数理解析研究所講究録
第 1797 巻 2012 年 115-143

115

A Survey of Conditional Expectations
on an Analogue of Wiener Space
By

Dong Hyun CHO *Abstract
Let $C^{r}[0,t]$ denote the space of -valued continuous functions on the interval $[0,t]$ , where is apositive integer, and for a partition $0=t_{0}<t_{1}<\cdots<t_{n}<t_{n+1}=t$ of $[0,t]$ , let
and:be given by $X_{n}(x);=(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$ and $X_{n+1}(x)=(X_{n}(x),x(t_{n+1}))=$$\mathbb{R}^{r}$$r$$X_{n}:C^{r}[0,t]arrow \mathbb{R}^{(n+1)r}$$X_{n+1}$$C^{r}[0,t]arrow \mathbb{R}^{(n+2)r}$$(x(t_{0}), \ldots,x(t_{n}),x(t_{n+1}))$.

In this survey paper, with the conditioning function, we introduce two simple formulasandfor conditional expectations of functions defined on $C^{r}[0,t]$ which is a generalization of the r-dimensional
Wiener space. As applications of the formulas, we evaluate the conditional expectations of functions
defined on $C^{r}[0,t]$ . Finally, we investigate that the conditional analytic Feynman integrals of some functionals can be applied to solve an integral equation which is formally equivalent to the Schr\"odinger partial
differential equation and to express the operator-valued Feynman integrals in terms the conditional Feynman integrals.
$X_{n}$$X_{n+1}$\S 1. Introduction and Preliminaries
Let $C_{0}[0,t]$ be the space of real-valued continuous functions on $[0,t]$ with $x(O)=0$ . It
is well-known that the space $C_{0}[0,t]$ equipped with the Wiener measure which is a probability space.. On the space, Yeh introduced an inversion formula that a conditional expectation
can be found by Fourier-transform [19], and also, in [20], [21], he obtained very useful results
including Kac-Feynman integral equation and the conditional Cameron-Martin translation theorem using the inversion formula. However Yeh’s inversion formula is very complicated in its
applications when the conditioning function is vector-valued. In [17], Park and Skoug derived
a simple fomula for conditional Wiener integrals on $C_{0}[0,t]$ with the conditioning function$X:C_{0}[0,t]arrow \mathbb{R}^{n+1}$
given by$x$

$X(x)=(x(t_{1}), \ldots,x(t_{n+1}))$

2010 Mathematics Subject Classification(s): $28C20$ .
Key Words: Analogue of Wiener measure, conditional analytic Feynman integral, conditional analytic Wiener
integral, Paley-Wiener-Zygmund integral, simple formula for conditional Wiener integral.
This research was supported by Basic Science Research Program through the National Research Foundation(NRF)
of Korea funded by the Ministry of Education, Science and Technology (2011-0004957).
*Department of Mathematics, Kyonggi University, Suwon 443-760, Korea.

116
where $0=t_{0}<t_{1}<\cdots<t_{n+1}=t$ is a partition of the interval $[0,t]$ . In their simple formulathey expressed the conditional Wiener integral directly in terms of ordinary Wiener integral.
Using the formula, they generalized Kac-Feynman formula and obtained a Cameron-Martin
type translation theorem for conditional Wiener integrals.
Let $C^{r}[0,t]$ denote the space of -valued continuous functions on the interval $[0,t]$ , whereandbe given by
is a positive integer, and let$\mathbb{R}^{r}$$X_{n+1};C^{r}[0,t]arrow \mathbb{R}^{(n+2)r}$$X_{n}:C^{r}[0,t]arrow \mathbb{R}^{(n+1)r}$$r$$X_{n}(x)=(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$and $X_{n+1}(x)=(x(t_{0}), \ldots,x(t_{n}),x(t_{n+1}))$ .

, we introduce two simple
andIn this survey paper, with the conditioning function
formulas for conditional expectations of functions defined on $C^{r}[0,t]$ which is a generalizationof the r-dimensional Wiener space. As applications of the formulas, we evaluate the conditional
expectations of functions defined on $C^{r}[0,t]$ . Finally, we investigate that the conditional analytic
Feynman integral of the functional$X_{n+1}$$X_{n}$$\exp\{\int_{0}^{t}\theta(s,x(s))dm_{L}(s)\}$can be applied to solve an integral equation which is formally equivalent to the Schr\"odinger
partial differential equation and to express the operator-valued Feynman integrals in terms theconditional Feynman integrals.
denote the sets of the complex numbers, theandThroughout this paper, letcomplex numbers with positive real parts and the nonzero complex numbers with nonnegativeof R.
be the Lebesgue measure on the Borel classreal parts, respectively. Let$(C[0,t],\mathcal{B}(C[0,t]))$. For aNow, we begin with introducing the probability measureonpositive real , let $C=C[0,t]$ be the space of all real-valued continuous functions on the closedwith $0=t_{0}<t_{1}<\cdots<t_{n}\leq t$ , letinterval $[0,t]$ with the supremum norm. For7:
be the function given by$\mathbb{C},$$\mathbb{C}_{+}$$\mathbb{C}_{+}^{\sim}$$\mathcal{B}(\mathbb{R})$$m_{L}$$w_{\varphi}$$t$$=(t_{0},t_{1}, \ldots,t_{n})$$C[0,t]arrow \mathbb{R}^{n+1}$$k_{t}$$J_{t}\prec x)=(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$For$B_{j}\in \mathcal{B}(\mathbb{R})(j=0,1, \cdots,n)$, the subset$J_{\vec{t}}^{-1}( \prod_{j=0}^{n}B_{j})$.

of $C[0,t]$ is called an interval and let

be the set of all such intervals. For a probability measure$\varphi$on$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\mathcal{I}$, let$m_{\varphi}[J_{\vec{t}}^{-1}( \prod_{j=0}^{n}B_{j})]$$=[ \prod_{j=1}^{n}\frac{1}{2\pi(t_{j}-t_{j-1})}]^{2}l\int_{B_{0}}\int_{\prod_{j=1}^{n}B_{j}}\exp\{-\frac{1}{2}\sum_{j=1}^{n}\frac{(u_{j}-u_{j-1})^{2}}{t_{j}-t_{j-1}}\}dm_{L}^{n}(u_{1}, \cdots,u_{n})d\varphi(uo)$.

on $C[0,t]$ coincides with the smallest -algebra generated by
The Borel -algebraon $(C[0,t], \mathcal{B}(C[0,t]))$ such thatand there exists a unique probability measure$\sigma$$\mathcal{B}(C[0,t])$$\mathcal{I}$$\sigma$$w_{\varphi}$$w_{\varphi}(I)=m_{\varphi}(I)$117
for all in . This measure
is called an analogue of the Wiener measure associated withthe probability measure$[18]$ . Let
be a positive integer and $C^{r}=C^{r}[0,t]$ the productspace of $C[0,t]$ with the product measure . Since $C[0,t]$ is a separable Banach space, we have$I$$\mathcal{I}$$w_{\varphi}$$\varphi[11],$$r$$w_{\varphi}^{r}$$\mathcal{B}(C^{r}[0,t])=\prod_{j=1}^{r}\mathcal{B}(C[0,t])$. This probability measure space$(C^{r}[0,t],\mathcal{B}(C^{r}[0,t]),w_{\varphi}^{r})$is called

an analogue of the r-dimensional Wiener space.
Lemma 1.1 ([11, Lemma 2.1]). If$f:\mathbb{R}^{n+1}arrow \mathbb{C}$is a Borel measurablefunction, then$\int_{C}f(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))dw_{\varphi}(x)$$=*[ \prod_{j=1}^{n}\frac{1}{2\pi(t_{j}-t_{j-1})}]^{Z}\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}f(uu, \ldots,u_{n})\exp\{-\frac{1}{2}\sum_{j=1}^{n}\frac{(u_{j}-u_{j-1})^{2}}{t_{j}-t_{j-1}}1\}$

.
where

$=*$

$dm_{L}^{n}(u_{1}, \ldots,u_{n})d\varphi(uo)$,means that either side exists, then both sides exist and they are equal.
$\iota f$

Let $\{e_{k};k=1,2, \ldots\}$ be a complete orthonormal subset of $L_{2}[0,t]$ such that eachbounded variation. For $f\in L_{2}[0,t]$ and $x\in C[0,t]$ , let$(f,x)= \lim_{narrow\infty}\sum_{k=1}^{n}\{f,e_{k}\}\int_{0}^{t}e_{k}(s)dx(s)$$e_{k}$

is of,

if the limit exists. Heredenotes the inner product over $L_{2}[0,t]$ . $(f,x)$ is called the PaleyWiener-Zygmund integral of $f$ according to . Note that
also denotes the inner productover Euclidean space unless otherwise specified.
Applying [11, Theorem 3.5], we can easily prove the following theorem.
$\{\cdot,$$\cdot\}$$x$

Theorem 1.2. Let$\{h_{1},h_{2}, \ldots,h_{n}\}$let $Z_{l}(x)=(h_{l},x)$ on $C[0,t]$ . Then
normal distribution. Moreover,$\langle\cdot,$$\cdot\rangle$

be an orthonormal system of $L_{2}[0,t]$ . For $l=1,2,$,
are independent and each has the standard
is Borel measumble, then$Z_{1},Z_{2},$ $\cdots,Z_{n}$$\cdots,n$$Z_{l}$$\iota ff:\mathbb{R}^{n}arrow \mathbb{R}$$\int_{C}f(Z_{1}(x),Z_{2}(x), \ldots,Z_{n}(x))dw_{\varphi}(x)$,$=*( \frac{1}{2\pi})^{\frac{n}{2}}\int_{\mathbb{R}^{n}}f(u_{1},u_{2}, \ldots,u_{n})\exp\{-\frac{1}{2}\sum_{j=1}^{n}u_{j}^{2}\}dm_{L}^{n}(u_{1},u_{2}, \ldots,u_{n})$where A means that either side exists then both sides exist and they are equal.
$\iota f$\S 2. Evaluation Formulas for Feynman Integrals
In this section, we establish the evaluation formulas for the analytic Feynman -integrals of
several kinds of functionals on the analogue of the r-dimensional Wiener space. For a function$w_{\varphi}^{r}$

118$F:C^{r}[0,t]arrow \mathbb{C}$$\mathbb{C}_{+}$

and, then we call$\lambda>0$$J_{\lambda}^{*}(F)$

, let

on
has the analytic extension$C^{r}[0,t]$with the parameter-integral of over$F^{\lambda}(x)=F(\lambda^{-\frac{1}{2}}x)$

the analytic Wiener$w_{\varphi}^{r}$. If$E[F^{t}]$$J_{t}^{*}(F)$$F$$\lambda$

and it is denoted by.

$E^{anw_{\lambda}}[F]=J_{\lambda}^{*}(F)$Further, if for a nonzero real $q,$

, thenhas the limit as 1 approaches $to-iq$ through-integral of over $C^{r}[0,t]$ with the parameter and denoted$\mathbb{C}_{+}$$E^{anw_{\lambda}}[F]$

it is called the analytic Feynmanby$w_{\varphi}^{r}$

$F$

$q$

$E^{anf_{q}}[F]= \lim_{\lambdaarrow-iq}E^{anw_{\lambda}}[F]$.

Now, we have the following theorem [15].

Theorem 2.1. Let $m1,$$\cdots,m_{r}$be positive integers and suppose that$\int_{\mathbb{R}}|u|^{m}d\varphi(u)<\infty$,

where $m= \max\{m_{1}, \ldots,m_{r}\}$ . For $x=(x1, \ldots,x_{\gamma})\in C^{r}[0,t]$ , let$F(x)= \sum_{j=1}^{r}\int_{0}^{t}(x_{j}(s))^{m_{j}}dm_{L}(s)$Then for$\lambda\in \mathbb{C}_{+},$$E^{anw_{\lambda}}[F]$.

exists and it is given by,$E^{anw_{\lambda}}[F]= \sum_{j=1}^{r}\lambda^{-\not\simeq}m\sum_{k=0}^{[\not\simeq^{m}]}\frac{t^{k+1}}{k+1}\frac{m_{j}!}{2^{k}k!(m_{j}-2k)!}\int_{\mathbb{R}}v_{j}^{m_{j}-2k}d\varphi(v_{j})$denotes the greatest integerfunction. Furthermore, for a nonzero real
whereexists and it is given by the right hand side of the equality above where is replaced by-iq.
$q,$$[\cdot]$$\lambda$

Example 2.2. Suppose that$\int_{\mathbb{R}}|u|^{2}d\varphi(u)<\infty$. For $x=(X1, \ldots,X_{\gamma})\in C^{r}[0,t]$ , let,$F_{l}(x)= \sum_{j=1}^{r}\int_{0}^{t}(x_{j}(s))^{l}dm_{L}(s)$$l=1,2$ .

For a nonzero real , we have by Theorem 2. 1$q$

$E^{anf_{q}}[F_{1}]=t( \frac{i}{q})^{1}\Sigma\sum_{j=1}^{r}\int_{\mathbb{R}}v_{j}d\varphi(v_{j})$and$E^{anf_{q}}[F_{2}]= \frac{i}{q}(t\sum_{j=1}^{r}\int_{\mathbb{R}}u_{j}^{2}d\varphi(v_{j})+\frac{n^{2}}{2})$.

$E^{anf_{q}}[F]$

119
Let
$L_{2}^{r}[0,t]$

be the class of all -valued Borel measures of bounded variation overthe space of all functions which forhave the form$\mathcal{M}=\mathcal{M}(L_{2}^{r}[0,t])$and$S_{w_{\varphi}}^{r}$

$F$

(2.1)

for
$\mathcal{M}$$\mathbb{C}$$\sigma\in \mathcal{M}$$F(x)= \int_{L_{2}^{r}[0,t]}\exp\{i\sum_{j=1}^{r}(v_{j},x_{j})\}d\sigma(u_{1}, \cdots,v_{r})$-a.e. $x=(xl, \ldots,x_{r})\in C^{r}[0,t]$ . Note thatis a Banach algebra which is equivalent towith the norm
, the total variation of.
Now, we have the following theorem [15].
$w_{\varphi}^{r}$$S_{w_{\varphi}}^{r}$$\sigma[11]$$\Vert F\Vert=\Vert\sigma\Vert$

Theorem 2.3. Let
given bybe given by (2.1). Then for$F\in S_{w_{\varphi}}^{r}$$\lambda\in \mathbb{C}_{+},$$E^{anw_{\lambda}}[F]= \int_{L_{2}^{r}[0,t]}\exp\{-\frac{1}{2\lambda}\sum_{j=1}^{r}\Vert v_{j}\Vert^{2}\}d\sigma Cv)$

$E^{anw_{\lambda}}[F]$exists and it is

,

where
. Furthermore, for nonzero real
exists and it is given by theright hand side of the equality above where is replaced by-iq.
$\vec{v}=(v_{1}, \ldots, v_{r})$$q,$$E^{anf_{q}}[F]$$\lambda$

Let $r=1$ andbe an orthonormal subset of $L_{2}[0,t]$ . For $1\leq p\leq\infty$ , let
the set of cylinder type functions having the form$\{e_{1}, \ldots,e_{l}\}$(2.2)$\mathcal{A}_{l}^{(p)}$be$F_{l}(x)=f((e_{1},x), \ldots,(e_{l},x))$fora.e. $x\in C[0,t]$ , where
theorem [15].
$w_{\varphi^{-}}$$f\in L_{p}(\mathbb{R}^{l})$is Borel measurable on

Theorem 2.4. Let
exists and it is given by$F_{l}\in \mathcal{A}_{l}^{(p)}(1\leq p\leq\infty)$$\mathbb{R}^{l}$. We now have the following

be given by (2.2). Then for$\lambda\in \mathbb{C}_{+},$$E^{anw_{\lambda}}[F_{l}]$.

