数理解析研究所講究録
第 1797 巻 2012 年 51-85

51Introduction to Analogue of Wiener Measure Spaceand Its Applications
By

Kun Sik RYU ,* Man Kyu IM ** and Ki Seong CHOI ***

Abstract
This talk is the improvement of our survey paper [42]. The contents of this talk consist of the following:(1) The definitions, notations and some well-known facts which are needed to understand this talk.
(2) Complex-valued, measure-valued and operator-valued analogue of Wiener measure and their exam-

ples.
(3) The translation theorem of analogue of Wiener measure and its applications.
(4) The integration formula of.
(5) The integration formula of.
(6) The measure of the set of all analogue of Wiener paths staying below a continuous differentiable
function.
(7) Relationship among the Bartle integral and the conditional expectations.
(8) The simple formula for conditional expectation.
(9) The measure-valued Feynman-Kac formula.
(10) Volterra integral equation for the measure-valued Feynman-Kac formula.
(11) Dobrakov’s integral with respect to the operator-valued analogue of Wiener measure.
(12) The operational calculus of analogue of Wiener functional.
(13) The theories of Fourier-Feynman transform.
$\exp\{\alpha\Vert x\Vert_{\infty}\}$$\exp\{\lambda\int_{0}^{t}x(s)^{2}ds\}$\S 1. Preliminaries
In this section, we present some notation, definitions and well-known facts which are needed
to understand the subsequent sections.
2010 Mathematics Subject Classification(s): Primary $28C20$ ; Secondary $28C35$
Key Words: analogue of Wiener measure, Bartle integral, the Bochner integral, Dobrakov integral, Volterra integral
equation, conditional expectation, measure-valued Feynman-Kac formula, integral transform*Department of Mathematics Education, Han Nam University, Daejon 306-791, Korea.
Department of Mathematics Education, Han Nam University, Daejon 306-791, Korea.
$***$
Department of Information Security, Kon Yang University, Nonsan 320-711, Korea.
$**$

52
(A)

let

of

Let be the real number field and the complex number field. For a natural number ,be the set of all Borel measurable subsets
be the n-times product space of . Let$\alpha_{2}=-1$ ,
. Let
and
the Lebesgue measure on the measurable space
and $\alpha_{4}=-i$ .
$\mathbb{R}$$\mathbb{R}^{n}$$\mathbb{R}$$n$$\mathbb{C}$$\mathbb{R}$$\mathcal{B}(\mathbb{R})$$\alpha_{1}=1,$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$m_{L}$$\alpha_{3}=i$

For a positive real number $a,b$ , let $C[a,b]$ be the space of all real-valued continuous. By the Stonefunctions on a closed bounded interval $[a,b]$ with the supremum norm
(B)$\Vert\cdot\Vert_{\infty}$

Weierstrass theorem,(1.1)$(C[a,b], \Vert\cdot\Vert_{\infty})$LetFor
and for$\mathcal{M}(\mathbb{R})$$p\in \mathbb{R}$is a real separable Banach space.

be the space of all finite complex-valued countably additive measure on, let
be the Dirac measure concentrated at with total mass one. For
is defined by, the total variation $|\mu|(E)$ on$p$$\delta_{p}$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

$\mu\in \mathcal{M}(\mathbb{R})$$E$$E\in \mathcal{B}(\mathbb{R})$(1.2)$| \mu|(E)=\sup\sum_{i=1}^{n}|\mu(E_{i})|$,. Thenof disjoint sets inwhere the supremum is taken over all finite sequences
and, by the Jordan decomposition theorem [16, p. 307, (19.13) Theorem], there areis inunique non-negative measures $\mu_{j}\in \mathcal{M}(\mathbb{R})(j=1,2,3,4)$ such that$|\mu|$$\mathcal{B}(\mathbb{R})$$\langle E_{i}\}$$\Lambda 4(\mathbb{R})$(1.3)$\mu=\sum_{j=1}^{4}\alpha_{j}\mu_{j}$.

is a complex Banach space.
By [10, Theorem 4.1.7],Letbe the space of all finite complex-valued measures$(\mathcal{M}(\mathbb{R}), \cdot |(\mathbb{R}))$$\mathcal{R}M(R)$absolutely continuous with respect to$m_{L}$$\mu$on$(\mathbb{R},\mathcal{B}(\mathbb{R}))$, that is, the Radon-Nikodim derivative$\frac{d|\mu|}{dm_{L}}$which areexists.

be the space ofLet
be a measure space. For a positive real number , letbeis -integrable. Let
complex-valued -measurable functions $f$ on such thatthe space of complex-valued -measurable functions $f$ on which are -essentially bounded.
andare equivalence classes of functions in
The elements of $L^{p}(X,\mu)$ and$|\mu|-a.e$is. Since, respectively, with the equivalence relation being defined byofis a Banach space and the dual spaceisomorphic to,such that
there is a function in. For
is isomorphic to(C)$(X, \mathcal{B},\mu)$$\mathcal{L}^{p}(X,\mu)$

$p$

$X$

$\mu$

$|f|^{p}$$\mathcal{L}^{\infty}(X,\mu)$$|\mu|$

$X$

$\mu$

$|\mu|$$\mathcal{L}^{p}(X,\mu)$$L^{\infty}(X,\mu)$$\mathcal{R}\mathcal{M}(\mathbb{R})$$\mathcal{L}^{\infty}(X,\mu)$$L^{1}(\mathbb{R},m_{L}),$$L^{\infty}(\mathbb{R},m_{L})$$x^{*}( \psi)=\int_{\mathbb{R}}\theta(s)d\mu(s)$for$\mathcal{R}\mathcal{M}(\mathbb{R})^{*}$$\mathcal{R}\mathcal{M}(\mathbb{R})$$\mu\in \mathcal{R}\mathcal{M}(\mathbb{R})$$L^{\infty}(\mathbb{R},m_{L})$$\theta$$x^{*}\in \mathcal{R}\mathcal{M}(\mathbb{R})^{*}$$\mathcal{R}\mathcal{M}(\mathbb{R})$.

the dual space of B. For a B-valued countably
Let be a complex Banach space and, the semivariationof on is given by
and for
additive measure on$B$$B^{*}$

$v$

(1.4)

$(X, \mathcal{B})$$E\in \mathcal{B}$$\Vert v\Vert(E)=\sup\{|x^{*}v|(E)|x^{*}\in B^{*}$

where $|x^{*}v|(E)$ is the total variation on$E$$\Vert v\Vert(E)$

and

$E$

$v$

$\Vert x^{*}\Vert_{B}*\leq 1\}$of the complex-valued measure$x^{*}v$.

53
be a complex Banach space anda complex measure space. A function$f:Xarrow B$ is said to be -measurable if there exists a sequenceof B-valued simple functions
with
(D)

Let

$B$

$(X, \mathcal{B},\mu)$$\{f_{n}\}$

$\mu$

(1.5)

$\lim_{narrow\infty}\Vert f_{n}-f\Vert_{B}=0|\mu|-a.e$

A function $f$ is said to be -weakly measurable if
Pettis’ measurability theorem [11],
$\mu$

(1.6)

$f$

is -measurable if and only if $f$ is$\mu$$|\mu|$$x^{*}f$.

is -measurable for each$\mu$$x^{*}\in B^{*}$. By the-essentially separably valued and $f$ is -weakly$\mu$measurable.

We say that $f$ is -Bochner integrable if there exists a sequenceof B-valued simple functions such thatconverges to $f$ in the norm sense in for a.e. and$\{f_{n}\rangle$

$\mu$

$B$

$\langle f_{n}\rangle$$|\mu|-$$\lim_{narrow\infty}\int_{X}\Vert f(t)-f_{n}(t)\Vert_{B}d|\mu|(t)=0$.

In this case, $( Bo)-\int_{X}f(t)d\mu(t)$ is defined by(1.7)$( Bo)-\int_{X}f(t)d\mu(t)=\lim_{narrow\infty}\int_{X}f_{n}(t)d\mu(t)$,where the limit means the limit in the norm sense. By [11], [4, p. 45, Theorem 2],
(1.8)

$f$

is -Bochner integrable if and only if$\mu$$\int_{X}\Vert f(t)\Vert_{B}d|\mu|(t)$is finite.

