A_{t}+A_{t}^{*}\cdot dA_{t}=a_{t}^{*}A_{t}dt+A_{t}^{*}a_{t}dt$$d(A_{t}^{*}A^{*})t$$=$$dA_{t}^{*}\cdot A_{t}*+A_{t}^{*}\cdot dA_{t}^{*}=2A_{t}^{*}a_{t}^{*}dt$,.

and, of course, is not differentiable.
has no meaning inOn the other hand,
distributions introduced in [11],
noise
white
admissible
of
the
idea
To study further we need[14], where conditional expectations and quantum martingales are discussed.
we putFor a -valued measurable function on$\mathcal{L}((E), (E)^{*})$$A_{t}A_{t}^{*}$$f$$\mathbb{C}$$\mathbb{R}^{n}$$|||f|||_{r}^{2}= \int_{\mathbb{R}^{n}}|f(t_{1}, \cdots, t_{n})|^{2}(1+t_{1}^{2})^{r}\cdots(1+t_{n})2rdt_{1}\cdots dt_{n}$,$r\in$.

$\mathbb{R}$.

For $\phi\sim(f_{n})\in(E)$ we put$||| \phi|||_{r,\beta}2=\sum_{n=0}^{\infty}n!e^{2}|\beta n||fn|||_{r}^{2}$,$r,\beta\in \mathbb{R}$

.

(7.6)

system of Hilbert spaces. Then we put$(A)= \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim_{\infty r,,\betaarrow}(A)_{r},\beta=\cap r,\beta(A)r,\beta$,$(A)^{*}=$ $\mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow\infty}(A)_{-r},-\beta=\bigcup_{\beta r},(A)_{-r,-\beta}$.

There holds an inclusion relation:,$(E)\subset(A)\subset(A)_{0,0=}L^{2}(E*\mu)\cong\Gamma(H\mathbb{C})\subset(A)^{*}\subset(E)^{*}$

where the injections are all continuous. A white noise distribution belonging to$admi_{\mathit{8}}sible$$(A)^{*}$

is called.

a continuous flow in
Proposition 7.2 The annihilation process. But it is not differentiable in either space.
as in$\mathcal{L}((A), (A))a\mathit{8}$$\{A_{t}\}i\mathit{8}$well$\mathcal{L}((A)^{*}, (A)^{*})$The proof is straightforward computation of norms and is omitted. Thus both$\mathcal{L}((A), (A))$ andare well defined and, with the canonical commutation
relation we obtain immediately$A_{t}A_{t}^{*}\in$$A_{t}A_{t}^{*}\in \mathcal{L}((A)^{*}, (A)^{*})$$A_{tt}A_{t}^{*}=A_{t}^{*}A+tI$

It is noteworthy that the right hand side belongs to
Therefore.

$At+A*$$d(A_{t}A_{t}^{*})=dA_{tt}*$.

$\mathcal{L}((E), (E)^{*})$and differentiable, see (7.4).

. $dA_{t}dt$ .
. $dA_{t}+dt=At$ . $dA_{t}^{*}+At+*$(7.7)

On the other hand, as an operator on $(A),$ $(7.4)$ becomes$d(A_{t}^{*}A_{t})=dA_{t}\cdot A_{t}^{*}+A_{t}\cdot dA_{t}^{*}=A_{t}^{*}\cdot dA_{t}+A_{t}^{*}\cdot dA_{t}$

.

(7.8)

208

The relations (7.3), (7.8), (7.5) and (7.7) form the quantum It\^o formula. They are sometimes
summarized as follows:$dA_{t}\cdot dA_{t}=dA_{t}^{*}\cdot dA_{t}=dA_{t}^{*}\cdot dA_{t}*=0$,$dA_{t}\cdot dA_{t}^{*}=dt$.

For the Brownian motion $B_{t}=A_{t}+A_{t}^{*}$ we have$d(B_{t}^{2})=d((A_{t}+A_{t}^{*})^{2})=dB_{t}\cdot B_{t}+B_{t}\cdot dB_{t}+dt=2B_{t}\cdot dB_{t}+dt$.

This is an essence of the classical It\^o formula: $(dB_{t})^{2}=dt$ .

References[1] V. P. Belavkin: A quantum nonadapted It\^o formula and stochastic
scale, J. Funct. Anal. 102 (1991), 414-447.

$analy_{\mathit{8}is}$in Fock[2] T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, no. 13,
Carleton University, Ottawa, 1975.
[3] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise,” Kluwer Academic, 1993.
[4] Z. Huang: Quantum white noises–White noise approach to quantum stochastic calculus,
Nagoya Math. J. 129 (1993), 23-42.
[5] R. L. Hudson and K. R. Parthasarathy: Quantum It\^o’s formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301-323.
[6] I. Kubo and S. Takenaka: Calculus on Gau8sian white noise I-IV, Proc. Japan Acad.
56A (1980), 376-380; 411-416; 57A (1981), 433-437; 58A (1982), 186-189.
[7] H.-H. Kuo: Lectures on white noise analysis, Soochow J. Math. 18 (1992), 229-300.
[8] J. M. Lindsay: Quantum and non-causal stochastic calculus, Probab. Theory Relat.
Fields 97 (1993), 65-80.
[9] P. A. Meyer: (‘Quantum Probability for Probabilists,” Lect. Notes in Math. Vol. 1538,
Springer-Verlag, 1993.
[10] N. Obata: (‘White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577,
Springer-Verlag, 1994.
[11] N. Obata: Condltional expectation in classical and quantum white noise calculi, 京都大学数理解析研究所講究録 923(1995), 154-190.
[12] N. Obata: Generalized quantum
processes on Fock space, Publ. RIMS Kyoto
Univ. 31 (1995), 667-702.
[13] N. Obata: Integral kernel operators on Fock space–Generalizations and applications toquantum dynamics, to appear in Acta Appl. Math.
[14] N. Obata: White noise approach to quantum martingales, to appear in “Proc. JapanRussia Symposium on Probability and Statistics (M. Fukushima and H. Nagai, eds.).”[15] N. Obata: Invitation to quantum white, to appear in “Proc. Encounter of Mathematics and Physics,” Seoul, 1996.
[16] K. R. Parthasarathy: “An Introduction to Quantum Stochastic Calculus,” Birkh\"auser,
1992.
$stocha\mathit{8}ti_{C}$$noi_{\mathit{8}}e$