$E^{anw_{\lambda}}[F_{l}]=( \frac{\lambda}{2\pi})^{2}\int_{\mathbb{R}^{l}}f(u\gamma\exp\{-\frac{\lambda}{2}\Vert\vec{u}\Vert^{2}\}dm_{L}^{l}(\vec{u})l$

Furthermore, for nonzero real

the above equality where$\lambda$

exists $\iota fp=1$ and it is given by the right hand side
is replaced by-iq.
$q,$$E^{anf_{q}}[F_{l}]$For a positive integer , let
the set of all functions on$l$$\hat{M}(\mathbb{R}^{l})$$\phi$

(2.3)

where

be the class of all complex Borel measures on
defined by$\mathcal{M}(\mathbb{R}^{l})$$\mathbb{R}^{l}$$\phi(\vec{u})=\int_{\mathbb{R}^{l}}\exp\{i\langle\vec{u},\vec{z}\}\}d\rho(z\gamma$$\rho\in\lambda 4(\mathbb{R}^{l})$.

,$\mathbb{R}^{l}$

and

120
Theorem 2.5. Let $\Phi(x)=\phi((e_{1},x), \ldots,(e_{l},x))$ for(2.3). Then forexists and it is given by$w_{\varphi}-a.e$$\lambda\in \mathbb{C}_{+},$. $x\in C[0,t]$ , where

Funhermore, for nonzero real $q,$

above equality whereis given by$E^{anw_{\lambda}}[\Phi]$$E^{anw_{\lambda}}[ \Phi]=\int_{\mathbb{R}^{l}}\exp\{-\frac{1}{2\lambda}\Vert\vec{z}\Vert^{2}\}d\rho(\vec{z})$$\lambda$$\phi$$E^{anf_{q}}[\Phi]$.

exists and it is given by the right hand side of theis replaced by-iq.

Theorem 2.6. Let $0=t_{0}<t_{1}<\cdots<t_{n}\leq t$ and$G(x)=f(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$

$G$

, for

where
given by$f\in L_{p}(\mathbb{R}^{n+1},\mathcal{B}(\mathbb{R}^{n+1}),m_{L}^{n}\otimes\varphi)(1\leq p\leq\infty)$

be given by$w_{\varphi}-a.e$. Then. $x\in C[0,t]$ ,$for/1\in \mathbb{C}_{+},$$E^{anw_{\lambda}}[G]$exists and it is$E^{anw_{\lambda}}[G]=( \frac{\Lambda}{2\pi})^{n}2\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}f(u0,u1, \cdots,u_{n})\exp\{-\frac{\prime t}{2}\sum_{j=1}^{n}(u_{j}-u_{J-1})^{2}\}dm_{L}^{n}(u_{1}, \ldots,u_{n})d\varphi(uo)$Furthermore, for nonzero real

the above equality where$\lambda$.

exists $\iota fp=1$ and it is given by the right hand side
is replaced by-iq.
$q,$$E^{anf_{q}}[G]$be a complex valued Borel measure on $[0,t]$ . Then $\eta=\mu+v$ can be decomposed
uniquely into the sum of a continuous measure and a discrete measure . Further, letdenote the Dirac measure with total mass 1 concentrated at . Let be a positive integer andwhich have the form
letbe the set of all -valued functions onLet$\eta$$v$$\mu$$p_{j}$$\mathcal{G}^{*}$$\theta$$\mathbb{C}$(2.4)$\delta_{p_{j}}$$r$$[0,\infty)\cross \mathbb{R}^{r}$$\theta(s,\vec{u})=\int_{\mathbb{R}^{r}}\exp\{i\langle\vec{u},\vec{v}\rangle\}d\sigma_{s}Cv)$

where $\{\sigma_{s};s\in[0,t]\}$ is the family fromsatisfying the following conditions:is a Borel measurable function of on $[0,t]$ ,$\mathcal{M}(\mathbb{R}^{r})$1. for each Borel subset of.
2.
Now, we have the following theorems [9].
$E$$\mathbb{R}^{r},$$\sigma_{s}(E)$$s$$\Vert\sigma_{s}\Vert\in L_{1}([0,t],\mathcal{B}([0,t]), |\eta|)$

Theorem 2.7. Let$m$and be two positive integers, and$k$$\eta=\mu+\sum_{j=1}^{m}w_{j}\delta_{p_{j}}$where $0<p1<\cdots<p_{m}<t$ and$w_{j}\in \mathbb{C}$

. Let

,

$\theta\in \mathcal{G}^{*}be$given by (2.4) and for $x\in C^{r}[0,t]$, set$F_{k}(x)=[ \int_{0}^{t}\theta(s,x(s))d\eta(s)]^{k}$121
Then for$\lambda>0$,$E[F_{k}^{\lambda}]= \sum H(k,\lambda,\vec{0},q0,q_{1}, \ldots,q_{m};j_{0}, \ldots,j_{m})q_{0}+q_{1}.+\cdot\cdot+q_{m}=kj_{0}+\cdot\cdot+jm=q_{0}$where$for\vec{v}_{m,j_{m}+1}\in \mathbb{R}^{r}$$H(k,\lambda,\vec{v}_{m,j_{m}+1},q0,q_{1}, \ldots,q_{m};j_{0}, \ldots,j_{m})$$=k!( \prod_{\alpha=1}^{m}\frac{w_{\alpha}^{q_{\alpha}}}{q_{\alpha}!})\int_{\Delta_{q_{0},j_{0},\cdots,jm}}\int_{\mathbb{R}^{kr}}\exp\{-\frac{1}{2\lambda}\sum_{\alpha=0}^{mj_{\alpha}}\sum_{\beta=1}^{+1}(s_{\alpha},\rho-s_{cx/3-1})\Vert\sum_{\gamma=\beta}^{j_{\alpha}+1}\vec{v}_{\alpha,\gamma}+$$\sum_{l=\alpha+1}^{m}\sum_{\gamma=1}^{j_{l}+1}\vec{v}_{l,\gamma\Vert^{2}\}\int_{\mathbb{R}^{r}}\exp\{1}i\lambda^{-}z\langle\vec{\eta},\sum_{\alpha=0}^{mj}\sum_{\beta=1}^{\alpha+1}\vec{v}_{\alpha/3}\}\}d\varphi^{r}(\vec{\eta})d(\prod_{\alpha=0\beta}^{m}\prod_{=1}^{j_{\alpha}}\sigma_{s_{a\beta}}\cross$$\prod_{(x=1}^{m}\sigma_{p_{\alpha}^{\alpha}}^{q})(\vec{v},\vec{z})d\mu^{q_{0}}(\vec{s})$with$s0,0=0$ ,$\vec{s}=(s_{0,1}, \ldots,s_{0,j_{0}},s_{1,1}, \ldots,s_{1,j_{1}}, \ldots,s_{m,1}, \cdots,s_{m,j_{m}})$$s_{\alpha},0=s_{\alpha-1,j_{\alpha-1}+1}=p_{\alpha}$for$\alpha=1,$ $\cdots,m$,$s_{m,j_{m}+1}=t$,,$\Delta_{q_{0};j_{0},\cdots,j_{m}}=\{\vec{s}:0<s_{0,1}<\cdots<s0,j_{0}<pl<s_{1,1}<\cdots<s_{1,j_{1}}<p_{2}$$<\cdots<p_{m-1}<s_{m-1,1}<\cdots<s_{m-1,j_{m-1}}<p_{m}<s_{m,1}<\cdots<s_{m,j_{m}}<t\}$ ,$\vec{v}=C_{0,1}v,$$\ldots,\vec{v}_{0,j_{0}},\vec{v}_{1,1},$ $\cdots,\vec{v}_{1,j_{1}},$$\cdots,\vec{v}_{m,1},$ $\ldots,\vec{v}_{m,j_{m}})$$\vec{z}=(\vec{z}1,1, \ldots,\vec{z}_{1,q_{1},\vec{Z}2,1,\ldots,\vec{Z}2,q_{2}}, \ldots,\vec{z}_{m,1}, \ldots,\vec{z}_{m,q_{m}})$$\vec{v}_{\alpha-1,j_{\alpha-1}+1}=\sum_{l=1}^{q_{a}}\vec{z}_{\alpha,l}$$for\alpha=1,$ $\cdots,m$

and

;

.

Corollary 2.8. Under the assumptions as given in Theorem 2.7 with one exception $\eta=\mu$ ,that is, assuming that has no discrete part, we have$\eta$$E[F_{k}^{\lambda}]=k!

\int_{\Delta_{k}}\int_{\mathbb{N}^{kr}}\exp\{-\frac{1}{2\lambda}\sum_{l=1}^{k}(s_{l}-s_{l-1})\Vert\sum_{\gamma=l}^{k}\vec{v}_{\gamma}\Vert^{2}\}\int_{\mathbb{R}^{r}}\exp\{i\lambda^{-\frac{1}{2}}\langle\vec{\eta},\sum_{l=1}^{k}\vec{v}_{l\rangle\}}$$d \varphi^{r}(\vec{\eta})d(\prod_{l=1}^{k}\sigma_{s_{l}})Cv)d\mu^{k}(\vec{s})$where $s_{0}=0,\vec{s}=(s_{1}, \cdots,s_{k}),$$\vec{v}=Cv\iota,$$\cdots,\vec{v}_{k})$and $\Delta_{k}=\{\vec{s} : 0<s_{1}<\cdots<s_{k}<t\}$ .

Corollary 2.9. Under the assumptions as given in Theorem 2.7 with one exception, that is, assuming that has no continuous part, we have$\sum_{j=1}^{m}w_{j}\delta_{p_{j}}$$E[F_{k}^{\lambda}]=k!

$\eta$\sum_{q_{1}+\cdots+q_{m}=k}(\prod_{\alpha=1}^{m}\frac{w_{\alpha}^{q_{\alpha}}}{q_{\alpha}!})\int_{\mathbb{R}^{kr}}\exp\{-\frac{1}{2\lambda}\sum_{\alpha=1}^{m}(p_{\alpha}-p_{\alpha-1})\Vert\sum_{\gamma=\alpha}^{m}\sum_{l=1}^{q_{\gamma}}\vec{z}_{\gamma,l}\Vert^{2}\}$$\eta=$

122$\cross\int_{\mathbb{R}^{r}}\exp\{$ $i \lambda^{-}21\langle\vec{\eta},\sum^{m}\sum^{\alpha}^{q}\vec{z}_{\alpha,l}\}\}d\varphi^{r}(\vec{\eta})d(\prod_{t=1}^{m}\sigma_{p_{\alpha}}^{q_{\alpha}})(\vec{z})$$\alpha=1l=1$

where $p0=0,$ $\vec{z}=(\vec{z}\cdots,\vec{z}\vec{z}, \cdots,\tilde{z}_{m,1}, \cdots,\vec{z}_{m,q_{m}})$ .
Theorem 2.10. Let be nomally distributed with mean vector and variance covariance, whereis the r-dimensional identity matrix. Then, under the assumptions andmatrixit is given byexists fornotations as given in Theorem 2.7,$\vec{0}$$\varphi^{r}$$\sigma^{2}I_{r}$$I_{r}$$E^{anw_{\lambda}}[F_{k}]$$\lambda\in \mathbb{C}_{+}and$$E^{anw_{1}}[F_{k}]= \sum_{qql}\ldots\sum_{+j_{m}=q_{0}}T(k,\lambda,\sigma,\vec{0},q0, \cdots,q_{m};j_{0}, \cdots,j_{m})$where$for\vec{v}_{m,j_{m}+1}\in \mathbb{R}^{r},$$T(k,,t,\sigma,\vec{v}_{m,j_{m}+1},q0, \cdots,q_{m};j_{0}, \cdots,j_{m})$$H(k,\lambda,\vec{v}_{m,j_{m}+1},q0,q_{1}, \cdots,q_{m};jo, \cdots,j_{m})$replacingis given by the expression of$\int_{\mathbb{R}^{r}}\exp\{i\lambda^{-z}1\langle\vec{\eta},\sum_{\alpha=0}^{m}\sum_{\beta=1}^{j_{\alpha}+1}\vec{v}_{\alpha,\beta}\rangle\}d\varphi^{r}(\vec{\eta})$. Furthermore, for nonzero realbygiven by the above equality replacing by-iq.
$\exp\{-\frac{\sigma^{2}}{2\lambda}\Vert\sum_{a=0}^{m}\sum_{\beta=1}^{j_{\alpha}+1}\vec{v}_{(t}p\Vert^{2}\}$$q,$$E^{anf_{q}}[F_{k}]$exists and it is$\lambda$

Theorem 2.11. Let the assumptions and notations be as given in Theorem 2.7 and let$F(x)= \exp\{\int_{0}^{t}\theta(s,x(s))d\eta(s)\}$for$x\in C^{r}[0,t]$. Then for$\lambda>0$, we have$E[F^{\lambda}]=1+ \sum_{k=1}^{\infty}\frac{1}{k!}E[F_{k^{1}}’]$

is as given in Theorem 2.7. Furthemore, under the assumptions as given in
where
is obtained by
Theorem 2.10,$E[F_{k}^{\lambda}]$$E^{anf_{q}}[F]$$E^{anf_{q}}[F]=1+ \sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[F_{k}]$for nonzero real , where$q$$E^{anf}q[F_{k}]$is as given in Theorem 2.10.