By [52, Corollary 2],(1.9) if $U$ is a bounded linear operator on Binto a Banach

integrable function, then $Uf$ is a$B_{1}$$B_{1}$

and $f$ is a $B-valued\mu$ -Bochner-valued -Bochner integrable function, and
$\mu$

.

$( Bo)-\int_{X}(Uf)(t)d\mu(t)=U((Bo)-\int_{X}f(t)d\mu(t))$

Theorem 1.1. Let
integrable fimction. Then for
and$(X, \mathcal{B},\mu)$be a complex measure space anda -Bochner$[f(t)](E)$ is a complex-valued -integrablefunction of$f:Xarrow \mathcal{M}(\mathbb{R})$$E\in \mathcal{B}(\mathbb{R}),$

$\mu$

$\mu$

$t$

(1.10)$[( Bo)-\int_{X}f(t)d\mu(t)](E)=\int_{X}[f(t)](E)d\mu(t)$.

Remark Consider a function $H$ on $[0,1]\cross[0,1]$ defined by $H(x,y)=,Y[0_{X}](y)$ . Then $H$
is-integrable on $[0,1]\cross[0,1]$ , so by the Fubini theorem, $H(x,y)$ is an-integrable$L^{\infty}([0,1],m_{L})$function of for all andis infor all $x\in[0,1]$ . But
has noessentially separable range, so
is not -Bochner integrable. Hence, in generally, the(1.10)
equality
is not tme in the theory of Bochner integral.
$m_{L}\cross m_{L}$$m_{L}$$x$$y$$H(x, \cdot)$$H(x, \cdot)$$H(x, \cdot)$$m_{L}$

54
be a complex Banach space and $(Y,C,v)$ a B-valued measure space. Let be aof complexcomplex-valued lvll-measurable function on , that is, there exists a sequence. We say that is v-Bartle integrable ifvalued simple functions witha.e. and theof simple functions such that} converges tothere exists a sequence {
(E)

Let

$B$

$g$

$Y$

$\langle g_{n}\rangle$$g$$\lim_{narrow\infty}|g_{n}-g|=0\Vert v\Vert-a.e$sequence$\langle\int g_{n}(s)dv(s)\rangle$$g\Vert v\Vert-$$\langle g_{n}$$g_{n}\rangle$

is Cauchy in the norm sense. In this case, $( Ba)-\int_{Y}g(s)dv(s)$ is definedby(1.11)$( Ba)-\int_{Y}g(s)dv(s)=\lim_{narrow\infty}\int g_{n}(s)dv(s)$,where the limit means the limit in the norm sense. By [13, Theorem 8],(1.12)

if $f$ is a v-measurable function which is lvll-essentially bounded, then $f$ is v-Bartle
integrable and$\Vert(Ba)-\int_{Y}f(s)dv(s)\Vert_{B}\leq(\Vert v\Vert-ess\sup|f(s)|)\Vert v\Vert(Y)$.

By [27, Theorem 2.4],(1.13)$g$is v-Bartle integrable if and only if for each$x^{*}\in B^{*},$ $g$each $E\in C$ , there is an element $( Ba)-\int_{E}g(s)dv(s)$ in$x^{*}[( Ba)-\int_{E}g(s)dv(s)]=\int_{E}g(s)dx^{*}v(s)$

$B$

is

$x^{*}v$-integrable, and forsuch thatfor$x^{*}\in B^{*}$.

By [13, Theorem 8],(1.14)

if$U$

is a bounded linear operator from$B$

into a Banach space$B_{1}$

and$g$is v-Bartle

integrable, then is Uv-Bartle integrable. In this case$g$$U[( Ba)-\int_{Y}g(s)dv(s)]=(Ba)-\int_{Y}g(s)dUv(s)$ .

By [13, Theorem 10],
(1.15)

if$\{f_{n}\rangle$

is a sequence of v-Bartle integrable functions which converges$\Vert v\Vert-a.e$. toand if is a v-Bartle integrable function such that $|f_{n}(s)|\leq g(s)\Vert v\Vert-a.e$. for all
$s$

$g$

natural numbers$n$then $f$ is v-Bartle integrable and for $E\in C$$( Ba)-\int_{E}f(s)dv(s)=\lim_{narrow\infty}(Ba)-\int_{E}f_{n}(s)dv(s)$

.

$f$

55
Let be a complex Banach space. Let
and $(Y,C)$ be two measurable spaces and letthe -algebra of sets in the space
generated by the family of rectangles $E\cross F$ for
all in and in . Let be a complex-valued measure on
and a B-valued measureon $(Y,C)$ . For in
, let
(F)

$B$

$\mathcal{B}\otimes C$$E$$(X, \mathcal{B})$$X\cross Y$$\sigma$$F$$\mathcal{B}$

$G$

$C$

$(X, \mathcal{B})$

$\mu$

$v$

$\mathcal{B}\otimes C$(1.16)$( \mu\cross v)(G)=(Ba)-\int_{Y}[\int_{X}\chi_{G}(u,v)d\mu(u)]dv(v)$.

By [ $20$ , Proposition 2], using the dominated convergence theorem in [21], Kluvanek provedthatis a B-valued measure on
and for,$n$$\mathcal{B}\otimes C$$\mu\cross v$(1.17)$G\in \mathcal{B}\otimes C$$( \mu\cross v)(G)=(Ba)-\int_{Y}[\int_{X}\chi_{G}(u,v)d\mu(u)]dv(v)$$=( Bo)-\int_{X}[(Ba)-\int_{Y}\chi_{G}(u,v)dv(v)]d\mu(u)$holds. Moreover, in [20, Proposition 3], he showed that
(1.18)$x^{*}(\mu\cross v)=\mu\cross(x^{*}v)$

for all.
When both measures and are complex-valued, a sufficient condition for validity of the
Fubini theorem is the integrability of the function with respect to. But, if is a vector
measure then the integrability of the function with respect tois no longer a sufficientcondition for the validity of the Fubini theorem. Indeed, we can find a counter example for thisfact in [20].
$x^{*}\in B^{*}$

$v$

$\mu$

$v$

$\mu\cross v$$\mu\cross v$

Theorem 1.2. Let be a sepamble complex Banach space,measure space and $(Y,C,v)$ a B-valued measure space. Let
and-Bartle integmble. Then$B$

a complex-valuedbe-measumble$(X, \mathcal{B},\mu)$$f:X\cross Yarrow \mathbb{C}$$\mathcal{B}\otimes C$$\mu\cross v$(1.19)

for.

$\Vert v\Vert-a.e$$v,$$f(u,v)$ is a$\mu$-integmble function of$u$(1.20)$\int_{X}f(u,v)d\mu(u)$(1.21)$( Ba)-\int_{X\cross Y}f(u,v)d\mu\cross v(u,u)=(Ba)-\int_{Y}[\int_{X}f(u,v)d\mu(u)]dv(v)$Moreover, $\iota ffor|\mu|-a.e$.

$u,$

is v-Bartle integrable and$f(u,v)$ is a v-Bartle integmble.

function of and $( Ba)-\int_{Y}f(u,v)dv(v)$$v$

is -Bochner integmble then$\mu$

(1.22)$( Ba)-\int_{X\cross Y}f(u,v)d\mu\cross v(u,\iota))=(Ba)-\int_{X}[(Ba)-\int_{Y}f(u,v)dv(v)]d\mu(u)$$=( Ba)-\int_{Y}[\int_{X}f(u,v)d\mu(u)]dv(u)$.