Theorem 2.12. Let the assumptions and notations be as given Theorem 2.7 and let $G_{k}(x)=$$F_{k}(x)\psi(x(t))$. Let $G(x)=$for
for $x\in C^{r}[0,t]$ , where$\exp\{\int_{0}’\theta(s,x(s))d\eta(s)\}\psi(x(t))$ . Then
, we havefor$\psi(u\gamma=\int_{\mathbb{R}^{r}}\exp\{i\langle\vec{u},\vec{v}\rangle\}dv\cap v$$v\in \mathcal{M}(\mathbb{R}^{r})$$\lambda>0$$E[G^{\lambda}]= \int_{\mathbb{R}^{r}}\int_{\mathbb{R}^{r}}\exp\{i\lambda^{-z}1\langle\vec{\eta},\vec{v}\}-\frac{t}{2\lambda}\Vert\vec{v}\Vert^{2}\}d\varphi^{r}(\vec{\eta})dvOv+\sum_{k=1}^{\infty}\frac{1}{k!}E[G_{k}^{\lambda}]$where$E[c_{k}^{\prime\iota_{]=\sum_{q0+q_{1}+\cdots+q_{m}=kj_{0}+}\sum_{+j_{m}=q_{0}}\int_{\mathbb{R}^{r}}H(k,\lambda,\vec{v}_{m,j_{m}+1,q0,q_{1},\cdots,q_{m};j_{0},\cdots,j_{m})}}}\ldots$$dvCv_{m,j_{m}+1})$.

123
Furthermore, under the assumptions and notations as given in Theorem 2.10, $E^{anf_{q}}[G]$ can beobtained by$E^{anf_{q}}[G]= \int_{\mathbb{R}^{r}}\exp\{\frac{t+\sigma^{2}}{2qi}\Vert\vec{v}\Vert^{2}\}dvCv)+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[G_{k}]$for nonzero real , where$q$$E^{anf_{q}}[G_{k}]= \sum_{q0+q_{1}+\cdots+q_{m}=kj_{0}+}\ldots\sum_{+j_{m}=q0}\int_{\mathbb{R}^{r}}T(k, -iq,\sigma,\vec{v}_{m,j_{m}+1,q_{0}}, \cdots,q_{m};j_{0}, \cdots,j_{m})$$dvCv_{m,j_{m}+1})$.

Corollary 2.13. Under the assumptions and notations as given in Theorem 2.12 with one, the Dimc measure concentmted at
exception, we have for a nonzero real$\vec{0}\in \mathbb{R}^{r}$$\varphi^{r}=\delta_{\vec{0}}$$q$$E^{anf_{q}}[G]= \int_{\mathbb{R}^{r}}\exp\{\frac{t}{2qi}\Vert\vec{v}\Vert^{2}\}dvCv)+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[G_{k}]$where$E^{anf_{q}}[G_{k}]= \sum_{q0+q_{1}+\cdots+q_{m}=kj_{0}+}..\sum_{+j_{m}=q0}\int_{\mathbb{R}^{r}}T(k, -iq,0,\vec{v}_{m,j_{m}}+1,q0, \cdots,q_{m};j_{0}, \cdots,j_{m})$$dvCv_{m,j_{m}+1})$

which is a main result of [14].
RemarkUnder the conditions as given in Corollaries 2.8 and 2.9, we can obtain more
simple expressions in Theorems 2.10, 2.11, 2.12 and Corollary 2.13.
, where $0\leq p\iota<\cdots<p_{m}\leq t$ , we can obtain all the results includingIfwith minor modifications.
, then using the following version of theIf-nomial formula [12,$\bullet$$\bullet$$\eta=\mu+\sum_{j=1}^{m}w_{j}\delta_{p_{j}}$$\eta$$\bullet$$\eta=\mu+\sum_{j=1}^{\infty}w_{j}\delta_{p_{j}}$$\aleph_{0}$

p.41]
(2.5)

$( \sum_{p=0}^{\infty}b_{p})^{n}=\sum_{h=0q_{0}+q_{1}+}^{\infty}..\sum_{+q_{h}=n,q_{h}\neq 0}\frac{n!}{q_{0}!q_{1}!\cdots q_{h}!}b_{0}^{q_{0}}b_{1}^{q_{1}}\cdots b_{h}^{q_{h}}$we can show that for$\lambda>0,$$E[G^{\lambda}]$,exists in Theorem 2.12.

\S 3. Simple formulas for conditional expectations
Letvalue space of

be integrable and let be a random vector on $C^{r}[0,t]$ assuming that theis a normed space with the Borel -algebra. Then, we have the conditional$F;C^{r}[0,t]arrow \mathbb{C}$
$X$

$X$

$\sigma$124
expectation $E[F|X]$ of given from a well known probability theory [16]. Further, there
exists a -integrable complex-valued function on the value space of such that $E[F|X](x)=$$(\psi oX)(x)$ for. $x\in C^{r}[0,t]$ , where is the probability distribution of . The function
$X$

$F$

$X$

$\psi$$P_{X}$$X$$P_{X}$$w_{\varphi}^{r}-a.e$$\psi$

is called the conditional -integral of given and it is also denoted by $E[F|X]$ .
Let $0=t_{0}<t_{1}<\cdots<t_{n}<t_{n+1}=t$ be a partition of the interval $[0,t]$ . Define the polygonal
functions by
$X$

$F$

$w_{\varphi}^{r}$$[x](s)=x(t_{j-1})+ \frac{s-t_{j-1}}{t_{j}-t_{j-1}}(x(t_{j})-x(t_{j-1}))$,$t_{j-1}\leq s\leq t_{j}$,for $x\in C^{r}[0,t]$ and$[ \vec{\xi}_{n+1}](s)=\xi_{j-1}+\frac{s-t_{j}}{t_{j}-t_{j-1}}(\xi_{j}-\xi_{j-1})$

for$\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n+1})\in \mathbb{R}^{(n+2)r}$,$t_{j-1}\leq s\leq t_{j-1}$, where $j=1,$ $\cdots,n+1$ .

Now, we have two simples formulas for the conditional

Theorem 3.1. Let$X_{n+1}$

(3.1)

:

,

$C^{r}[0,t]arrow \mathbb{R}^{(n+2)r}$$w_{\varphi}^{r}$-integrals on $C^{r}[0,t][4,6]$ .

be given by$X_{n+1}(x)=(x(t_{0}),x(t_{1}), \cdots,x(t_{n+1}))$and let$F:C^{r}[0,t]arrow \mathbb{C}$be$w_{\varphi}^{r}$-integrable. Then for a Borel subset

$B$

of

$\mathbb{R}^{(n+2)r}$$\int_{x_{n+1(B)}^{-1}}F(x)dw_{\varphi}^{r}(x)=\int_{B}E[F(x-[x]+[\vec{\xi}_{n+1}])]dP_{X_{n+1}}(\vec{\xi}_{n+1})$where$b_{n+1}$$b_{n+1}$-a.

is the probability distribution of$X_{n+1}$on$(\mathbb{R}^{(n+2)r}, \mathcal{B}(\mathbb{R}^{(n+2)r}))$we have,, so that we have for$e.\vec{\xi}_{n+1}\in \mathbb{R}^{(n+2)r}$$E[F|X_{n+1}](\vec{\xi}_{n+1})=E[F(x-[x]+[\vec{\xi}_{n+1}])]$

Theorem 3.2. Let$X_{n};C^{r}[0,t]arrow \mathbb{R}^{(n+1)r}$(3.2)

be given by$X_{n}(x)=(x(t_{0}),x(t_{1}), \cdots,x(t_{n}))$Moreover let$F$

be integrable on $C^{r}[0,t]$ and. Then we have for any$(\mathbb{R}^{(n+1)r},\mathcal{B}(\mathbb{R}^{(n+1)r}))$.

$P_{X_{n}}$.

a probability distribution of$X_{n}$

defind on$\vec{\xi}_{n}=(\xi_{0},\xi_{1}, \cdots,\xi_{n})\in \mathbb{R}^{(n+1)r}$$E[F|X_{n}]( \vec{\xi}_{n})=[\frac{1}{2\pi(t-t_{n})}]^{2}\int_{\mathbb{R}^{r}}E[F(x-[x]’+[\vec{\xi}_{n+1}])]\exp\{-\frac{\Vert\xi_{n+1}-\xi_{n}\Vert^{2}}{2(t-t_{n})}\}dm_{L}^{r}(\xi_{n+1})$

for R-a.e.

$\vec{\xi}_{n}\in \mathbb{R}^{(n+1)r}$, where $\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n},\xi_{n+1})$.

125
Theorem 3.3. Let $r=1,$ $F_{m}(x)= \int_{0}^{t}(x(u))^{m}dm_{L}(u)(m\in \mathbb{N})forx\in C[0,t]$ and suppose thatoo. Then-a. .
is -integrable. Moreover,exists forand it is given by$\int_{\mathbb{R}}|u|^{m}d\varphi(u)<$$F_{m}$$E[F_{m}|X_{n+1}](\vec{\xi}_{n+1})$$w_{\varphi}$$P_{X_{n+1}}$$e$$\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n},\xi_{n+1})\in \mathbb{R}^{n+2}$$E[F_{m}|X_{n+1}]( \vec{\xi}_{n+1})=\sum_{j=1}^{n+1}\sum_{k=0}^{[\frac{m}{2}]_{m}}\sum_{l=0}^{-2k}\frac{m!(l+k)!(t_{j}-t_{j-1})^{k+1}\xi_{j-1}^{m-2k-l}(\xi_{j}-\xi_{j-1})^{l}}{2^{k}l!(m-2k-l)!(l+2k+1)!}$where$[\cdot]$

denotes the greatest integerfunction.

Example 3.4. For $m=1,2,3$ , let $F_{m}(x)= \int_{0}^{t}(x(u))^{m}dm_{L}(u)$ for $x\in C[0,t]$ and suppose that
, we have by Theo. Then fora.e.

$\int_{\mathbb{R}}|u|^{m}d\varphi(u)<\infty$$P_{X_{n+1^{-}}}$$\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n},\xi_{n+1})\in \mathbb{R}^{n+2}$

rem 3.3$E[F_{1}|X_{n+1}]( \vec{\xi}_{n+1})=\frac{1}{2}\sum_{j=1}^{n+1}(t_{j}-t_{j-1})(\xi_{j}+\xi_{j-1})$

which can be also obtained by an application of Corollary 4.5 in [11]. We also have
$E[F_{2}|X_{n+1}]( \vec{\xi}_{n+1})=\frac{1}{6}\sum_{j=1}^{n+1}(t_{j}-t_{j-1})(t_{j}-t_{j-1}+2\xi_{j}^{2}+2\xi_{j}\xi_{j-1}+2\xi_{j-1}^{2})$which is the result given by Corollary 4.10 of [11]. Moreover we have

$E[F_{3}|X_{n+1}]( \vec{\xi}_{n+1})=\frac{1}{4}\sum_{j=1}^{n+1}(t_{j}-t_{j-1})[(t_{j}-t_{j-1})(\xi_{j}+\xi_{j-1})$$+\xi_{j}^{3}+\xi_{j}^{2}\xi_{j-1}+\xi_{j}\xi_{j-1}^{2}+\xi_{j-1}^{3}]$$P_{X_{n}}$.

Theorem 3.5. Under the conditions and notations as given in Theorem 3.3, we have for-a.
$e.\vec{\xi}_{n}=(\xi_{0},\xi_{1}, \cdots,\xi_{n})\in \mathbb{R}^{n+1}$$E[F_{m}|X_{n}]( \vec{\xi}_{n})=\sum_{j=1}^{n}\sum_{k=0}^{Tm}\sum_{l=0}^{-2k}\frac{m!(l+k)!(t_{j}-t_{j-1})^{k+1}\xi_{j-1}^{m-2k-l}(\xi_{j}-\xi_{j-1})^{l}}{2^{k}l!(m-2k-l)!(l+2k+1)!}[^{m}]$$+ \sum_{k=0}^{[\frac{m}{2}][\frac{m-2k}{\sum_{l=0}^{2}}]}\frac{m!(2l+k)!\xi_{n}^{m-2k-2l}(t-t_{n})^{l+k+1}}{2^{l+k}l!(m-2k-2l)!(2l+2k+1)!}$.

Example 3.6. For $m=1,2,3$ , let $F_{m}(x)= \int_{0}^{t}(x(u))^{m}dm_{L}(u)$ for $x\in C[0,t]$ and suppose thatoo. Moreover, let
and$\int_{\mathbb{R}}|u|^{m}d\varphi(u)<$$Z_{1}(x)= \frac{1}{t}F_{1}(x)$$Z_{2}(x)= \sum_{j=1}^{n+1}\frac{1}{t_{j}-t_{j-1}}\int_{t_{j-1}}^{t_{j}}x(u)dm_{L}(u)$126
for $x\in C[0,t]$ . Then for$P_{X_{n}}-a.e.\vec{\xi}_{n}=(\xi_{0},\xi_{1}, \cdots,\xi_{n})\in \mathbb{R}^{n+1}$, we have.

$E[F_{1}|X_{n}]( \tilde{\xi}_{n})=\frac{1}{2}\sum_{j=1}^{n}(t_{j}-t_{j-1})(\xi_{j}+\xi_{j-1})+(t-t_{n})\xi_{n}$Hence we have
$E[Z_{1}|X_{n}]( \vec{\xi}_{n})=\frac{1}{2t}\sum_{j=1}^{n}(t_{j}-t_{j-1})(\xi_{j}+\xi_{j-1})+\frac{1}{t}(t-t_{n})\xi_{n}$and$E[Z_{2}|X_{n}]( \vec{\xi}_{n})=\frac{1}{2}\sum_{j=1}^{n}(\xi_{j}+\xi_{j-1})+\xi_{n}$which are also given by Theorems 4.3 and 4.6 in [11], respectively. Further, we have
$E[F_{2}|X_{n}]( \vec{\xi}_{n})=\frac{1}{6}\sum_{j=1}^{n}(t_{j}-t_{j-1})(t_{j}-t_{j-1}+2\xi_{j}^{2}+2\xi_{j}\xi_{j-1}+2\xi_{j-1}^{2})$$+(t-t_{n}) \xi_{n}^{2}+\frac{1}{2}(t-t_{n})^{2}$

which is also given by Theorem 4.8 in [11]. Finally, we have
$E[F_{3}|X_{n}]( \vec{\xi}_{n})=\frac{1}{4}\sum_{j=1}^{n}(t_{j}-t_{j-1})[(t_{j}-t_{j-1})(\xi_{j}+\xi_{j-1})+\xi_{j}^{3}+\xi_{j}^{2}\xi_{j-1}+\xi_{j}\xi_{j-1}^{2}$$+ \xi_{j-1}^{3}]+(t-t_{n})\xi_{n}^{3}+\frac{3}{2}(t-t_{n})^{2}\xi_{n}$.

\S 4. Evaluation formulas for an analogue of conditional Feynman integrals, let
For a function :and, whereandand are given by (3.1) and (3.2), respectively. Supposethatexists for each
. By the definition of conditional -integral and Theorem 3.1,
we have for$F$$C^{r}[0,t]arrow \mathbb{C}$$X_{n}^{\lambda}(x)=X_{n}(\lambda^{-1/2_{X)}}$$E[F^{\lambda}]$$X_{n+1}$$\lambda>0$$F^{\lambda}(x)=F(\lambda^{-1/2}x),$ $X_{n+1}^{\lambda}(x)=X_{n+1}(\lambda^{-1/2}x)$$X_{n}$$\lambda>0$$w_{\varphi}^{r}$$\lambda>0$$E[F^{\lambda}|X_{n+1}^{t}j](\vec{\xi}_{n+1})=E[F(\lambda^{-1/2}(x-[x])+[\vec{\xi}_{n+1}])]$

for$P_{X_{n+1}^{\lambda}}$-a.e.

, where$\vec{\xi}_{n+1}\in \mathbb{R}^{(n+2)r}$$(\mathbb{R}^{(n+2)r},\mathcal{B}(\mathbb{R}^{(n+2)r}))$$P_{X_{n+1}^{\lambda}}$is the probability distribution of$X_{n+1}^{\lambda}$. Moreover, we can obtain from Theorem 3.2$E[F^{\lambda}|X_{n}^{\lambda}](\tilde{\xi}_{n})$(4.1)$=[ \frac{\lambda}{2\pi(t-t_{n})}]^{r}2\int_{\mathbb{R}^{r}}E[F(\lambda^{-\Sigma}(x-[x])1+[\vec{\xi}_{n+1}])]\exp\{-\frac{\Lambda\Vert\xi_{n+1}-\xi_{n}\Vert^{2}}{2(t-t_{n})}\}$$dm_{L}^{r}(\xi_{n+1})$

defined on

127
for
and

$P_{X_{n}^{\lambda}}$-a.e.

, where $\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n}, \xi_{n+1})$ for$\vec{\xi}_{n}=(\xi_{0},\xi_{1}, \cdots,\xi_{n})\in \mathbb{R}^{(n+1)r}$is the probability distribution ofonhas an analytic extension
then it is called the conditional analytic Wiener -integral ofand denoted by$X_{n}^{\prime 1}$$P_{X_{n}^{\lambda}}$$E[F(\lambda^{-1/2}(x-[x])+[\vec{\xi}_{n+1}])]$$J_{\lambda}^{*}(F)(\vec{\xi}_{n+1})$$w_{\varphi}^{r}$$F$$\xi_{n+1}\in \mathbb{R}^{r}$,. If, foron
as a function of ,$(\mathbb{R}^{(n+1)r}, \mathcal{B}(\mathbb{R}^{(n+1)r}))$$\vec{\xi}_{n+1}\in \mathbb{R}^{(n+2)r}$$\lambda$$\mathbb{C}_{+}$

given$X_{n+1}$

with parameter ,1$E^{anw_{\lambda}}[F|X_{n+1}](\vec{\xi}_{n+1})=J_{t}^{*}(F)(\vec{\xi}_{n+1})$. If for a non-zero real

has a limit as approaches to$-iq$ through, then it is called the conditional analytic Feynman-integral of given
with parameter and denoted byfor$\vec{\xi}_{n+1}\in \mathbb{R}^{(n+2)r}$$q,$$E^{anw_{\lambda}}[F|X_{n+1}](\vec{\xi}_{n+1})$$\lambda$$F$$\mathbb{C}_{+}$$w_{\varphi}^{r}$$X_{n+1}$$q$.

$E^{anf_{q}}[F|X_{n+1}]( \vec{\xi}_{n+1})=\lim_{\lambdaarrow-iq}E^{anw_{\lambda}}[F|X_{n+1}](\vec{\xi}_{n+1})$

Similarly, we define
andusing (4.1).
Throughout the remainder of this section, we will assume $r=1$ .
$E^{anw_{\lambda}}[F|X_{n}](\vec{\xi}_{n})$$E^{anf_{q}}[F|X_{n}](\vec{\xi}_{n})$Theorem 4.1. Let
be given by (3.1). Then, under the assumptions and notations asgiven in Theorem 3.3,
exists for
-a.
.
for
Moreover, for a non-zem real
exists and it is given by$X_{n+1}$$E^{anw_{\lambda}}[F_{m}|X_{n+1}](\vec{\xi}_{n+1})$$q,$$\lambda\in \mathbb{C}_{+}and$$e.\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$P_{X_{n+1}}$$E^{anf_{q}}[F_{m}|X_{n+1}](\vec{\xi}_{n+1})$$E^{anf_{q}}[F_{m}|X_{n+1}](\vec{\xi}_{n+1})$$= \sum_{j=1}^{n+1}\sum_{k=0}^{Zm}\sum_{l=0}^{-2k}(\frac{i}{q}I^{k}\frac{m!(l+k)!(t_{j}-t_{j-1})^{k+1}\xi_{j-1}^{m-2k-l}(\xi_{j}-\xi_{j-1})^{l}}{2^{k}l!(m-2k-l)!(l+2k+1)!}[^{m}]$Theorem 4.2. Let
in Theorem 3.5,
a non-zero real$X_{n}$

be given by (3.2). Then, under the assumptions and notations as given
exists for. Moreover, forfor -a.
exists and it is given by$E^{anw_{\lambda}}[F_{m}|X_{n}](\vec{\xi}_{n})$

$q,$

.

$\lambda\in \mathbb{C}_{+}and$$P_{X_{n}}$$e.\vec{\xi}_{n}\in \mathbb{R}^{n+1}$$E^{anf_{q}}[F_{m}|X_{n}](\vec{\xi}_{n})$$E^{anf_{q}}[F_{m}|X_{n}](\vec{\xi}_{n})$$= \sum_{j=1}^{n}\sum_{k=0}^{[\frac{m}{2}]_{m}}\sum_{l=0}^{-2k}(\frac{i}{q})^{k}\frac{m!(l+k)!(t_{j}-t_{j-1})^{k+1}\xi_{j-1}^{m-2k-l}(\xi_{j}-\xi_{j-1})^{l}}{2^{k}l!(m-2k-l)!(l+2k+1)!}$$+ \sum_{k=0}^{[\frac{m}{2}][\frac{m-2k}{\sum_{l=0}^{2}}]}(\frac{i}{q})^{k+l}\frac{m!(2l+k)!\xi_{n}^{m-2k-2l}(t-t_{n})^{l+k+1}}{2^{l+k}l!(m-2k-2l)!(2l+2k+1)!}$For each $j=1,$ $\cdots,n+1$ , let$\alpha_{j}(u)=\frac{1}{\sqrt{t_{j}-t_{j-1}}}\chi(t_{j-1},t_{j}](u)$

. Let

$V$

.

be the subspace of $L_{2}[0,t]$generated byand letdenote the orthogonal complement of . Let and, respectively. Let be in
be the orthogonal projections from $L_{2}[0,t]$ to and
, let
For
and
be the measures defined byand$\{\alpha_{1}, \cdots,\alpha_{n+1}\}$$V^{\perp}$

$V$

$V$

$\vec{\xi}_{n}=(\xi_{0}, \cdots,\xi_{n})\in \mathbb{R}^{n+1}$$V^{\perp}$$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$\sigma_{\vec{\xi}_{n}}$(4.2)$\sigma$$\sigma_{\vec{\xi}_{n+1}}$$\frac{d\sigma_{\tilde{\xi}_{n}}}{d\sigma}(v)=\exp\{i\sum_{j=1}^{n}(\mathcal{P}v)(t_{j})(\xi_{j}-\xi_{j-1})\}$$\mathcal{P}$$\mathcal{P}^{\perp}$$\Lambda t(L_{2}[0,t])$

.

128
and
(4.3)

$\frac{d\sigma_{\vec{\xi}_{n+1}}}{d\sigma}(v)=\exp\{i(v, [\vec{\xi}_{n+1}])\}$

respectively, where $v\in L_{2}[0,t]$ . Moreover, let
(4.4)

for

$F_{\vec{\xi}_{n+1}}\in S_{w_{\varphi}}$,be defined by$F_{\tilde{\xi}_{n+1}}(x)= \int_{L_{2}[0,t]}\exp\{i(v,x)\}d(\sigma_{\tilde{\xi}_{n+1}}\circ(\mathcal{P}^{\perp})^{-1})(v)$a.e. $x\in C[0,t]$ .
We now have the following theorems [7].
$w_{\varphi^{-}}$

Theorem 4.3. Let$l\in \mathbb{C}_{+},$$E^{anw_{\lambda}}[F|X_{n+1}](\vec{\xi}_{n+1})$

(4.5)

and
exists for$X_{n+1}$$F\in S_{w_{\varphi}}$

be given by (2.1) and (3.1), respectively. Then, forand it is given by$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$E^{anw_{\lambda}}[F|X_{n+1}](\vec{\xi}_{n+1})=E^{anw_{\lambda}}[F_{\vec{\xi}_{n+1}}]$$= \int_{L_{2}[0,t]}\exp\{-\frac{1}{2\lambda}\Vert v\Vert^{2}\}d(\sigma_{\vec{\xi}_{n+1}}o(\mathcal{P}^{\perp})^{-1})(v)$

where
$q,$

$\sigma_{\vec{\xi}_{n+1}}$

and$F_{\vec{\xi}_{n+1}}$$E^{anf_{q}}[F|X_{n+1}](\vec{\xi}_{n+1})$

Theorem 4.4. Let$\lambda\in \mathbb{C}_{+},$are given by (4.3) and (4.4), respectively. Moreover, for a non-zem realis given by (4.5) replacing$F\in S_{w_{\varphi}}$$E^{anw_{\lambda}}[F|X_{n}](\vec{\xi}_{n})$

(4.6)

,

and

exists for$X_{n}$$t$by-iq.

be given by (2.1) and (3.2), respectively. Then, forand it is given by$\vec{\xi}_{n}\in \mathbb{R}^{n+1}$$E^{anw_{\lambda}}[F|X_{n}]( \vec{\xi}_{n})=\int_{L_{2}[0,t]}\exp\{-\frac{1}{2\lambda}[\Vert \mathcal{P}^{\perp}v\Vert^{2}+(t-t_{n})[(\mathcal{P}v)(t)]^{2}]\}d\sigma_{\tilde{\xi}_{n}}(v)$whereis given by (4.2). Moreover, for a non-zem real
replacing by-iq.
$\sigma_{\vec{\xi}_{n}}$$q,$$E^{anf}q[F|X_{n}](\vec{\xi}_{n})$,is given by (4.6)$\lambda$

Corollary 4.5. Let
and be related by (2.1) and suppose thaton . Then for a non-zem real , we have
$F\in S_{w_{\varphi}}$$\sigma$$\sigma$

is concentmted$V^{\perp}$$q$$E^{anf_{q}}[F|X_{n}]( \vec{\xi}_{n})=E^{anf_{q}}[F|X_{n+1}](\vec{\xi}_{n+1})=E^{anf}q[F]=\int_{L_{2}[0,t]}\exp\{\frac{1}{2qi}\Vert v\Vert^{2}\}d\sigma(v)$for$a.e.\vec{\xi}_{n}\in \mathbb{R}^{n+1}$

and for$a.e.\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$.

Remark. Because $L_{2}[0,t]=V\oplus V^{\perp}$ andtrivial. For $k=1,2,$ , let$V$

is of finite dimensional, the space$V^{\perp}$

is non-$\cdots$$h_{k}(t)= \frac{(-1)^{j+1}2^{2k-1}}{(t_{j}-t_{j-1})^{2k-1}}(t-\frac{t_{j-1}+t_{j}}{2})^{2k-1}$

Then we haveinso thatdimensional.
$V^{\perp}$$t_{j-1}\leq t\leq t_{j}$$(j=1, \cdots,n+1)$ .

for $k=1,2,$ . Indeed, every finite linear combination of theis at least uncountable and of infinite dimensional since $L_{2}[0,t]$ is of infinite$\mathcal{P}h_{k}=0$$V^{\perp}$

if$\cdots$$h_{k}sa\infty$

129
Throughout this paper, letbe an orthonormal subset of $L_{2}[0,t]$ such thatare independent. Note that such an orthonormal set can be obtained fromtheas given in the above remark. Let
be the orthonormal set obtained fromby the Gram-Schmidt orthonormalization process. Now, for $k=1,$, let$\{v_{1},v_{2}, \cdots,v_{l}\}$$\{\mathcal{P}^{\perp}v_{1}, \cdots, \mathcal{P}^{\perp}v_{l}\}$$h_{k}s$$\{e_{1}, \cdots,e_{l}\}$$\{\mathcal{P}^{\perp}v_{1}, \cdots, \mathcal{P}^{\perp}v_{l}\}$$\cdots,l$$\mathcal{P}^{\perp}v_{k}=\sum_{j=1}^{l}\alpha_{kj}e_{j}$be the linear combinations of the(4.7)$e_{j}s$and let

$A=\{\begin{array}{llll}\alpha_{11}\alpha_{12} \cdots \cdots \alpha_{ll}\alpha_{2l}\alpha_{22} \cdots \cdots \alpha_{2l}\vdots \vdots\ddots \vdots\vdots \vdots \ddots \vdots\alpha_{ll} \alpha_{l2} \cdots \alpha_{ll}\end{array}\}$be the coefficient matrix of the combinations. We can also regard:
given by$A$

as the linear transformation$\mathbb{R}^{l}arrow \mathbb{R}^{l}$$T_{A}$(4.8)$T_{A}(\vec{z})=\vec{z}A$

where is arbitrary row-vector inand$\vec{z}$$\mathbb{R}^{l}$,. Note that is invertible so that$A$$\vec{\xi}_{n+1}=(\xi_{0},\xi_{1}, \cdots,\xi_{n},\xi_{n+1})\in \mathbb{R}^{n+2}$$\vec{\xi}_{n}=(\xi_{0},\xi_{1}, \cdots,\xi_{n})\in \mathbb{R}^{n+1}$$\vec{v}_{\vec{\xi}_{n}}=((\mathcal{P})(t_{j})(\xi_{j}-\xi_{j-1}),$$T_{A}$

is an isomorphism. For, let$\cdots,$ $\sum_{j=1}^{n}(\mathcal{P}v_{l})(t_{j})(\xi_{j}-\xi_{j-1}))$and$\vec{v}_{\vec{\xi}_{n+1}}=((vl, [\vec{\xi}_{n+1}]), \cdots,(v_{l}, [\vec{\xi}_{n+1}]))$.

Further, for $s\in[0,t]$ let
(4.9)

For

$\rho\in \mathcal{M}(\mathbb{R}^{l})$$(\mathcal{P}\vec{v})(s)=((\mathcal{P}v_{1})(s), \cdots, (\mathcal{P}_{l}\ddagger))(s))$, let$\rho_{\vec{\xi}_{n+1}}$

and$\rho_{\tilde{\xi}_{n}}$be the complex Borel measures on$\frac{d\rho_{\tilde{\xi}_{n+1}}}{d\rho}(\gamma z=\exp\{i\{\vec{v}_{\vec{\xi}_{n+1}},\vec{z}\rangle\}$and(4.10).