56
Let X and be (real or complex) Banach spaces and denote by $L(X,Y)$ the Banach
space of all bounded linear operator from X to Y. Let be a non-empty set and a -algebraof subsets of . We say that a set function $m:\mathcal{B}arrow L(X,Y)$ is an operator-valued countably
additive in the strong operator topology if for every in X the set function $\mathcal{B}\ni E\mapsto m(E)x\in X$
is a countable additive vector measure. We define a non negative set function へ，which is called
the semivariation of the measure by equality
(G)

$Y$

$T$

$\mathcal{B}$$\sigma$

$T$

$x$

$m$

$m$

$\hat{m}(E)=\sup\{\Vert\sum_{i=1}^{n}m(E\cap E_{i})x_{i}||E_{i}\in \mathcal{B},$ $x_{i}\in X$

and

with

$E_{i}\cap E_{j}=\emptyset$

for $i=1,2,$$|x_{i}|\leq 1$$\ldots,n$for $i\neq j$ }.

of is finite. Let
We say that is an integrable subset in if the semivariation
set of all integrable subsets of . From [12], we have following theorem.
$E$

$E$

$\hat{m}(E)$$\mathcal{B}$$\mathcal{K}$be the$T$(for
Theorem 1.3 -Theorem). Let contains no subspace isomorphic to the spaceexample let be a weakly complete Banach space). Then the semivariation is continuous on$(E_{n})=0$ .
then, that is,is a sequence ofdecreasing subsets in with$Y$$(^{*}$$c_{0}$

へ

$Y$

$m$

へ

$\mathcal{K}$$\mathcal{K}$$\iota f\langle E_{n}\rangle$A$\mathcal{K}$$\lim_{narrow\infty}E_{n}=\emptyset$-simple function on

simple integrable function$T$$\lim_{narrow\infty}m$with values in X is called the simple integrable function. For any$\psi=\sum_{k=1}^{n}x_{k}\gamma_{E_{k}}$, let $\int_{E}\psi dm=\sum_{k=1}^{n}m(E_{k}\cap E)x_{k}$ .

of simple integrable
A function $f:Tarrow X$ is called measurable if there is a sequence
for each $t\in T$ .
functions such that
A measurable function $f:Tarrow X$ is said to be Dobrakov integrable if there is a sequenceof simple integrable functions converging almost everywhere to $f$ . In this case, theintegral of the function $f$ on a set in is defined by the equality$\{f_{n}\}$$\lim_{narrow\infty}f_{n}(t)=f(t)$$\hat{m}$$(f_{n}\}$$E$$\mathcal{K}$$( D)-\int_{E}fdm=\lim_{narrow\infty}\int_{E}f_{n}dm$

Here this limit is uniform with respect to
(H)

Let

$(\Omega,\mathcal{B},\mu)$$E\in \mathcal{K}$be a measure space. Let.

.

$X:\Omegaarrow \mathbb{R}^{n+1}$

be a measurable function and $F$ a. Let $P_{X}(A)=\mu(X^{-1}(A))$ for. Thenis. By the Radon-Nikodym theorem, there is a function $E^{\mu}(F|X)$ , uniquea measure on
up to -null sets such that$\mathbb{C}$-valued integrable function on$(\Omega,\mathcal{B},\mu)$$A\in \mathcal{B}(\mathbb{R}^{n+1})$$P_{X}$$\mathcal{B}(\mathbb{R}^{n+1})$$\mu$$\int_{X^{-1}(A)}Fd\mu=\int_{A}E^{\mu}(F|X)dP_{X}$for$A\in \mathcal{B}(\mathbb{R}^{n+1})$. This function$E^{\mu}(F|X)$

is called the conditional expectation of$F$given .
$X$

57
Let be inand be a complex-valued Borel measure on $[a,b]$ . A complex-valued
Borel measurable function on(oris said to belong to
) if
(I)

$\mathcal{M}(\mathbb{R})$$\varphi$$\eta$$[a,b]\cross \mathbb{R}$$\theta$(1.23)$L_{\varphi;\infty,1;\eta}$$L_{\varphi;\infty,1;\eta}^{t}$$\Vert\theta\Vert_{\varphi;\infty,1;\eta}=\int_{[a,b]}\Vert\theta(s, \cdot)\Vert_{\varphi;\infty}d|\eta|(s)$is finite, where,$\Vert\theta(0, \cdot)\Vert_{\varphi;\infty}=\inf\{\lambda>0||\varphi|(\{\xi\in \mathbb{R}||\theta(0,\xi)|>_{/}l\})=0\}$$\Vert\theta(s, \cdot)\Vert_{\varphi;\infty}=\inf\{\lambda>0|m_{L}(\{\xi\in \mathbb{R}||\theta(s,\xi)|>\lambda\})=0\}$

If is bounded Borel measurable then is in$\theta$$\theta$For(J)$\theta\in L^{\infty}(\mathbb{R},m_{L})$, we consider an operator(1.24)

for$E\in \mathcal{B}(\mathbb{R})$$M_{\theta}$.

.

$M_{\theta}$

frominto itself by$\mathcal{R}\Lambda 4(\mathbb{R})$$[M_{\theta}( \mu)](E)=\int_{E}\frac{d\mu}{dm_{L}}(\xi)\theta(\xi)dm_{L}(\xi)$and$\mu\in \mathcal{R}\mathcal{M}(\mathbb{R})$. Then(1.25)

so$L_{\varphi;\infty,1;\eta}$$(0<s\leq t)$$\frac{dM_{\theta}(\sqrt r)}{dm_{L}}(\xi)=\frac{d\mu}{dm_{L}}(\xi)\theta(\xi)$,

is well-defined. Since,$|M_{\theta}( \mu)|(\mathbb{R})\leq\int_{\mathbb{R}}|\frac{d\mu}{dm_{L}}(\xi)||\theta(\xi)|dm_{L}(\xi)\leq\Vert\theta\Vert_{\infty}|\mu|(\mathbb{R})$$M_{\theta}$

is a bounded linear operator.
For $s>0$ , let(1.26)

for$P_{s}(E)= \int_{E}\frac{1}{\sqrt{2\pi s}}\exp\{-\frac{u^{2}}{2s}\}dm_{L}(u)$$E\in \mathcal{B}(\mathbb{R})$. For $s>0$ , we consider an operator$S_{s}$

from$\mathcal{R}\mathcal{M}(\mathbb{R})$into itself defined by.

(1.27)$[S_{s}( \mu)](E)=(\mu*P_{s})(E)=\frac{1}{\sqrt{2\pi s}}l_{\mathbb{R}}[\int_{E}\exp\{-\frac{(u-v)^{2}}{2s}\}dm_{L}(u)]d\mu(v)$Then$\frac{dS_{s}(\mu)}{dm_{L}}(\xi)=\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}\exp\{-\frac{(\xi-v)^{2}}{2s}\}d\mu(v)$so is well-defined. It is not hard to show thatnorm
of is less than or equals one.
$S_{s}$$\Vert S_{s}\Vert$$S_{s}$,

is a bounded linear operator and the operator$S_{s}$

Let and be two positive real numbers. Then by the Chapman-Kolmogorov equation in[19] and the classical Fubini theorem, we have$s_{1}$(1.28)$s_{2}$$S_{s_{1}}\circ S_{s_{2}}=S_{s_{1}+s_{2}}$.

58
For $s>0,$$E\in \mathcal{B}(\mathbb{R})$$\varphi\in \mathcal{M}(\mathbb{R})$-essentially bounded function on$\theta$$[T(s, \varphi,\theta)](E)=\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}[\int_{E}\theta(v)\exp\{-\frac{(u-v)^{2}}{2s}\}dm_{L}(u)]d\varphi(v)$$T(s,\varphi,\theta)\in \mathcal{R}\mathcal{M}(\mathbb{R})$(1.30)(K)$|\varphi|$

and$\varphi$

is a measure on

if the integral.

and$\frac{dT(s,\varphi,\theta)}{dm_{L}}(u)=\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}\theta(v)\exp\{-\frac{(u-v)^{2}}{2s}\}d\varphi(v)$Let$(\mathbb{R},\mathcal{B}(\mathbb{R}))$, let(1.29)Then, a Borel measurable$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\int_{C[a,b]}F(\lambda^{-1}x)d\omega_{\varphi}(x)$.

and $F:C[a,b]arrow \mathbb{R}$ a measurable function. For all$\lambda>0$,

exists, then we denote$\int_{C[a,b]}F(_{j}t^{-1}x)d\omega_{\varphi}(x)=J(\lambda)$