$\frac{d\rho_{\vec{\xi}_{n}}}{d\rho}(\gamma z=\exp\{i\{\vec{v}_{\vec{\xi}_{n}},\vec{z}\}\}$.

$\mathbb{R}^{l}$defined by

130
Now we define$\psi_{\xi_{n+1}}\in\hat{M}(\mathbb{R}^{l})$by$\phi_{\vec{\xi}_{n+1}}(u\gamma=\int_{\mathbb{R}^{l}}\exp\{i\{\vec{u},\tilde{z})\}z$

where$T_{A}^{-1}$is the inverse transformation of(4.11)

for$T_{A}$,, and define$\Phi_{\vec{\xi}_{n+1}}(x)=\phi_{\vec{\xi}_{n+1}}((e1,x), \cdots,(e_{l},x))$$w_{\varphi}-a.e$. $x\in C[0,t]$ . We now have the following theorems.

be given by (2.3) and let $\Phi(x)=\phi((v_{1},x), \cdots, (v_{l},x))$ for
Theorem 4.6. Let$a.e$ . $x\in C[0,t]$ . Then we have for$\phi\in\hat{M}(\mathbb{R}^{l})$$w_{\varphi^{-}}$$\lambda\in \mathbb{C}_{+}$(4.12)$E^{anw}{}^{t}[ \Phi|X_{n+1}](\vec{\xi}_{n+1})=E^{anw_{\lambda}}[\Phi_{\vec{\xi}_{n+1}}]=\int_{\mathbb{R}^{l}}\exp\{-\frac{1}{2\Lambda}\Vert\vec{z}\Vert^{2}\}Z$, where$for\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$

a non-zem real$q,$

$T_{A}$

and$\Phi_{\tilde{\xi}_{n+1}}$

are given by (4.8) and (4.11), respectively. Moreover, for$E^{anf}q[\Phi|X_{n+1}](\vec{\xi}_{n+1})$

is given by (4.12) replacing$j\downarrow$by-iq.

Theorem 4.7. Let the assumptions and notations be as given in Theorem 4.6. Then we havefor$\lambda\in \mathbb{C}_{+}$(4.13)$E^{anw_{\lambda}}[ \Phi|X_{n}](\vec{\xi}_{n})=\int_{\mathbb{R}^{l}}\exp\{-\frac{1}{2\lambda}[\Vert z\Vert^{2}+(t-t_{n})\langle(\mathcal{P}\vec{v})(t),\vec{z}\rangle^{2}]\}d\rho_{\vec{\xi}_{n}}(z\gamma$, wherefor
non-zem real$\vec{\xi}_{n}\in \mathbb{R}^{n+1}$$q,$$\mathcal{P}\vec{v}$and$\rho_{\vec{\xi}_{n}}$$E^{anf}q[\Phi|X_{n}](\vec{\xi}_{n})$are given by (4.9) and (4.10), respectively. Moreover, for a

is given by (4.13) replacing$\Lambda$

by-iq.

Corollary 4.8. Let the assumptions and notations be as given in Theorem 4.7 and suppose$k=1,$. Then for a non-zem real , we havethat$v_{k}\in V^{\perp}for$$\cdots,l$$q$$E^{anf_{q}}[ \Phi|X_{n}](\vec{\xi}_{n})=E^{anf}q[\Phi|X_{n+1}](\vec{\xi}_{n+1})=E^{anf_{q}}[\Phi]=\int_{\mathbb{R}^{l}}\exp\{\frac{1}{2qi}\Vert\vec{z}\Vert^{2}\}d\rho(z\gamma$

for$a.e.\vec{\xi}_{n}\in \mathbb{R}^{n+1}$

and for$a.e.\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$.

For convenience, we introduce useful notations from the Gram-Schmidt orthonormalization
process. For $v\in L_{2}[0,t]$ , we obtain an orthonormal set $\{e_{1}, \cdots, e_{l},e_{l+1}\}$ as follows; let(4.14)

for $j=1,$$c_{j}(v)=\{$ $\langle v,e_{j}\}\sqrt{||v||^{2}-\sum_{k=1}^{l}\langle v,e_{k}\}^{2}}$$\cdots,l$for $j=l+1$

and$e_{l+1}= \frac{1}{c_{l+1}(v)}[v-\sum_{j=1}^{l}c_{j}(v)e_{j}]$131
if $c_{l+1}(v)\neq 0$ . Then we have
(4.15)$v= \sum_{j=1}^{l+1}c_{j}(v)e_{j}$and$\Vert u\Vert^{2}=\sum_{j=1}^{l+1}[c_{j}(v)]^{2}$.

Note that the equalities in (4.15) hold trivially for the case $c_{l+1}(v)=0$ . Using the above notations
we have the following theorem.

Theorem 4.9. Let the assumptions and notations be as given in Theorems 4.3 and 4.6.
let
.
for
$x\in C[0,t]$ , whereandare given by (4.4) and (4.11), respectively. Then for
we have
Further, let $\Psi(x)=F(x)\Phi(x)$ and for$F_{\vec{\xi}_{n+1}}$(4. 16)$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$\Psi_{\vec{\xi}_{n+1}}(x)=F_{\vec{\xi}_{n+1}}(x)\Phi_{\vec{\xi}_{n+1}}(x)$$w_{\varphi}-a.e$$\lambda\in \mathbb{C}_{+}$$\Phi_{\vec{\xi}_{n+1}}$$E^{anw_{\lambda}}[ \Psi|X_{n+1}](\vec{\xi}_{n+1})=E^{anw_{1}},[\Psi_{\vec{\xi}_{n+1}}]=\int_{L_{2}[0,t]}\int_{\mathbb{R}^{l}}\exp\{-\frac{1}{2\lambda}[\Vert v\Vert^{2}+2\{\vec{c}(v)$,$\vec{Z}\rangle+\Vert_{\vec{Z}}\Vert^{2}]\}Z$

for
non-zem real$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$q,$

are given by (4.14). Further, for a
is given by (4.16) replacing by-iq.

, where $\vec{c}(v)=(c_{1}(v), \cdots,c_{l}(v))$ and the$E^{anf_{q}}[\Psi|X_{n+1}](\vec{\xi}_{n+1})$$c_{j}s$$\lambda$

Theorem 4.10. Let the assumptions and notations be as given in Theorems 4.4, 4.7 and 4.9.
Then forhave$\lambda\in \mathbb{C}_{+}we$(4.17)$E^{anw_{\lambda}}[ \Psi|x_{n}](\vec{\xi}_{n})=\int_{L_{2}[0,t]}\int_{\mathbb{R}^{l}}\exp\{-\frac{1}{2\lambda}[\Vert \mathcal{P}^{\perp}v\Vert^{2}+2\{zZ$$+(t-t_{n})[(\mathcal{P}v)(t)+\langle v(t),\vec{z}\}]^{2}]\}d\rho_{\tilde{\xi}_{n}}(z\gamma d\sigma_{\vec{\xi}_{n}}(v)$$for\vec{\xi}_{n}\in \mathbb{R}^{n+1}$, wherefor a non-zero real$q,$$E^{anf}q[\Psi|X_{n}](\vec{\xi}_{n})$Let $1\leq p\leq\infty$ and$F_{l}$$w_{\varphi}-a.e$(4.19)$c_{j}s$$\lambda$

be given by(4.18)

forand theare given by (4.14). Further,is given by (4.17) replacing by-iq.

$\vec{c}(\mathcal{P}^{\perp}v)=(c_{1}(\mathcal{P}^{\perp}v), \cdots,c_{l}(\mathcal{P}^{\perp}v))$$F_{l}(x)=f((v_{1},x), \cdots,(v_{l},x))$. $x\in C[0,t]$ , where$f\in L_{p}(\mathbb{R}^{l})$. For$\vec{\xi}_{n}\in \mathbb{R}^{n+1}$and$f_{\vec{\xi}_{n}}(u\gamma=f(\mathcal{T}_{A}(u\gamma+\vec{v}_{\vec{\xi}_{n}})$and(4.20)$f_{\vec{\xi}_{n+1}}(u\gamma=f(\mathcal{T}_{A}(\vec{u})+\vec{v}_{\vec{\xi}_{n+1}})$,$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$

let

132
. Note that means any row-vector
where is the linear transformation given byis the transpose of the matrix given by (4.7). Moreover letin and$\mathcal{T}_{A}(u\gamma=\vec{u}A^{T}$$\mathcal{T}_{A}$$\mathbb{R}^{l}$$A^{T}$$A$(4.21)

for$w_{\varphi^{-}}$$\vec{u}$$F_{l,\vec{\xi}_{n+1}}(x)=f_{\vec{\xi}_{n+1}}((e_{1},x), \cdots,(e_{l},x))$a.e. $x\in C[0,t]$ . Using the above notations, we have the following theorems [7].

Theorem 4.11. Let $1\leq p\leq\infty$ and(4.22)$F_{l}$

be given by (4.18). Then for$\lambda\in \mathbb{C}_{+}we$have$E^{anw_{\lambda}}[F_{l}|X_{n+1}](\vec{\xi}_{n+1})=E^{anw_{\lambda}}[F_{l,\tilde{\xi}_{n+1}}]$$=( \frac{\lambda}{2\pi})^{l}2\int_{\mathbb{R}^{l}}f_{\vec{\xi}_{n+1}}(u\gamma\exp\{-\frac{\lambda}{2}\Vert\vec{u}\Vert^{2}\}dm_{L}^{l}(u\gamma$

for$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$$p=1$ , then, where$f_{\vec{\xi}_{n+1}}$

andfor a non-zem real$q,$

$F_{l,\tilde{\xi}_{n+1}}$

are given by (4.20) and (4.21), respectively. Further,$E^{anf_{q}}[F_{l}|X_{n+1}](\vec{\xi}_{n+1})$

is given by (4.22) replacing$\lambda$$\iota f$

by-iq.

Theorem 4.12. Let the assumptions and notations be as given in Theorem 4.11. Then forhave$\lambda\in \mathbb{C}_{+}we$(4.23)$E^{anw_{\lambda}}[F_{l}|X_{n}]( \vec{\xi}_{n})=(\frac{\lambda}{2\pi})^{l}2[\frac{1}{1+(t-t_{n})\Vert \mathcal{T}_{A}^{-1}((\mathcal{P}v\gamma(t))\Vert^{2}}]^{1}2\int_{\mathbb{R}^{l}}f_{\vec{\xi}_{n}}(u\gamma$$\cross\exp\{-\frac{\lambda}{2}[\Vert\vec{u}\Vert^{2}-\frac{(t-t_{n})\langle \mathcal{T}_{A}^{-1}((P\vec{v})(t)),\vec{u}\rangle^{2}}{1+(t-t_{n})||\mathcal{T}_{A}^{-1}((\mathcal{P}v\gamma(t))\Vert^{2}}]\}dm_{L}^{l}(u\gamma$

for$\vec{\xi}_{n}\in \mathbb{R}^{n+1}$, where$P\tilde{v}$then for a non-zem real

Corollary 4.13. Let
we have forand$f_{\vec{\xi}_{n}}$are given by (4.9) and (4.19), respectively. Further, $\iota fp=1$ ,$E^{anf_{q}}[F_{l}|X_{n}](\vec{\xi}_{n})$$q,$$F_{l}$

is given by (4.23) replacing

be given by (4.18) and suppose that$\lambda$

by-iq.

$v_{k}\in V^{\perp}for$$k=1,$ $\cdots,l$. Then$\lambda\in \mathbb{C}_{+}$$E^{anw_{\lambda}}[F_{l}|X_{n}](\vec{\xi}_{n})=E^{anw_{\lambda}}[F_{l}|X_{n+1}](\vec{\xi}_{n+1})$$=E^{anw_{\lambda}}[F_{l}]=( \frac{\lambda}{2\pi})^{l}2\int_{\mathbb{R}^{l}}f(u\gamma\exp\{-\frac{\lambda}{2}\Vert\vec{u}\Vert^{2}\}dm_{L}^{l}(u\gamma$

for$a.e.\vec{\xi}_{n}\in \mathbb{R}^{n+1}$

and for$a.e.\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}.$

Funher, $\iota fp=1$ , then for a non-zem real we have$E^{anf_{q}}[F_{l}|X_{n}](\vec{\xi}_{n})=E^{anf_{q}}[F_{l}|X_{n+1}](\vec{\xi}_{n+1})$.

$=E^{anf_{q}}[F_{l}]=( \frac{q}{2\pi i})^{\frac{l}{2}}\int_{\mathbb{R}^{l}}f(u\gamma\exp\{\frac{qi}{2}\Vert\vec{u}\Vert^{2}\}dm_{L}^{l}(u\gamma$

133
Theorem 4.14. Let $G_{l}=FF_{l}$ , where andMoreover, forlet(4.21),
andrespectively. Then we have for$F$$F_{l}$$\vec{\xi}_{n+1}\in \mathbb{R}^{n+2}$

are given by (2.1) and (4.18), respectively.
where
and
are given by (4.4)$G_{l,\vec{\xi}_{n+1}}=F_{\vec{\xi}_{n+1}}F_{l,\vec{\xi}_{n+1}’}$$F_{\vec{\xi}_{n+1}}$$F_{l,\vec{\xi}_{n+1}}$$\lambda\in \mathbb{C}_{+}$(4.24)$E^{anw_{\Lambda}}[G_{l}|X_{n+1}](\vec{\xi}_{n+1})=E^{anw_{\lambda}}[G_{l,\tilde{\xi}_{n+1}}]$$=( \frac{\lambda}{2\pi})^{l}2\int_{L_{2}[0,t]}\int_{\mathbb{R}^{l}}f_{\vec{\xi}_{n+1}}(u\gamma\exp\{\frac{1}{2\lambda}[\sum_{j=1}^{l}[\lambda iu_{j}$$+c_{j}(v)]^{2}-\Vert v\Vert^{2}]\}dm_{L}^{l}(u\gamma_{d(\sigma_{\vec{\xi}_{n+1}}\circ(\mathcal{P}^{\perp})^{-1})(v)}$

where

, and

,

are given by (4.3), (4.14) and (4.20), respectively. Furthemore, $\iota fp=1$ , then for any non-zem realis given by(4.24) replacing by-iq.
$\vec{u}=(u_{1}, \cdots,u_{l})$$\sigma_{\vec{\xi}_{n+1}’}c_{j}(v)$and$f_{\vec{\xi}_{n+1}}$$E^{anf_{q}}[G_{l}|X_{n+1}](\vec{\xi}_{n+1})$$q,$$\lambda$

Theorem 4.15. Let the assumptions and notations be given as in Theorems 4.12 and 4.14.
Then forhave$\lambda\in \mathbb{C}_{+}we$(4.25)$E^{anw_{\lambda}}[G_{l}|X_{n}](\vec{\xi}_{n})$$=( \frac{\lambda}{2\pi})^{l}2[\frac{1}{1+(t-t_{n})\Vert \mathcal{T}_{A}^{-1}((\mathcal{P}\vec{v})(t))\Vert^{2}}]^{1}z\int_{l_{Q}[0,t]}\int_{\mathbb{R}^{l}}f_{\vec{\xi}_{n}}(\vec{u})\exp\{\frac{1}{2\lambda}$$\cross[\sum_{j=1}^{l}[\lambda iu_{j}+c_{j}(\mathcal{P}^{\perp}v)]^{2}-\Vert \mathcal{P}^{\perp}v\Vert^{2}+\frac{t-t_{n}}{1+(t-t_{n})||\mathcal{T}_{A}^{-1}((\mathcal{P}\vec{u})(t))\Vert^{2}}$$\cross[i[(\mathcal{P}v)(t)-\langle\vec{c}(\mathcal{P}^{\perp}v),\mathcal{T}_{A}^{-1}((\mathcal{P}^{\neg}v)(t))\}]+\lambda\langle \mathcal{T}_{A}^{-1}((\mathcal{P}\vec{v})(t)),\vec{u}\}]^{2}]\}dm_{L}^{l}u\gamma d\sigma_{\vec{\xi}_{n}}(v)$for$\vec{\xi}_{n}\in \mathbb{R}^{n+1}$$p=1$ , then, where$\iota f$$\sigma_{\vec{\xi}_{n}}$for a non-zem real

Remark. Since
then we haveandis given by (4.2). Further,is given by (4.25) replacing by-iq.

$\vec{c}(\mathcal{P}^{\perp}v)=(c_{1}(\mathcal{P}^{\perp}v), \cdots,c_{l}(\mathcal{P}^{\perp}v))$$q,$$E^{anf_{q}}[G_{l}|X_{n}](\vec{\xi}_{n})$$\lambda$,
is concentrated on. If, thenfor $k=1,$
andso that. Further, the coefficient matnix given by (4.7) is
the identity matrix and hence
which implies. In each case, we canobtain more simple equations in each theorem of this section.
$\mathcal{P}^{\perp}v=v$$\sigma_{\vec{\xi}_{n+1}}=\sigma_{\vec{\xi}_{n}}=\sigma$$\mathcal{P}v_{k}=0$$\mathcal{P}^{\perp}v_{k}=v_{k}$

, if
for
so that we also haveand$\mathcal{P}v=0$$v\in V^{\perp}$$F_{\vec{\xi}_{n+1}}=F$$v_{k}\in V^{\perp}$$(\mathcal{P}\vec{u})(t)=\vec{0}$$f_{\vec{\xi}_{n+1}}=f_{\vec{\xi}_{n}}=f$$V^{\perp}$$\sigma\in \mathcal{M}(L_{2}[0,t])$$\cdots,l$$A$$F_{l,\vec{\xi}_{n+1}}=F_{l}$\S 5. Time-dependent and time-independent conditional Feynman integrals
Throughout this section, let $0=t_{0}<t_{1}<\cdots<t_{n-1}<t_{n}=t$ be a partition of $[0,t]$ , let$X_{t}(x)=(x(t_{0}), \cdots,x(t_{n-1}),x(t_{n}))$ and $Y_{t}(x)=(x(t_{0}), \cdots,x(t_{n-1}))$ for $x\in C^{r}[0,t]$ . Our first theorem
in this section is a time-dependent version of Theorem 2.7 above; its proof is included in [9].

134
Theorem 5.1. Let anddescribed in Theorem 2.7, where$\eta=\mu+\sum_{l=1}^{n}\sum_{j=1}^{r_{l}}w_{l,j}\delta_{p_{l,j}}$$F_{k}$

be the function and the measure as

for all $(l,j)$ and $0=t_{0}<p_{1,1}<p_{1,2}<\cdots<p_{1,r_{1}}<$$t_{1}<p_{2,1}<\cdots<p_{2,r_{2}}<t_{2}<\cdots<t_{n-1}<p_{n,1}<\cdots<p_{n,r_{n}}<t_{n}=t$ . Then for$w_{l,j}\in \mathbb{C}$$\lambda\in \mathbb{C}_{+}and$$\vec{\xi}_{n+1}\in \mathbb{R}^{(n+1)r},$$E^{anw_{\lambda}}[F_{k}|X_{t}](\vec{\xi}_{n+1})$(5.1)

is given by$E^{anw_{\lambda}}[F_{k}|X_{t}]( \vec{\xi}_{n+1})=k!\sum_{q_{1}+\cdots+q_{n}=k}\prod_{l=1}^{n}A(l,\lambda,\vec{\xi}_{n+1},q_{l})$where$A(l,\lambda,\vec{\xi}_{n+1},q_{l})$$= \sum_{m_{l,0}+m_{l,1}+\cdots+m_{l,r_{l}}=q_{l}}(\prod_{j=1}^{r_{l}}\frac{w_{l,j}^{m_{l,j}}}{m_{l,j}!})\sum_{j_{0}+j_{1}+\cdots+j_{r_{l}}=m_{l,0}}\int_{\Delta_{m_{l,0},j_{0},\cdots,jr_{l}}}\int_{\mathbb{R}^{q_{l’}}}$$\exp\{i\sum_{u=0}^{r_{l}j_{u}}\sum_{v=1}^{+1}([\vec{\xi}_{n+1}](s_{l,u,v}),\tilde{v}_{l,u,v}\}-\frac{1}{2\lambda}\sum_{u=0}^{r_{l}j_{u}}\sum_{v=1}^{+1}(s_{l,u,v}-s_{l,u,v-1})\Vert\sum_{\beta=u+1}^{r_{l}}$$\sum_{\gamma=1}^{j_{\beta}+1}\frac{t_{l}-s_{l\beta,\gamma}}{t_{l}-t_{l-1}}\vec{v}_{l\beta,\gamma}+\sum_{\gamma=v}^{j_{u}+1}\frac{t_{l}-s_{l,u,\gamma}}{t_{l}-t_{l-1}}\vec{v}_{l,u,\gamma}+\sum_{\gamma=1}^{v-1}\frac{t_{l-1}-s_{l,u,\gamma}}{t_{l}-t_{l-1}}\vec{v}_{l,u,\gamma}+\sum_{\beta=0}^{u-1}$$\sum_{\gamma=1}^{j_{\beta}+1}\frac{t_{l-1}-s_{l\beta,\gamma}}{t_{l}-t_{l-1}}\vec{v}_{l\beta,\gamma}\Vert^{2}\}d(\prod_{u=0v}^{r_{l}}\prod_{=1}^{j_{u}}\sigma_{s_{l,u,v}}\cross\prod_{u=1}^{r_{l}}\sigma_{p_{lu}^{l,u}}^{m},)Cv_{l},\vec{h}_{l})d\mu^{n_{l,0}}(\vec{s}_{l})$with$s_{l,0,0}=t_{l-1}$,$s_{l,u,0}=p_{l,u}=s_{l,u-1,j_{u-1}+1}$

for $u=1,$$\vec{s}_{l}=(s_{l,0,1}, \ldots,s_{l,0,j_{0}}, \ldots,s_{l,r_{l},1}, \ldots,s_{l,r_{l},j_{r_{l}}})$$\cdots,r_{l}$,$s_{l,r_{l},j_{r_{l}}+1}=t_{l}$,,$\Delta_{m_{l,0};j_{0},\cdots,j_{r_{l}}}=\{\vec{s}_{l}:t_{l-1}<s_{l,0,1}<\cdots<s_{l,0,j_{0}}<p_{l,1}<s_{l,1,1}<\cdots<s_{l,1,jl}<p_{l,2}$$<\cdots<p_{l,r_{l}}<s_{l,r_{l},1}<\cdots<s_{l,r_{l},j_{r_{l}}}<t_{l}\}$,,$\vec{v}_{l}=(\neg v_{l,0,1}, \ldots,\vec{v}_{l,0,j_{0}},\vec{v}_{l,1,1}, \cdots,\vec{v}_{l,1,j_{1}}, \ldots,\vec{v}_{l,r_{l},1}, \ldots,\vec{v}_{l,r_{l},j_{r_{l}}})$$\vec{v}_{l,r_{l},j_{r_{l}}+1}=\vec{0}\in \mathbb{R}^{r}$,;$\vec{h}_{l}=(\vec{h}_{l,1,1}, \ldots,\vec{h}_{l,1,m_{l,1}},\vec{h}_{l,2,1}, \ldots,\vec{h}_{l,2,m_{l,2}}, \ldots,\vec{h}_{l,r_{l},1}, \ldots,\vec{h}_{l,r_{l},m_{l,r_{l}}})$$\vec{v}_{l,u-1,j_{u-1}+1}=\sum_{v=1}^{m_{l,u}}\vec{h}_{l,u,v}$

for $u=1,$

Funhemore, for nonzem real

replacing$\lambda$$q,$$\ldots,r_{l}$.

$E^{anf_{q}}[F_{k}|X_{t}](\vec{\xi}_{n+1})$

is given by the right hand side of (5.1)

by-iq.

By the same method as used in the proof of Theorem 2.11, we can prove the following
theorem.

135
Theorem 5.2. Let the assumptions and notations be as given in Theorem 5. 1, and let be
as given in Theorem 2.11. Then for nonzem real andexists$F$$E^{anf_{q}}[F|X_{t}](\vec{\xi}_{n+1})$$\vec{\xi}_{n+1}\in \mathbb{R}^{(n+1)r},$$q$

and it is given by$E^{anf_{q}}[F|X_{t}]( \vec{\xi}_{n+1})=1+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[F_{k}|X_{t}](\vec{\xi}_{n+1})$

where$E^{anf_{q}}[F_{k}|X_{t}](\vec{\xi}_{n+1})$Let$\mathbb{R}^{(n+1)r}$is as given in Theorem 5.1.

$\psi(u\gamma=\int_{\mathbb{R}^{r}}\exp\{i\langle\vec{u},\vec{v}\}\}dvCv)$

and

for

$v\in \mathcal{M}(\mathbb{R}^{r})$. Then we have for$\vec{\xi}_{n+1}=(\vec{\xi}_{0},\vec{\xi}_{1}, \cdots,\vec{\xi}_{n})\in$$\lambda>0$(5.2)$\psi(A^{-z}(x(t)-1[x](t))+[\vec{\xi}_{n+1}](t))=\psi(\vec{\xi}_{n})$.

By (5.2) and Theorem 5.2, we have the following theorem.

Theorem 5.3. Let
and be as given in Theorem 2.12. Then, under the assumptions
as given in Theorem 5.2, we have for nonzem real and$G$$G_{k}$$q$$\vec{\xi}_{n+1}=(\vec{\xi}0,\vec{\xi}_{1}, \cdots,\vec{\xi}_{n})\in$$\mathbb{R}^{(n+1)r}andnotations$$E^{anf_{q}}[G_{k}|X_{t}](\vec{\xi}_{n+1})=\psi(\vec{\xi}_{n})E^{anf_{q}}[F_{k}|X_{t}](\vec{\xi}_{n+1})$

and$E^{anf_{q}}[G|X_{t}]( \vec{\xi}_{n+1})=\psi(\vec{\xi}_{n})E^{anf_{q}}[F|X_{t}](\vec{\xi}_{n+1})=\psi(\vec{\xi}_{n})+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[G_{k}|X_{t}](\vec{\xi}_{n+1})$.

Now we obtain a time-independent version of Theorem 2.7. To do his, we need the following
lemma [9].

Lemma 5.4. For$\lambda>0,$$\vec{v}\in \mathbb{R}^{r}$and$\vec{\xi}=(\vec{\xi}_{0},\vec{\xi}_{1}, \cdots,\vec{\xi}_{n-1})\in \mathbb{R}^{nr}$let$\Psi(n,\lambda,\vec{v},\vec{\xi})=[\frac{\lambda}{2\pi(t-t_{n-1})}]^{r}z\int_{\mathbb{R}^{r}}\exp\{i\sum_{u=0}^{r_{n}j_{u}}\sum_{v=1}^{+1}\langle[(\vec{\xi},\vec{\xi}_{n})](s_{n,u,v})$,$\vec{v}_{n,u,v}\}+i\{\vec{\xi}_{n},\vec{v}\rangle-\frac{\lambda}{2}\frac{\Vert\vec{\xi}_{n}-\vec{\xi}_{n-1}\Vert^{2}}{t-t_{n-1}}\}dm_{L}^{r}(\vec{\xi}_{n})$where$(\vec{\xi},\vec{\xi}_{n})=(\vec{\xi}_{0},\vec{\xi}_{1}, \cdots,\vec{\xi}_{n-1},\vec{\xi}_{n})\in \mathbb{R}^{(n+1)r},$$0,1,$ $\cdots,r_{n};v=1,2,$ $\cdots,j_{u}+1$(5.3)$t_{n-1}\leq s_{n,u,v}\leq t_{n}=t$and$\vec{v}_{n,u,v}\in \mathbb{R}^{r}$. Then we have$\Psi(n,A,\vec{v},\vec{\xi})=\exp\{i\{\vec{\xi}_{n-1},\vec{v}+\sum_{u=0}^{r_{n}j_{u}}\sum_{v=1}^{+1}\vec{v}_{n,u,v}\}$$- \frac{t-t_{n-1}}{2\lambda}\Vert\vec{v}+\sum_{u=0}^{r_{n}j_{u}}\sum_{v=1}^{+1}\frac{s_{n,u,v}-t_{n-1}}{t-t_{n-1}}\vec{v}_{n,u,v}\Vert^{2}\}$.

for $u=$136
Theorem 5.5. Let the assumptions and notations be as given in Theorem 5.1. Then foris given byand$\vec{\xi}=(\vec{\xi}_{0},\vec{\xi}_{1}, \cdots,\vec{\xi}_{n-1})\in \mathbb{R}^{nr},$ $E^{anw_{\lambda}}[F_{k}|Y_{t}](\vec{\xi})$$\lambda\in \mathbb{C}_{\dagger}$$E^{anw_{\lambda}}[F_{k}| Y_{t}](\vec{\xi})=k!\sum_{q_{1}+\cdots+q_{n}=k}[\prod_{l=1}^{n-1}A(l,\lambda,\vec{\xi},q_{l})]B(n,\lambda,\vec{0},\vec{\xi},q_{n})$where for$\vec{v}\in \mathbb{R}^{r},$$B(n,\lambda,\vec{v},\vec{\xi}, q_{n})$is given by the expression $ofA(n,l,\vec{\xi}, q_{n})$ replacing$\exp\{i\sum_{u=0}^{r_{n}j}\sum_{v=1}^{u+1}\langle[\vec{\xi}](s_{n,u,v}),\vec{v}_{n,u,v}\}\}$

which is given by (5.3). Funhemore, for a nonzem real
given by the above expression replacing by-iq.
by$\Psi(n,\lambda,\vec{v},\vec{\xi})$$q,$$E^{an[q}[F_{k}|Y_{t}](\vec{\xi})$

is$\lambda$

Theorem 5.6. Let the assumptions and notations be as given in Theorem 5.5 and let $F(x)=$for $x\in C^{r}[0,t]$ . Then for nonzem real andexists and it is given by$\exp\{\int_{0}^{t}\theta(s,x(s))d\eta(s)\}$$q$$\vec{\xi}\in \mathbb{R}^{nr},$$E^{anf}q[F|Y_{t}](\vec{\xi})$$E^{an\int_{q}}[F| Y_{t}](\vec{\xi})=1+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[F_{k}|Y_{t}](\vec{\xi})$

where$E^{anf_{q}}[F_{k}|Y_{l}](\vec{\xi})$

is as given in Theorem 5.5.