If there exists a function
all real $l>0$ , then we write$J^{*}(\lambda)$

analytic in the half-plane$\mathbb{C}^{+}$

such that$J(\lambda)=J^{*}(\lambda)$

for almost$\int_{C[a,b]}^{u\downarrow anw_{\lambda}}F(x)d\omega_{\varphi}(x)=J^{*}(/l)$

is the analytic analogue of Wiener integral of over $C[a,b]$ with paramand we call thatexists, then we set
eter 1, and for non-zero real number , if the limit$F$$J^{*}(\lambda)$$q$$\lambdaarrow iq\lim_{\lambda\in C^{+}}J^{*}(,t)$$\lambdaarrow iq\lim_{\Lambda\in C^{+}}J^{*}(\lambda)=\int_{C[a,b]}^{ananf_{q}}F(x)d\omega_{\varphi}(x)$and we say that the limit is the analytic analogue of Feynman integral of .
$F$

Notation. For$\lambda\in \mathbb{C}^{+}$

and $y\in C[a,b]$ let$(T_{an,\lambda}F)(y)= \int_{C[a,b]}^{ananw_{\lambda}}F(x+y)d\omega_{\varphi}(x)$and given a number$p$such that $1\leq p\leq\infty,$$p$,

and $p’$ will always be related by$\underline{1}+\underline{1},$$p$$\{H_{n}\}$and $H$ be analogue of Wiener measurable functions such that for each $\rho>0$ ,$\lim_{narrow\infty}\int_{C[a,b]}|H_{n}(py)-H(py))|^{2}d\omega_{\varphi}(y)=0$

Then we write(1.31)$\lim_{narrow\infty}(w_{\varphi,s}^{2})H_{n}^{a}\approx^{n}H$

.

$p$

$=1$

. Let

59
and we call $H$ the scale invaniant limit in the mean of order 2 ofover $C[a,b]$ . We define asimilar definition for any real number instead of . Let be non-zero real number. For $1<p\leq 2$
we define the analytic Fourier-Feynman transform of , which we denote by, by the$H_{n}$$n$$q$$L^{p}$$F$$T_{an,q}^{(p)}F$

formula$(T_{\mathfrak{U}1,q}^{(p)}F)(y)=$$\lim_{\lambda\in \mathbb{C}^{+},\lambdaarrow-iq}(w_{\varphi,s}^{p’})(T_{an,\Lambda}F)(y)$whenever this limit exists. Let
be a functional on analogue of Wiener space such thatexists infor s-almost every . We define theanalytic analogue of FourierFeynman transform of , which we denote by, as that functional (if it exists) on analogueof Wiener space such that
$F$$(T_{an,\lambda}F)(y)$$L^{1}$$\mathbb{C}^{+}$

$y$

$F$

$T_{an,q}^{(1)}F$$(T_{an,q}^{(1)}F)(y)=$$\lim_{A\in \mathbb{C}^{+},\lambdaarrow-iq}(T_{an,\lambda}F)(y)$for s-almost every . For each natural number and a partition $a=t_{0}<t_{1}<\cdots<t_{n}=b$ , let
be the collection of functions $F:C[0,t]arrow \mathbb{R}$ satisfying (1) and (2) below:(1) $f$ is a measurable function on
.
$n$

$y$

$\mathcal{A}_{n}$$\mathbb{R}^{n+1}$(2) $F(x)\approx f(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))an$.

Let $(-1)!\downarrow=1!!=1,$ $(2n)!!=(2n)(2n-2)\cdots 2,$ $(2n-1)!!=(2n-1)(2n-3)\cdots 3\cdot 1$ for(L)a natural number . Let$n$$\prod_{p=k}^{n}c_{p}=c_{k}c_{k+1}\cdots c_{n}$if $n\geq k$ and$\prod_{p=k}^{n}c_{p}=1$

if $n<k$ .

By the elementary calculus for integral and the properties of Gamma functions, for a positive
real number and for a non-negative integer , we have the following equality.
$A$(1.32)$m$$\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi A}}u^{m}\exp\{-\frac{(u-u_{o})^{2}}{2A}\}dm_{L}(u)=\sum_{k=0}^{m}[_{Z}](_{2k}^{m})A^{k}(2k-1)!!u_{0}^{m-2k}$$= \sum_{k=0}^{[\frac{m}{2}]}\frac{m!A^{k}}{(m-2k)!(2k)!!}u_{0}^{m-2k}$.

Here is the Gauss symbol.
Using Dirichilet’s integral in [14] and the change of vaniables theorem, we can show thefollowing equality.
$[\cdot]$fi(1.33)$\int_{\Delta_{n}^{t}}m_{L})(s,s, \ldots,s_{n})=t^{n+\Sigma_{j=1}^{n}k_{j}}\frac{j=1}{(n+\sum_{j=1}^{n}k_{j})!}$

where
and $s_{0}=0$ .

$k_{1},k_{2},$ $\ldots,k_{n}$$t\}$$k_{j}!$.

are all non-negative integers, $\Delta_{n}^{t}=\{(s,s, \ldots,s_{n})|0<s1<s2<\cdots<s_{n}\leq$60
For a natural number(1.34)$n$, let$\sum_{k,n}’p(k_{1},k_{2}, \ldots,k_{n})=\sum^{1}\sum_{=k_{n}=0k_{n-1}0}^{2-k_{n}}\sum_{k_{n-2}=0k_{1}}^{3-k_{n}}\sum_{=0}^{j=2}p(k_{1},k_{2}, \ldots,k_{n})-k_{n-1}\ldots n-\Sigma^{n}k_{j}$For $1\leq u\leq n-1,$$k_{n-u}$.

moves from to $(u+1)- \sum_{p=1}^{u}k_{n-(p-1)}$ , so$0$$(u+2)- \sum_{p=1}^{u+1}k_{n-(p-1)}=[(u+1)-(\sum_{p=1}^{u}k_{n-(p-1)})-k_{n-u}]+1\geq 1$

Hence $2-k_{n},$ $3-(k_{n}+k_{n-1}),$is well-defined.

$\ldots,n-\sum_{p=2}^{n}k_{p}$.

are all large than or equal 1 which implies that$\sum_{k,n}’p(k_{1},k_{2}, \ldots,k_{n})$\S 2. The complex-Valued Analogue of Wiener Measure$\omega_{\varphi}$

In this section, we will introduce a complex-valued analogue of Wiener measureon$C[a,b]$ and we will give some examples of it.
Let be a non-negative integer. Forwith $a=t_{0}<t_{1}<\cdots<t_{n}\leq b$ , let$C[a,b]arrow \mathbb{R}^{n+1}$
be a function withJE, :$\omega_{\varphi}$$\vec{t}=(t_{0},t_{1}, \ldots,t_{n})$$n$$J_{\vec{t}}(x)=(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$

For $B_{j}\in \mathcal{B}(\mathbb{R})(j=0,1,2, \ldots,n)$ , the subset$J^{\vec{-}1}( \prod_{j=0}^{n}B_{j})$.

of $C[a,b]$ is called an interval and let

be the set of all intervals. For a non-negative finite Borel measure$m_{\varphi}(J_{\vec{t}}^{-1}( \prod_{j=0}^{n}B_{j}))=\int[\int_{n}n+1;\tilde{t};u0,u_{1},$$\varphi$on$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\mathcal{I}$, let$\ldots,0$

where$W(n+1; \vec{t};u_{0},u_{1}, \ldots,u_{n})=(\prod_{j=1}^{n}\frac{1}{\sqrt{2\pi(t_{j}-t_{j-1})}})\exp\{-\frac{1}{2}\sum_{j=1}^{n}\frac{(u_{j}-u_{j-1})^{2}}{t_{j}-t_{j-1}}\}$.

of all Borel subsets in $C[a,b]$ , coincides with the smallest a-algebraon $(C[a,b],\mathcal{B}(C[a,b]))$ such thatgenerated by and there exists a unique positive measure
for all in .
Then the set$\mathcal{B}(C[a,b])$$\mathcal{I}$$\omega_{\varphi}$$\omega_{\varphi}(D=m_{\varphi}(I)$For$\varphi\in \mathcal{M}(\mathbb{R})$$I$$\mathcal{I}$with the Jordan decomposition$\varphi=\sum_{j=1}^{4}\alpha_{j}\varphi_{j}$$\omega_{\varphi}=\sum_{j=1}^{4}\alpha_{j}\omega_{\varphi_{j}}$. We say thatis the complex-valued analogue of Wiener measure on $(C[a,b],\mathcal{B}(C[a,b]))$ , associated with. If is a Dirac measure at the origin in then is the classical Wiener measure.
By the change of variables formula, we can easily prove the following theorem.