Theorem 5.7. Under the assumptions and notations as given in Theorems 5.3, 5.5 and 5.6,
we have for nonzem real and$\vec{\xi}=(\vec{\xi}0,\vec{\xi}_{1}, \cdots,\vec{\xi}_{n-1})\in \mathbb{R}^{nr}$$q$$E^{anf_{q}}[G_{k}| Y_{t}](\vec{\xi})=k!\sum_{q_{1}+\cdots+q_{n}=k}[\prod_{l=1}^{n-1}A(l, -iq,\vec{\xi},q_{l})]\int_{\mathbb{R}^{r}}B(n, -iq,\vec{v},\vec{\xi},q_{n})dvCv)$

and$E^{anf_{q}}[G| Y_{t}](\vec{\xi})=\int_{\mathbb{R}^{r}}\exp\{i\langle\vec{\xi}_{n-1},\vec{v}\}+\frac{t-t_{n-1}}{2qi}\Vert\vec{v}\Vert^{2}\}dvCv)+\sum_{k=1}^{\infty}\frac{1}{k!}E^{anf_{q}}[G_{k}|Y_{t}](\vec{\xi})$.

and are defined on the r-dimensional Wiener space, then we
If
Remarkincan obtain the same results in Theorems 5.1, 5.2, 5.3, 5.5, 5.6 and 5.7 withand .
the expressions of, we can obtain more simple expressions in Theorems
If $\eta=\mu$ or
5.1, 5.2, 5.3, 5.5, 5.6 and 5.7.
, we can obtain all the results including in
If some of theare in the set
Theorem 5.1 with minor modifications.
are , then, using (2.5), we can show
and some of the
Ifin Theorem 5.7 exist.
in Theorem 5.3 andthat$\bullet$$G_{k},$$F_{k},$$F$$G$$\vec{\xi}_{0}=\vec{0}\in \mathbb{R}^{r}$$\vec{\xi}_{n+1}$$\bullet$$\bullet$$\bullet$$\vec{\xi}$$\eta=\sum_{l=1}^{n}\sum_{j=1}^{r_{l}}w_{l,j}\delta_{p_{l,j}}$$p_{l,j}s$$\{t_{0},t_{1}, \cdots,t_{n}\}$$\eta=\mu+\sum_{l=1}^{n}\sum_{j=1}^{r_{l}}w_{l,j}\delta_{p_{l,j}}$$E^{anf_{q}}[G|X_{t}]$$\eta$$r_{l}s$$E^{anf_{q}}[G|Y_{t}]$$\infty$

137\S 6. Application to an integral equation
In this section, we present a solution of an integral equation including the integral equation
which is formally equivalent to the Schr\"odinger partial differential equation.
Let $t\in(O,\infty)$ and, oobe the function given by$\theta:[0$$)\cross \mathbb{R}^{r}arrow \mathbb{C}$(6.1)$\theta(s,\vec{u})=\int_{\mathbb{R}^{r}}\exp\{i\{\vec{u},\vec{v}\}\}d\sigma_{s}Cv)$

where$\{\sigma_{s} :s\in[0, \infty)\}$is the family from

1. For each Borel subset of2.
.
Furthermore, for a function :$E$$\mathbb{R}^{r},$$\sigma_{s}(E)$$\mathcal{M}(\mathbb{R}^{r})$satisfying the following conditions;is a Borel measurable function of on $[0,t]$ ,$s$$\Vert\sigma_{s}\Vert\in L_{1}[0,t]$

$F$

(6.2)

$C^{r}[0,t]arrow \mathbb{C}$and$\lambda>0$, let$I_{F}^{\lambda}(\vec{\xi})=E[F(\lambda^{-}z(x-1[x])+[\vec{\xi}])]$,and the polygonal functions are taken over the partition $0<t$ . In the following
theorem, if we replace by with $0<s\leq t$ , then we will assume that the expectation is takenover $C^{r}[0,s]$ and the polygonal functions are taken over the partition $0<s$ . We now have thefollowing theorem [5].

where$\vec{\xi}\in \mathbb{R}^{2r}$$F$

Theorem 6.1. For$t\in$$F_{s}$( , oo) let$O$$F_{t}(x)= \exp\{\int_{0}^{t}\theta(u,x(u))dm_{L}(u)\}$for$x\in C^{r}[0,t]$ ,where is given by (6.1). Further, for$\theta$$(t,\vec{\xi}_{1},A)\in(O, \infty)\cross \mathbb{R}^{r}\cross(0, \infty)$

, let

,

$H(t, \vec{\xi}_{1},\lambda)=(\frac{\lambda}{2\pi t})^{r}2\int_{\mathbb{R}^{r}}\exp\{-\frac{\lambda}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}I_{F_{t}}^{\lambda}(\vec{\xi}_{0},\vec{\xi}_{1})d\varphi^{r}(\vec{\xi}_{0})$is given by (6.2) in the sense of the conditioning fimction
being given by $X_{t}(x)=$$(x(O),x(t))$ on $C^{r}[0,t]$ . Then
and $H$ satisfies the following integml equation:
where$X_{t}$$I_{F_{t}}^{\lambda}$$F_{t}\in S_{w_{\varphi}}^{r}$(6.3)$H(t, \vec{\xi}_{1/}t)=(\frac{\lambda}{2\pi t})^{\frac{r}{2}}\int_{\mathbb{R}^{r}}\exp\{-\frac{\lambda}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}d\varphi^{r}(\vec{\xi}_{0})+\int_{0}^{t}[\frac{\lambda}{2\pi(t-s)}]^{Z}r$$\cross\int_{\mathbb{R}^{r}}\theta(s,$for$u \gamma H(s,\vec{u},\lambda)\exp\{-\frac{\lambda}{2(t-s)}\Vert\vec{\xi}_{1}-\vec{u}\Vert_{2}^{2}\}dm_{L}^{r}(u\gamma dm_{L}(s)$$(t,\vec{\xi}_{1},\lambda)\in(0, \infty)\cross \mathbb{R}^{r}\cross(0, \infty)$.

Theorem 6.2. Let the assumptions and notations be as given in Theorem 6.1. Moreover, for$(t,\vec{\xi}_{1/}l)\in(0,\infty)\cross \mathbb{R}^{r}\cross \mathbb{C}_{+}let$$H(t, \vec{\xi}_{1},\lambda)=(\frac{\lambda}{2\pi t})^{r}2\int_{\mathbb{R}^{r}}\exp\{-\frac{\lambda}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}E^{anw_{\lambda}}[F_{t}|X_{t}](\vec{\xi}_{0},\vec{\xi}_{1})d\varphi^{r}(\vec{\xi}_{0})$Then$H(t,\vec{\xi}_{1},\lambda)$

satisfies the integml equation (6.3) as given in Theorem 6.1.

.

138
For a function $f$ defined on$\mathbb{R}^{r}$, we adopt the following notation:$\overline{\int}_{\mathbb{R}^{r}}zdm_{L}^{r}z=\lim_{Aarrow\infty}\int_{\mathbb{R}^{r}}z\exp\{-\frac{1}{2A}||\vec{z}\Vert^{2}\}dm_{L}^{r}(\gamma Z$

if the limit exists. Using this notation, we have the following theorem.
Theorem 6.3. Let the assumptions and notations be as given Theorem 6.2. Moreover, for( , oo)let$(t,\vec{\xi}_{1},q)\in$$O$$\cross \mathbb{R}^{r}\cross(\mathbb{R}-\{0\})$$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}2\int_{\mathbb{R}^{r}}\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}E^{anf_{q}}[F_{t}|X_{t}](\vec{\xi}_{0},\vec{\xi}_{1})d\varphi^{r}(\vec{\xi}_{0})$Then$H(t,\vec{\xi}_{1}, -iq)$.

satisfies the following integml equation:$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}2\int_{\mathbb{R}^{r}}\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}0\Vert^{2}\}d\varphi^{r}(\vec{\xi}_{0})+\int_{0}^{t}[\frac{q}{2\pi i(t-s)}]^{r}2$$\cross\overline{\int}_{\mathbb{R}^{r}}\theta(s,u\gamma H(s,\vec{u}, -iq)\exp\{\frac{iq}{2(t-s)}\Vert\vec{\xi}_{1}-\vec{u}\Vert^{2}\}dm_{L}^{r}(u\gamma dm_{L}(s)$.

Corollary 6.4. Under the assumptions and notations as given in Theorem 6.3, if is theis exactly the r-dimensional Wiener measure on the
Dimc measure concentmted at , then, and we have
Borel class of$\varphi^{r}$$\vec{0}$$w_{\varphi}^{r}$$C_{0}^{r}[0,t]$$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}2\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}\Vert^{2}\}E^{anf_{q}}[F_{t}|X_{t}](\vec{0},\vec{\xi}_{1})$so thatsatisfies thefollowing integral equation which is formally equivalent to the
Schrodinger partial differential equation:$H(t,\vec{\xi}_{1}, -iq)$$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}z\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}\Vert^{2}\}+\int_{0}^{t}[\frac{q}{2\pi i(t-s)}]^{\frac{r}{2}}$$\cross\overline{\int}_{\mathbb{R}^{r}}\theta(s,u\gamma H(s,\vec{u}, -iq)\exp\{\frac{iq}{2(t-s)}\Vert\vec{\xi}_{1}-\vec{u}\Vert^{2}\}dm_{L}^{r}(u\gamma dm_{L}(s)$which is the integml equation as given in Theorem 6 of [10].

Theorem 6.5. Let the assumptions and notations be as given Theorem 6.3. Suppose that, that is,
has the pmbability density on . Moreover, for
let$\varphi^{r}\ll m_{L}$$\varphi^{r}$$\psi$$\mathbb{R}^{r}$$(t,\vec{\xi}_{1},q)\in(O,\infty)\cross \mathbb{R}^{r}\cross$$(\mathbb{R}-\{0\})$$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}2\int_{\mathbb{R}^{r}}\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}q$

.

139
Then

$H(t,\vec{\xi}_{1}, -iq)$satisfies the following integml eqnation:$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{\frac{r}{2}}\int_{\mathbb{R}^{r}}\psi(\vec{\xi}_{0})\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}dm_{L}^{r}(\vec{\xi}_{0})+\int_{0}^{t}[\frac{q}{2\pi i(t-s)}]^{r}2$$\cross\overline{\int}_{\mathbb{R}^{r}}\theta(s,\overline{u}2H(s,\vec{u}, -iq)\exp\{\frac{iq}{2(t-s)}\Vert\vec{\xi}_{1}-\vec{u}\Vert^{2}\}dm_{L}^{r}(u\gamma dm_{L}(s)$which is formally equivalent to the Schrodinger partial differential equation.
Remark If is Lebesgue measurable, then we can take a Borel measurable function withfor, so that we can assume that is Borel measurable. Furthermore,
a.e.
since
, we have by the dominated convergence theorem$\psi$$\psi_{1}$$\psi(u\gamma=\psi_{1}(\vec{u})$$\vec{u}\in \mathbb{R}^{r}$$m_{L^{-}}^{r}$$\psi$$\psi\in L_{1}(\mathbb{R}^{r})$$\overline{\int}_{\mathbb{R}^{r}}\psi(\vec{\xi}_{0})\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}dm_{L}^{r}(\vec{\xi}_{0})=\int_{\mathbb{R}^{r}}\psi(\vec{\xi}_{0})\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}dm_{L}^{r}(\vec{\xi}_{0})$so that$H(t,\vec{\xi}_{1}, -iq)$satisfies the following integral equation:$H(t, \vec{\xi}_{1}, -iq)=(\frac{q}{2\pi it})^{r}z\overline{\int}_{\mathbb{R}^{r}}\psi(\vec{\xi}_{0})\exp\{\frac{iq}{2t}\Vert\vec{\xi}_{1}-\vec{\xi}_{0}\Vert^{2}\}dm_{L}^{r}(\vec{\xi}_{0})+\int_{0}^{t}[\frac{q}{2\pi i(t-s)}]^{r}2$$\cross\overline{\int}_{\mathbb{R}^{r}}\theta(s,u\gamma H(s,\vec{u}, -iq)\exp\{\frac{iq}{2(t-s)}\Vert\vec{\xi}_{1}-\vec{u}\Vert^{2}\}dm_{L}^{r}(u\gamma_{dm_{L}(s)}$which is formally equivalent to the Schr\"odinger partial differential equation [1, 3, 12].