$\omega_{\varphi}$$\varphi$, let$\varphi$$\delta_{0}$$\mathbb{R}$$\omega_{\varphi}$

61
Theorem 2.1 (The Wiener Integration Formula). If
pnction then the following equality holds.

is a Borel measurable$f:\mathbb{R}^{n+1}arrow \mathbb{C}$$\int_{C[a,b]}f(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))d\omega_{\varphi}(x)$$=* \int_{\mathbb{R}^{n+1}};\vec{t};0,1\cdot\cdot 2,$

where $=*means$ that$\iota f$$\ldots$,

one side exists then both sides exist and the two values are equal.

Remark. Let $\varphi\in M(R)$ .
(1) It is not hard to show that(2)$\omega_{\varphi}(C[a,b])=\varphi(\mathbb{R})$$\omega_{\varphi}$

has no atoms.

.

(3) Let $J_{t}:C[a,b]arrow \mathbb{C}$ be a function with $J_{t}(x)=x(t)$ . Then for $E$ in$[S_{t}(\varphi)](E)$$\mathcal{B}(\mathbb{R}),$$\omega_{\varphi}(J_{t}^{-1}(E))=$.

Example 2.2. Let $\varphi\in M(R)$ .
(1) Let $I=\{x\in C[0,t]|x(0)\in B\}$ where is in. Then$0\leqs\leq t$ ,(2) Suppose that $f(u)=u$ is -integrable. Then for$B$$\mathcal{B}(\mathbb{R})$$\omega_{\varphi}(I)=\varphi(B)$.

$\varphi$$\int_{C[0,t]}x(s)d\omega_{\varphi}(x)=\int_{\mathbb{R}}ud\varphi(u)$

If

then

$\varphi=\delta_{p}$$\sigma^{2}$then$\int_{C[0,t]}x(s)d\omega_{\varphi}(x)=p$.

and if has a normal distribution with mean$\varphi$$\alpha$and vaniation.

$\int_{C[0,t]}x(s)d\omega_{\varphi}(x)=\alpha$(3) Suppose that $g(u)=u^{2}$ is -integrable. Then for $0\leq s\leq t$ ,$\varphi$.

$\int_{C[0,t]}x(s)^{2}d\omega_{\varphi}(x)=\int_{\mathbb{R}}u^{2}d\varphi(u)+s\varphi(\mathbb{R})$If$\varphi=\delta_{p}$variancethen $\int_{C[0,t]}x(s)^{2}d\omega_{\varphi}(x)=p^{2}+s$ and if$\sigma^{2}$$\varphi$

has a normal distribution with mean

and

then
$\int_{C[0,t]}x(s)^{2}d\omega_{\varphi}(x)=\alpha^{2}+\sigma^{2}+s$(4) Let$\alpha$$\mathcal{F}(\varphi)$be the Fourier transform of a measure , that is,$\varphi$.

$[ \mathcal{F}(\varphi)](\xi)=\int_{\mathbb{R}}\exp\{i\xi u\}d\varphi(u)$Then for $0\leq s\leq t$ ,$\int_{C[0,t]}\exp\{i\xi x(s)\}d\omega_{\varphi}(x)=\exp\{-\frac{s\xi^{2}}{2}\}[\mathcal{F}(\varphi)](\xi)$

.

.

62
If

$\varphi=\delta_{p}$then $\int_{C[0,t]}\exp\{i\xi x(s)\}d\omega_{\varphi}(x)=\exp\{-\frac{s\xi^{2}}{2}+ip\xi\}$ and ifwith mean$\alpha$and variance$\sigma^{2}$$\varphi$

has a normal distribution

then$\int_{C[0,t]}\exp\{i\xi x(s)\}d\omega_{\varphi}(x)=\exp\{-\frac{(s+\sigma^{2})\xi^{2}}{2}+i\alpha\xi\}$.

be a function with $J_{s}(x)=x(s)$ . We assume that
Let $0<s\leq t$ be given and letconverges
converges to weakly. By calculation similar as in this example, since
pointwise, so by the continuity
converges topointwise,toweakly.
converges totheorem in [1, Theorem 12-5A, $p273$ ],$k(u)=u^{2}$,(5) We assume thatis -integrable. For$J_{s};C[0,t]arrow \mathbb{R}$$\{\mathcal{F}(\varphi_{n})\}$$\{\varphi_{n}\}$$\varphi$$\mathcal{F}(\omega_{\varphi}(J_{s}^{-1}(\cdot)))$$\{\mathcal{F}(\omega_{\varphi_{n}}(J_{s}^{-1}(\cdot)))\}$$\mathcal{F}(\varphi)$$\omega_{\varphi}(J_{s}^{-1}(\cdot))$$\langle\omega_{\varphi_{n}}(J_{s}^{-1}(\cdot))\rangle$$0\leq s_{1},s_{2}\leq t$$\varphi$$\int_{C[0,t]}x(s_{1})x(s_{2})d\omega_{\varphi}(x)=(\min\{s_{1},s_{2}\})\varphi(\mathbb{R})+\int_{\mathbb{R}}u^{2}d\varphi(u)$If.

then $\int_{C[0,t]}x(s_{1})x(s_{2})d\omega_{\varphi}(x)=\min\{s_{1},s_{2}\}+p^{2}$ and if has a normal distribution with$\varphi=\delta_{p}$$\varphi$mean and variance$\sigma^{2}$$\alpha$.

$\int C[0,t]^{x(s_{1})x(s)d\omega_{\varphi}(x)=\min\{s_{1},s_{2}\}+\alpha^{2}+\sigma^{2}}2$(6) For $0\leq S1<S2\leq s3<S4\leq t$ and for$\alpha,$$\beta\in \mathbb{R}$.

, using the change of variable formula, wehave$\varphi(\mathbb{R})\omega_{\varphi}(\{x\in C[0,t]|x(s_{2})-x(s_{1})\leq\alpha$and $x(s_{4})-x(s_{3})\leq\beta\})$$=\omega_{\varphi}(\{x\in C[0,t]|x(s_{2})-x(s_{1})\leq\alpha\})\cdot\omega_{\varphi}(\{x\in C[0,t]|x(s_{4})-x(s_{3})\leq\beta\})$Hence, if$\varphi$.

is a probability measure then $x(s2)-x(s_{1})$ and $x(s)-x(s)$ are independent.

Theorem 2.3. For$\varphi\in \mathcal{M}(\mathbb{R}),$$|\omega_{\varphi}|=\omega_{|\varphi|}$on $(C[a,b],\mathcal{B}(C[a,b]))$ .

We consider a set
are both measures on $(C[a,b],\mathcal{B}(C[a,b])),$
and$\mathcal{A}=\{E\in \mathcal{B}(C[a,b])||\omega_{\varphi}|(E)=\omega_{|\varphi|}(E)\}$$|\omega_{\varphi}|$$\omega_{|\varphi|}$. Then we have.
on$|\omega_{\varphi}|=\omega_{|\varphi|}$$\mathcal{I}\subset \mathcal{A}$. Since$\mathcal{B}(C[a,b])$

Theorem 2.4. If a sequenceof non-negative finite measures, converges to in theconverges toin the total variation
sense of total variation norm then a sequence$\{\varphi_{n}\rangle$$\varphi$$\{\omega_{\varphi_{n}}\rangle$$\omega_{\varphi}$norm.
does not conof measures on $C[a,b]$ such that
From [2], we can find a sequence
verges to weakly even though every finite dimensional measures of converges to somefinite dimensional measure of weakly. Here, we want to find the conditions such thatconverges to weakly.
converges to weakly whenever$(P_{n}\rangle$$\langle P_{n}\rangle$$P$$P_{n}$$P$$\{\omega_{\varphi_{n}}\}$$\langle\varphi_{n}\}$$\omega_{\varphi}$$\varphi$

63
Lemma 2.5. Let $X:[a,b]\cross C[a,b]arrow \mathbb{R}$ be a function with $X(s,x)=x(s)$ . Then for $a<t_{1}\leq$$b$

and for$\epsilon>0$,$\omega_{\varphi}(\{x|\sup\{x(s)-x(a)|a\leq s\leq t_{1}\}\geq\epsilon\})\leq\frac{1}{\epsilon}\sqrt{\frac{2t_{1}}{\pi}}\exp\{-\frac{\epsilon^{2}}{2t_{1}}\}$

Lemma 2.6. For$\epsilon>0$

and$\lambda>0$.