\S 7. Operator-valued function space integral
In this section we investigate that the conditional analytic Feynman integrals of functionals on $C^{r}[0,t]$ can be applied to express the operator-valued Feynman integrals in terms theconditional Feynman integrals.
Throughout this section, let $t>0$ be fixed, :be given by $X_{t}(x)=(x(0),x(t))$ .
Further, for
, let
and
and for a function $F:C^{r}[0,t]arrow \mathbb{C}$, let. Now we define the analytic operator-valued function space integral.
$X_{t}$$\lambda>0$$\vec{\xi}\in \mathbb{R}^{r}$$C^{r}[0,t]arrow \mathbb{R}^{2r}$$X_{t}^{\lambda,\vec{\xi}}(x)=X_{t}(\lambda^{-}zx+\vec{\xi})1$$F^{\lambda,\vec{\xi}}(x)=F(\lambda^{-}2x+\vec{\xi})1$

Definition 7.1. Let
let

$F$

:

$C^{r}[0,t]arrow \mathbb{C}$$\psi_{t}^{\prime 1,\vec{\xi}}(x)=\psi(\lambda^{-\frac{1}{2}}x(t)+\vec{\xi})$

be a function. For any$\lambda>0,$$\psi$

in$L_{1}(\mathbb{R}^{r})$and$\vec{\xi}$in$\mathbb{R}^{r}$,and$(I_{\lambda}(F) \psi)(\vec{\xi})=\int_{C^{r}}F^{\lambda,\vec{\xi}}(x)\psi_{t}^{\lambda,\vec{\xi}}(x)dw_{\varphi}^{r}(x)$.

as a function of and if the correspondence
gives an element, we say that the operator-valued function space integral
exists.
Next suppose that there exists an -valued function which is weakly analytic inand agrees
Ifof$I_{\lambda}(F)\psi$

is in$L_{\infty}(\mathbb{R}^{r})$$\vec{\xi}$$\mathcal{L}\equiv \mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$\psiarrow I_{\lambda}(F)\psi$$I_{\lambda}(F)$$\mathcal{L}$$\mathbb{C}_{+}$

140
and is called the analytic. Then this -valued function is denoted bywithon
operator-valued Wiener integral of associated with parameter . Finally, for a nonzero realin such that for every insuppose that there exists an operator
is called the. Then, as approaches $to-iq$ throughconverges weakly toanalytic operator-valued Feynman integral of with parameter .
$(0,\infty)$$I_{\lambda}(F)$$\mathcal{L}$$I_{\lambda}^{an}(F)$$F$$\lambda$$J_{q}^{an}(F)$$q$$\mathcal{L}$$\lambda$$J_{q}^{an}(F)\psi$$L_{1}(\mathbb{R}^{r}),$$\psi$$J_{q}^{an}(F)$$\mathbb{C}_{+}$$F$$I_{\lambda}^{an}(F)\psi$$q$Note that in Definition 7.], the weak limit and the weak analyticity are based on the weak*.
induced by its pre-dual
topology on$L_{1}(\mathbb{R}^{r})[2,8,13]$$L_{\infty}(\mathbb{R}^{r})$Lemma 7.2. Let. Thento$m_{L}^{r}$

and
and

$\lambda>0$$P_{x^{l\vec{f}}},\ll m_{L}^{2r}$$\vec{\xi}\in \mathbb{R}^{r}$. Suppose that$\varphi^{r}$is absolutely continuous with respect$\frac{dP_{X_{t}^{\lambda,\vec{\xi}}}}{dm_{L}^{2r}}(\vec{\eta}_{1},\vec{\eta}_{2})=(\frac{\lambda}{\sqrt{2\pi t}})^{r}\exp\{-\frac{\lambda\Vert\vec{\eta}_{2}-\vec{\eta}_{1}\Vert^{2}}{2t}\}\frac{d\varphi^{r}}{dm_{L}^{r}}(\lambda^{z^{I}}(\vec{\eta}_{1}-\vec{\xi}))$for$m_{L}^{2r}-a.e$.

.

$(\vec{\eta}_{1},\vec{\eta}_{2})\in \mathbb{R}^{2r}$,Theorem 7.3. Let the assumptions and notations be as in Lemma 7.2. For, and for each bounded subset
exists on
suppose thatof ,.
and allthere exists
suchfor allsatisfying the following conditions;
Funher, suppose that there exists afu nction on$F;C^{r}[0,t]arrow \mathbb{C}$$t\in\Omega$$that|E^{anw_{\lambda}}[F|X_{t}](\vec{\eta}_{1}, \vec{\eta}_{2})|\leq M_{\Omega}$$M_{\Omega}\geq 0$$\Psi$(i) for each$\Omega$$\mathbb{C}_{+}\cross \mathbb{R}^{2r}$$E^{anw_{\lambda}}[F|X_{t}](\vec{\eta}_{1},\vec{\eta}_{2})$$\mathbb{C}_{+}$$(\vec{\eta}_{1},\vec{\eta}_{2})\in \mathbb{R}^{2r}$$\mathbb{C}_{+}\cross \mathbb{R}^{r}$,$\lambda>0and\tilde{\eta}\in \mathbb{R}^{r},$$\Psi(\lambda,\vec{\eta})=\frac{d\varphi^{r}}{dm_{L}^{r}}(\lambda^{1}2\vec{\eta})$

is analytic onafunction of , and.
(iii) for each bounded subset ofis bounded on(ii) for each$\vec{\eta}\in \mathbb{R}^{r},$$\Omega$

Then for$\lambda\in \mathbb{C}_{+}$$\mathbb{C}_{+},$$\Omega\cross \mathbb{R}^{r}$$\Psi$, the analytic opemtor-valued Wiener integml$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$(7.1)$\lambda$$\mathbb{C}_{+}as$$\Psi(\lambda,\vec{\eta})$$I_{\lambda}^{an}(F)$exists as an element ofand is given by$(I_{\lambda}^{an}(F) \psi)(\vec{\xi})=(\frac{\lambda}{\sqrt{2\pi t}})^{r}\int_{\mathbb{R}^{2r}}E^{anw_{\lambda}}[F|X_{t}](\vec{\eta}_{1},\vec{\eta}_{2})\psi(\vec{\eta}_{2})$$\cross\Psi(\lambda,\vec{\eta}_{1}-\vec{\xi})\exp\{-\frac{\lambda\Vert\vec{\eta}_{2}-\vec{\eta}\iota\Vert^{2}}{2t}\}dm_{L}^{2r}(\vec{\eta}_{1},\vec{\eta}_{2})$. In addition, suppose that for a nonzem real and. Moreover, suppose that can be extended toexists forwith the following two additional conditions;(ii)’ for eachis continuous at $\lambda=-iq$ as afunction of , andsatisfying(iii)’ there exists afiunctionfor$\mathbb{R}^{r},$$\psi\in L_{1}(\mathbb{R}^{r})$and$m_{L}^{r}-a.e.\vec{\xi}\in \mathbb{R}^{r}$$q$$\Psi$$(\vec{\eta}_{1},\vec{\eta}_{2})\in \mathbb{R}^{2r}$$E^{anf_{q}}[F|X_{t}](\vec{\eta}\iota,\vec{\eta}_{2})$$(\mathbb{C}_{+}\cup\{-iq\})\cross \mathbb{R}^{r}$$\vec{\eta}\in \mathbb{R}^{r},$$\lambda$$\Psi(\lambda,\vec{\eta})$$\Phi_{q}\in L_{1}(\mathbb{R}^{r})$$|\Psi(\lambda,\vec{\eta})|\leq|\Phi_{q}(\vec{\eta})|$where$\Omega_{\epsilon}=\{\lambda\in \mathbb{C}_{+}:|\lambda+iq|<\epsilon\}$for all$(\lambda,\vec{\eta})\in\Omega_{\epsilon}\cross \mathbb{R}^{r}$for some real$\epsilon>0$.

,$\vec{\xi}\in$141Then as an element of
the analytic opemtor-valued Feynman integml
exists and it is given by (7.1) replacing and, respectively.
by-iq and$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$\lambda$

Example 7.4. For$\vec{\eta}=(\eta_{1}, \cdots,\eta_{r})\in \mathbb{R}^{r}$$\frac{d\varphi^{r}}{dm_{L}^{r}}(\vec{\eta})=(\frac{1}{\pi}I\prod_{j=1}^{rr}\frac{1}{1+\eta_{j}^{2}}$

and

and

$J_{q}^{an}(F)$$E^{anf_{q}}$$E^{anw_{\lambda}}$$\lambda\in \mathbb{C}_{+}^{\sim}$, let$\Psi(\lambda,\vec{\eta})=(\frac{1}{\pi})\prod_{j=1}^{rr}\frac{1}{1+,t\eta_{j}^{2}}$.

Thenis a probability measure on the Borel class ofand the condition (i) of Theorem
7.3 is satisfied. Further, foris analytic onand continuous on
because, satisfying conditions (ii) and (ii)’ of Theorem 7.3. Now we have forfor$\mathbb{R}^{r}$$\varphi^{r}$$\vec{\eta}\in \mathbb{R}^{r},$$1+\lambda\eta_{j}^{2}\neq 0$$\Psi(\lambda,\vec{\eta})$$\mathbb{C}_{+}$$\mathbb{C}_{+}^{\sim}$$\lambda\in \mathbb{C}_{+}^{\sim}$$(\lambda,\eta_{j})\in \mathbb{C}_{+}^{\sim}\cross \mathbb{R}$$| \frac{1}{1+\lambda\eta_{j}^{2}}|^{2}=\frac{1}{(1+\eta_{j}^{2}{\rm Re}\lambda)^{2}+(\eta_{j}^{2}{\rm Im}\lambda)^{2}}\leq\min\{1,$$\frac{1}{1+\eta_{j}^{4}({\rm Im}\lambda)^{2}}\}$

which satisfies conditions (iii) and (iii)’ of Theorem 7.3. Let be a function satisfying theassumptions of Theorem 7.3. Applying Theorem 7.3 to , for any nonzero real , the analytic
operator-valued Feynman integral
exists as an element ofand it isgiven by
$F$

$F$

$q$

$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$J_{q}^{an}(F)$$(J_{q}^{an}(F) \psi)(\vec{\xi})=(\frac{q}{\pi i\sqrt{2\pi t}})^{r}\int_{\mathbb{R}^{2r}}E^{anf_{q}}[F|X_{t}](\vec{\eta}_{1},\vec{\eta}_{2})\psi(\vec{\eta}_{2})$$\cross[\prod_{j=1}^{r}\frac{1}{1-iq(\eta_{1,j}-\xi_{j})^{2}}]\exp\{\frac{iq\Vert\vec{\eta}_{2}-\vec{\eta}_{1}\Vert^{2}}{2t}\}dm_{L}^{2r}(\vec{\eta}_{1},\tilde{\eta}_{2})$for
distribution, whereand. Note that the probability
having the above density for $r=1$ is known as the Cauchy distribution.

$\varphi$$\vec{\eta}_{1}=(\eta_{1,1}, \cdots,\eta_{1,r})$$\vec{\xi}=(\xi_{1}, \cdots , \xi_{r})\in \mathbb{R}^{r}$$\psi\in L_{1}(\mathbb{R}^{r})$Using the same method as used in the proof of Theorem 7.3, we can easily prove the following theorem.
Theorem 7.5. If the conditions (iii) and $(iii)’$ in Theorem 7.3 are replaced by the one condition that for each bounded subset of
there exists afimction
satisfying$\Omega$(7.2)$\Phi_{\Omega}\in L_{1}(\mathbb{R}^{r})$$\mathbb{C}_{+}$$|\Psi(\lambda,\vec{\eta})|\leq|\Phi_{\Omega}(\vec{\eta})|$for all$(/l,\vec{\eta})\in\Omega\cross \mathbb{R}^{r}$,then we have all the results of Theorem 7.3.
Theorem 7.6. Let
(7.3)

for

$F\in S_{w_{\varphi}}^{r}$

be given by (2.1). Then we have for$\lambda\in \mathbb{C}_{+}$$E^{anw_{\lambda}}[F|X_{t}]( \vec{\eta}_{1},\vec{\eta}_{2})=\int_{L_{2}^{r}[0,t]}\exp\{-\frac{1}{2At}[t\Vert\vec{v}\Vert_{-}\Vert\vec{V}_{t}\Vert^{2}]+\frac{i}{t}\{\vec{\eta}_{2}-\vec{\eta}_{1},\vec{V}_{t}\}\}d\sigma Cv)$

where
. Moreover, for any nonzem realis given by the right hand side of (7.3) replacing by-iq.

$(\vec{\eta}_{1},\vec{\eta}_{2})\in \mathbb{R}^{2r}$$E^{anf_{q}}[F|X_{t}]$,$\vec{V}_{t}=(\int_{0}^{t}v_{1}(s)ds, \cdots,\int_{0}^{t}v_{r}(s)ds)$$\lambda$

$q$

,

142
,
For
given by (2.1), we know from Theorem 7.6 that $E^{anw}[F|X_{t}]$ is bounded by
wheredenotes the total variation of . Combining Lemma 7.2, Theorems 7.3, 7.5 and 7.6,
we have the following theorem [8].
$\Vert\sigma\Vert$$F\in S_{w_{\varphi}}^{r}$$\Vert\sigma\Vert$$\sigma$. Suppose
Theorem 7.7. Let the assumptions be as given in Lemma 7.2 and letthat for a nonzem real there exists afunction on
satisfying the con$(ii)’$$(iii)’$
7.3,
(i),
(ii),
ditions
and either (iii) andof Theorem 7.3 or (7.2) of Theof Theorem
orem 7.5. Then the analytic opemtor-valued Feynman integml
exists as an element of, respectively,
and it is given by (7.1) replacing andby-iq andwhere
is given by Theorem 7.6.
$F\in S_{w_{\varphi}}^{r}$$\Psi$$q$$(\mathbb{C}_{+}\cup\{-iq\})\cross \mathbb{R}^{r}$$J_{q}^{an}(F)$$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$\lambda$$E^{anf}q$$E^{anw_{\lambda}}$$E^{anf_{q}}[F|X_{t}]$Theorem 7.8. Let the assumptions and notations be as given in Theorem 7.6. Furthermore,
suppose that is normally distributed with the mean vector and the variance-covariance ma, where is the r-dimensional identity matrix. Then for, the analytic opemtortrix
valuedintegmlexists as an element ofand it is given by$\vec{0}$$\varphi^{r}$$\alpha^{2}I_{r}$$\prime t\in \mathbb{C}_{+}$$I_{r}$$W\iota ener$(7.4)$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$I_{\lambda}^{an}(F)$$(I_{\lambda}^{an}(F) \psi)(\vec{\xi})=[\frac{\text{ノ}t}{2\pi(t+\alpha^{2})}]^{r}z\int_{L_{2}^{r}[0,t]}\exp\{-\frac{1}{2At}[t\Vert\vec{v}\Vert^{2}$$- \Vert\tilde{V}_{t}\Vert^{2}]\}\int_{\mathbb{R}^{r}}\psi(\vec{\eta})H(\lambda,\vec{\xi},\vec{\eta},$ $\frac{1}{t}\vec{V}_{t})dm_{L}^{r}(\vec{\eta})d\sigma\subset\dot{v})$for$\psi\in L_{1}(\mathbb{R}^{r})$and$m_{L}^{r}-a.e.\vec{\xi}\in \mathbb{R}^{r}$, where$H( \lambda,\vec{\xi},\vec{\eta},\vec{\zeta})=\exp\{-\frac{\lambda}{2\alpha^{2}}\Vert\vec{\xi}-\vec{\eta}\Vert^{2}-\frac{t\alpha^{2}}{2\lambda(t+\alpha^{2})}\Vert\vec{\zeta}+\frac{\lambda i}{\alpha^{2}}(\vec{\xi}-\vec{\eta})\Vert^{2}\}$for$\lambda\in \mathbb{C}_{+}^{\sim}and$$\vec{\zeta}\in \mathbb{R}^{r}$. Furthermore, for any nonzem real

Feynman integml
replacing by-iq.

$J_{q}^{an}(F)$exists as an element of$q$

the analytic opemtor-valued
and it is given by (7.4)$\mathcal{L}(L_{1}(\mathbb{R}^{r}),L_{\infty}(\mathbb{R}^{r}))$$\lambda$

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