,$\omega_{\varphi}(\{x|\sup_{\epsilon}|x(t)-x(\epsilon)|\leq jt\})=\omega_{\varphi}(\{x|\sup_{\epsilon 0\leq t\leq 0<t<Z}|x(t)-x(0)|\leq/\iota\})^{2}$.

Corollary 2.7.
$\omega_{\varphi}(\{x|\sup_{0\leq s\leq t_{1}}|x(s)-x(t_{1})|>\lambda\})\leq\frac{1}{\lambda}\sqrt{\frac{t_{1}}{\pi}}e^{t_{1}}-L^{2}(2-\frac{1}{\lambda}\sqrt{\frac{t_{1}}{\pi}}e^{-\frac{A^{2}}{t_{1}}})$.

Corollary 2.8. For each positive and , there exists a with $0<\delta<1$ such that for $sl,S2$$\delta$$\epsilon$$\eta$

in $[a,b]$$\omega_{\varphi}(\{x| \sup |_{X(S)-X(S2}1)|\geq\epsilon\})\leq\eta$.

$|s_{1}-s_{2}|<\delta$

From [2], we find the following theorem.

Theorem 2.9. The sequence
positive there exists a compact set
only if$\{P_{n}\}$$\epsilon$$K$of pmbability measures on $C[a,b]$ is tight, that is, forsuch thatfor all natuml number , if and(i) for each positive , there exists an$\eta$(ii) for each positivethat for $n\geq n0$ ,$\epsilon$$P_{n}(K)>1-\epsilon$$n$such that $P_{n}(\{x||x(a)|>\alpha\})\leq\eta$ for all andand , there exists a with $0<\delta<1$ and a natuml number no such$n$$\alpha$$\delta$$\eta$$P_{n}( \{x|\sup_{|s_{1}-s_{2}|<\delta}|x(s_{12})-x(s)|\geq\epsilon\})\leq\eta$.

From [2], we can find a sequenceof measures on $C[a,b]$ such that
does not converges to weakly even though every finite dimensional measures of converges to somefinite dimensional measure of weakly. Here, we want to find the conditions such thatconverges to
weakly wheneverconverges to weakly.
$\langle P_{n}\}$$\langle P_{n}\rangle$$P$$P_{n}$$P$$\langle\omega_{\varphi_{n}}\rangle$$\{\varphi_{n}\rangle$$\omega_{\varphi}$$\varphi$

Theorem 2.10. Let
mensional distributions ofconverges to weakly.
$P_{n},$$P$$P_{n}$

be pmbability measures on $(C[a,b],\mathcal{B}(C[a,b]))$ . If the finite diconverge weakly to those of , andis tight, then$P$$\iota f\{P_{n}\}$$\langle P_{n}\}$$P$

Theorem 2.11. Suppose$\{\varphi_{n}\}$is tight. Then$\{\omega_{\varphi_{n}}\}$is also tight.

Lemma 2.12. Let $f$ :be bounded continuous. Let$tor$ inwith $t_{0}=a<t_{1}<\cdots<t_{n}\leq b$ and$(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$ . Suppose
converges to weakly. Then$\mathbb{R}^{n+1}arrow \mathbb{R}$$\mathbb{R}^{n+1}$

be a veca function with$\vec{t}=(t0,t_{1}, \ldots,t_{n})$$J_{\tilde{t}}:C[a,b]arrow \mathbb{R}^{n}$$\{\varphi_{n}\rangle$$\varphi$$\lim_{marrow\infty}\int_{\mathbb{R}^{n+1}}f(u_{0},u_{1}, \ldots,u_{n})d\omega_{\varphi_{m}}J_{\vec{t}}^{-1}(x)=\lim_{marrow\infty}\int_{C[a,b]}f(J_{\vec{t}}(x))d\omega_{\varphi_{m}}(x)$$J_{\vec{t}}(x)=$64$= \lim_{marrow\infty}\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}f(u_{0},u_{1}, \ldots,u_{n})\frac{\exp\{-\sum_{j=1}^{n}\frac{(u_{j}-u_{j-1})^{2}}{2(t_{j}-t_{j-1})}\}}{n}d\prod m_{L}(u_{1}, \ldots,u_{n})d\varphi_{m}(uo)n$$j=1$$\prod_{j=1}\sqrt{2\pi(t_{j}-t_{j-1})}$$= \int_{C[a,b]}f(J_{t}\prec x))d\omega_{\varphi}(x)$Theorem 2.13. If$\{\varphi_{n}\}$.

is tight and$\{\varphi_{n}\}$converges to$\varphi$weakly, then$\{\omega_{\varphi_{n}}\rangle$

converges to$\omega_{\varphi}$weakly.

andRemark The referee point out the following facts: for $y\in C[a,b]$ , there are$x\in C_{0}[a,b]$ with $y=\alpha+x$ where $\alpha=y(a)$ and $x=y-\alpha\in C_{0}[a,b]$ . Let. By
be a function with $\psi(y)=(\alpha,x)$ as in above. Then
. Using thisTwo norm theorem [26], is a homeomophism. So we havefacts, we can easily prove the following corollary.
$\alpha\in \mathbb{R}$$\psi;C[a,b]arrow \mathbb{R}\oplus C_{0}[a,b]$$\Vert y\Vert_{\infty}\leq|\alpha|+\Vert x\Vert_{\infty}\equiv\Vert(\alpha,x)\Vert=\Vert\psi(y)\Vert$$\omega_{\varphi}=(\varphi\cross m_{\omega})0\psi^{-1}$$\psi$

Corollary 2.14. Let $f$ be in$L^{1}(\mathbb{R})$and set $\varphi(E)=\int_{E}f(x)dm_{L}(x)$ where $f>0$ andBorel subset of . For any integrablefunction$\mathbb{R}$

(2.1)

$F$

$E$

is a

,

$\int_{C[a,b]\cross C[a,b]}F(x,y)d\omega_{\varphi}\cross\omega_{\varphi}(x,y)$$= \int_{C[a,b]\cross C[a,b]}F(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta)d\omega_{\varphi}\cross\omega_{\varphi}(x,y)$

for all real number$\theta\iota f$

and only$\iota f$

the function $f(x)$ has the form $Ae^{-aP}$ where

$A$

and

$a$

,

are

positive constants.

\S 3. A Translation Theorem on$(C[a,b],\mathcal{B}(C[a,b]),\omega_{\varphi})$and the Paley-Wiener-Zygmund Integral
It is well-known fact that there is no quasi-invariant probability measure on the infinite dimensional vector space [49]. So, there is no quasi-invariant probability measure on $C_{0}[a,b]$
or $C[a,b]$ . In 1944, under the some assumptions, Cameron and Martin established a translation theorem on $(C_{0}[a,b],m_{w})$ in [5]. In this section, we will prove a translation theorem onunder the similar assumptions to Cameron’s assumptions. From these concepts,
we will show that the Paley-Wiener-Zygmund integral is well-defined -a.e.
By either the similar method as in the proof of Cameron and Martin’s translation theorem
on $C_{0}[a,b]$ in [5] or Remark 2, we can prove the following theorem.
$(C[a,b],\omega_{\varphi})$$\omega_{\varphi}$

Theorem 3.1 (The Translation Theorem onbounded variation. Let$\alpha\in \mathbb{R}$$(C[a,b],\mathcal{B}(C[a,b]),\omega_{\varphi})$). Let $h\in C[a,b]$ and ofand set $x o(s)=\int_{a}^{s}h(u)dm_{L}(u)+\alpha$for $a\leq s\leq b$. Let $L:C[a,b]arrow$

65
afunction with $L(x)=x+x_{0}$ and a pmbability measure ona measure onsuch that
andfor$C[a,b]$ be$\varphi$$\varphi_{\alpha}(B)=\varphi(B+\alpha)$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\omega_{\varphi}$-integmble then $F(x+x_{0})$ is$\omega_{\varphi}$-integmble$ofx$$B\in \mathcal{B}(\mathbb{R})$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\varphi_{\alpha}\ll\varphi$. Let. Then$F\equiv 1$$\iota fF$

be
is

and

$\int_{C[a,b]}F(y)d\omega_{\varphi}(y)=e^{-}1z^{\Vert h\Vert_{2}^{2}}\int_{C[a,b]}F(x+x_{0})e^{-\int_{a}^{b}h(u)dx(u)}\frac{d\varphi_{\alpha}}{d\varphi}(x(0))d\omega_{\varphi}(x)$Putting$\varphi_{\alpha}$.

in Theorem 3.1, we have the following corollary.

Corollary 3.2. Under the assumptions in Theorem 3.1,.

$\int_{C[a,b]}\exp\{-\int_{a}^{b}h(u)dx(u)\}d\omega_{\varphi}(x)=\exp\{-\frac{1}{2}\Vert h\Vert_{2}^{2}\}$

Replacing by in Corollary 3.2, by the uniqueness theorem for analytic extension in thetheory of complex analysis, we have the following corollary.
$h$$\lambda h$Corollary 3.3. Under the assumptions in Theorem 3.1, for all$\lambda\in \mathbb{C}$,$\int_{C[a,b]}\exp\{-,t\int_{a}^{b}h(u)dx(u)\}d\omega_{\varphi}(x)=\exp\{-\frac{A^{2}}{2}\Vert h\Vert_{2}^{2}\}$.

Theorem 3.4. Consider a mndom variable $X:C[a,b]arrow \mathbb{R}$ with $X(x)= \int_{a}^{b}h(u)dx(u)$ under the assumptions in Theorem 3.1. Thenthe variation.

$X$

has a normal distribution with the mean $zem$ and$\Vert h\Vert_{2}^{2}$

By the same method as in the proof of [50, Theorem 29.7], we can prove the following
theorem.

Theorem 3.5. Let$\{h_{1},h_{2}, \ldots,h_{n}\}$variation. For $i=1,2,$
each$X_{i}$$\ldots,n$be an orthonormal system such that each

, let $X_{i}(x)= \int_{a}^{b}h_{i}(s)dx(s)$.

Thenhas the standard normal distribution. Moreover, if$X_{1},X_{2},$ $\ldots,X_{n}$$f:\mathbb{R}^{n}arrow \mathbb{R}$$h_{i}$

is of bounded

are independent,

is Borel measumble,$\int_{C[a,b]}f(X_{1}(x),X_{2}(x), \ldots,X_{n}(x))d\omega_{\varphi}(x)$$=*(2 \pi)^{-\frac{n}{2}\int_{\mathbb{R}^{n}}2}f(u1,u2, \ldots,u_{n})\exp\{-\frac{1}{2}\sum_{j=1}^{n}u_{J}\}d\prod_{i=1}^{n}m_{L1}(u,u, \ldots,u_{n})$where $=*means$ that$\iota f$

one side exists then both sides exist and the two values are equal.

Let $\{e_{k}|k=1,2, \ldots\}$ be a complete orthonormal set in $L^{2}([a,b],m_{L})$ such that eachbounded variation. For $f$ in $L^{2}([a,b],m_{L})$ and in $C[a,b]$ , let$x$$\int_{a}^{b_{\text{へ}}}f(s)dx(s)=\lim_{narrow\infty}\int_{a}^{b}[\sum_{k=1}^{n}\int_{a}^{b}f(u)e_{k}(u)dm_{L}(u)e_{k}(v)]dx(v)$$e_{k}$

is of

66
if the limit exists.
$\int_{a}^{b}f(s)\hat{d}x(s)$is called the Paley-Wiener-Zygmund integral of $f$ according to . By the routine$x$method in the theory of Wiener space, we can prove that the integral$\int_{a}^{b_{\text{へ}}}f(s)dx(s)$is indepen-

dent on the orthonormal set $\{e_{k}|k=1,2, \ldots\}$ and the Paley-Wiener-Zygmund integral exists$\omega_{\varphi}-a.e.x\in C[a,b]$.

Remark In 1980, Cameron and Storvick introduced the definitions and some related theories of the spaces $S,S’$ and $S”$ of Wiener functionals. If we replace $(C_{0}[a,b],m_{w})$ by
which are similar to Cameron andin their paper, we can prove various results on
Storvick‘s results in [7].
$(C[a,b],\omega_{\varphi})$$(C[a,b], \omega_{\varphi})$\S 4. The Generalized Femique‘s Theorem for Analogue of Wiener Measure Space
From Lemma 2.5, we have the following lemma.

Lemma 4.1.
(4.1)

$m_{\varphi}( \{x\in C|\sup_{0\leq s\leq 1}|x(s)-x(0)|\geq K\})\leq\frac{1}{K}\sqrt{\frac{2}{\pi}}\exp\{-\frac{K^{2}}{2}\}$for a positive real number .
$K$

In this section, we investigate the existence of the integral

for two positive real numbers$\alpha,$$p$

Theorem 4.2. For $0<p<2,$
real number . If $p=2$ then$\alpha$.

$\int_{C}\exp\{\alpha(\sup_{0\leq s\leq l}|x(s)-x(O)|)^{p}\}dm_{\varphi}(x)$$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)-x(0)|^{p}\}dm_{\varphi}(x)$Theorem 4.3. $IfO<p<1$ and$\int_{\mathbb{R}}\exp\{2\alpha|u|^{p}\}d\varphi(u)$, then $\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{p}\}dm_{\varphi}(x)$ is$\alpha$Theorem 4.4. If $1\leq p<2$ andis finitefor all positiveis finite for$0< \alpha<\frac{1}{2}$.

is finite for some positive real number

finite.

$\int_{\mathbb{R}}\exp\{2^{p}\alpha|u|^{p}\}d\varphi(u)$is finite, then$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{p}\}dm_{\varphi}(x)$is finite.

$\int_{C}\exp\{\alpha(\sup_{0\leq s\leq 1}|x(s)|)^{p}\}dm_{\varphi}(x)$67
Theorem 4.5. If$\alpha<\frac{1}{2}$and$\int_{\mathbb{R}}\exp\{4\alpha|u|^{2}\}d\varphi(u)$is finite then$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{2}\}dm_{\varphi}(x)$is finite.

Remark If $p>2$ and $\alpha>0$ then by Theorem 2.1,
(4.2)$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{p}\}dm_{\varphi}(x)\geq\int_{C}\exp\{\alpha|x(1)|^{p}\}dm_{\varphi}(x)$$= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\int_{\mathbb{R}}\exp\{\alpha|u1|^{p_{-\frac{1}{2}(u_{1}-u0}})^{2}\}dm_{L}(u1)d\varphi(u_{0})$$\geq\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\int_{|u_{1}|\leq l}\exp\{\alpha|u_{1}|^{p}-\frac{1}{2}(u_{1}-u_{0})^{2}\}dm_{L}(u_{1})d\varphi(u_{0})$$+ \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\int_{|u_{1}|\geq l}\exp\{\alpha|u_{1}|^{2}-\frac{1}{2}(u_{1}-uo)^{2}\}dm_{L}(u_{1})d\varphi(uo)$$=+\infty$

Remark Suppose
theorems above,$\varphi=\delta_{0}$, that is,$(C,m_{\varphi})$is the concrete Wiener measure space. Then by the$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{p}\}dm_{\varphi}(x)$$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{2}\}dm_{\varphi}(x)$$+\infty$.

is finite foris finite for $0<p<2$ and all real number and$\alpha<\frac{1}{2}$$\alpha$. Moreover,$\int_{C}\exp\{\alpha\sup_{0\leq s\leq 1}|x(s)|^{p}\}dm_{\varphi}(x)=$for $p>2$ and $\alpha>0$ .

\S 5. An Integration Formula for Analogue of Wiener Measure
In this section, we investigate the integral of functionals such as $F(x)=( \int_{0}^{t}x(s)^{2}dm_{L}(s))^{n}$
and $G(x)= \exp\{\lambda\int_{0}^{t}x(s)^{2}dm_{L}(s)\}$ and we give some corollaries, follows from our results.

Lemma 5.1. Let $0=s_{0}<s_{1}<S2<\cdots<s_{n}=t$ . Suppose(5.1)$u_{0}^{2n}$is -integmble. Then$\varphi$$\int\prod_{C[0,t]j=1}^{n}x(s_{j})^{2}d\omega_{\varphi}(x)$$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n^{n}}k_{j}-1)\}$$= \sum_{k,n}’\{\frac{l=2j=n+2-lj=n+2-l}{nn}\prod^{n}(s_{j}-s_{j-1})^{k_{j}}\}\int_{\mathbb{R}}u_{0}^{2n-2\Sigma_{j=1}^{n}k_{j}}d\varphi(uo)$$(n- \sum_{j=1}^{n}k_{j})!(2n-2\sum_{j=1}k_{j}-1)!!\prod_{j=1}k_{j}!^{j=1}$.

68
Theorem 5.2. Let $F(x)=( \int_{0}^{t}x(s)^{2}dm_{L}(s))^{n}$ on $C[0,t]$ where

pose
(5.2)

$u_{0}^{2n}$$n$is a natuml number. Sup-is -integrable. Then$\varphi$$\int_{C[0,t]}F(x)d\omega_{\varphi}(x)$$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n}k_{j}-1)\}t^{n+\Sigma_{j=1}^{n}k_{j}}n$.
$=n! \sum_{k,n}’\frac{t=2j=n+2-lj=n+2-l}{(n+\sum_{j=1}^{n}k_{j})!(n-\sum_{j=1}^{n}k_{j})!(2n-2\sum_{j=1}^{n}k_{j}-1)!!\prod_{j=1}^{n}(2k_{j}-1)!!}\int_{\mathbb{R}}u_{0}^{2n-2\Sigma_{j=1}^{n}k_{j}}d\varphi(u_{0})$In Theorem 5.2, by putting $t=1$ and$\varphi=\delta_{0}$concrete Wiener measure on $C_{0}[0,t]$ , that is,

and $\int_{\mathbb{R}}0=1$ if, the Dirac measure at the origin$\omega_{\varphi}=m_{w},$$n= \sum_{j=1}^{n}k_{j}$$0\in \mathbb{R},$$\int_{\mathbb{R}}0=0$ if$\omega_{\varphi}$is the$n \neq\sum_{j=1}^{n}k_{j}$. So, we have the following corollary.

Corollary 5.3. Let $F(x)=( \int_{0}^{1}x(s)^{2}dm_{L}(s))^{n}$ on $C_{0}[0,1]$ . Then$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n}k_{j}-1)\}n$(5.3)$\int_{C[0,1]}F(x)dm_{w}(x)=\frac{1}{2^{n}(2n-1)!!}\sum_{k,n}’\frac{l=2j=n+2-lj=n+2-l}{\prod_{j=1}^{n}(2k_{j}-1)!!}$

Theorem 5.4. Suppose$At< \frac{1}{2}$and$\exp\{u^{2n}\}$is -integrable on

Let $G(x)= \exp\{\lambda\int_{0}^{t}x(s)^{2}dm_{L}(s)\}$ on $C[0,t]$ . Then $G(x)$ is(5.4)

for all natuml number .

$\mathbb{R}$$n$$\varphi$$\omega_{\varphi}$-integmble and$\int_{C[0,t]}G(x)d\omega_{\varphi}(x)$$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n}k_{j}-1)\}t^{n+\Sigma_{j=I}^{n}k_{j}}n$$= \varphi(\mathbb{R})+\sum\lambda^{n}\sum’\frac{l=2j=n+2-lj=n+2-l}{nnnn}\infty$$n=1$$k,n(n+ \sum_{j=1}k_{j})!(n-\sum_{j=1}k_{j})!(2n-2\sum_{j=1}k_{j}-1)!!\prod_{j=1}(2k_{j}-1)!!$$\cross\int_{\mathbb{R}}u_{0}^{2n-2\Sigma_{j=1}^{n}k_{j}}d\varphi(uo)$.

In Theorem 3.1 of section 4, by putting $h(u)=0$ on $[0,t]$ , if$F(x+\alpha)$ is-integrable and$F$$\omega_{\varphi-a}$$\int_{C[0,t]}F(x)d\omega_{\varphi-\alpha}(x)=\int_{C[0,t]}F(x+\alpha)d\omega_{\varphi}(x)$Using this, we can prove the following corollary.

.

is$\omega_{\varphi-a}$-integrable then69
Corollary 5.5. Suppose$\mathcal{X}<\frac{1}{2}$and $\exp\{u^{2n}\}$ is$\varphi-\alpha$-integrable where

Let $G(x)= \exp\{\lambda\int_{0}^{t}(x(s)+\alpha)^{2}dm_{L}(s)\}$ on $C[0,t]$ . Then $G(x)$ is(5.5)$\omega_{\varphi}$$\alpha$is a real number.

-integmble and$\int_{C[0,t]}G(x)d\omega_{\varphi}(x)$$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n^{n}}k_{j}-1)\}t^{n+\Sigma_{j=1}^{n}k_{j}}$$= \varphi(\mathbb{R})+\sum\lambda^{n}\sum’\frac{l=2j=n+2-lj=n+2-l}{nnnn}\infty$$n=1$$k,n(n+ \sum_{j=1}k_{j})!(n-\sum_{j=1}k_{j})!(2n-2\sum_{j=1}k_{j}-1)!!\prod_{j=1}(2k_{j}-1)!!$$\cross\int_{\mathbb{R}}u_{0}^{2n-2\Sigma_{j=1}^{n}k_{j}}d\varphi_{-(f}(u_{0})$In Theorem5.4, putting $t=1$ and$\varphi=\delta_{0}$, we have the following corollary by [4].

Corollary 5.6. For any positive real number$l$,$\int_{C_{0}[0,1]}\exp\{-\lambda\int_{0}^{1}x(s)^{2}dm_{L}(s)\}dm_{w}(x)$$\prod\{(l-\sum^{n}k_{j})(2l-2\sum^{n^{n}}k_{j}-1)\}$$=1+ \sum_{n=1}^{\infty}/\frac{t^{n}}{n!2^{n}(2n-1)!!}\sum_{k,n}’\frac{l=2j=n+2-lj=n+2-l}{\prod_{j=1}^{n}(2k_{j}-1)!!}=(\cosh\sqrt{2,t})^{-}21$.

\S 6. Probabilities of Analogue of Wiener Paths
Crossing Continuously Differentiable Curves
In this section, we give the analogue of Wiener measureof $\{x\in C[0, T]|x(O)<f(O)$and $x(s0)\geq f(so)$ for some $s_{0}\in[0, T]\}$ by use of integral equation techniques. This result is a
generalization of Park and Paranjape‘s 1974 result [31].
Let $T>0$ be given andthe standard Wiener measure on the space $C_{0}[0, T]$ of all contin$x(O)=0$
uous functions with
. From [45] and [46], we can found the following equations: for$m_{\varphi}$$m_{w}$$x$$b\geq 0$(6.1),$m_{w}(\{x\in C_{0}[0,$ $T]| \sup_{0\leq t\leq T}x(t)\geq b\})=2\int_{b/\sqrt{T}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{u^{2}}{2}}du$

and
(6.2)

$m_{w}(\{x\in C_{0}[0,$$T]| \sup_{0\leq t\leq T}(x(t)-at)\geq b\})$$= \int_{(aT+b)/\sqrt{T}}+_{u^{2}}\infty\frac{1}{\sqrt{2\pi}}e^{-}\tau du+e^{-2ab}\int_{-\infty}(aT-b)/_{u^{2}}\sqrt{T}\frac{1}{\sqrt{2\pi}}e^{-}\tau du$.

70
In 1974, Park and Paranjape proved the following theorem [31].

$\frac{C}{t^{p}}$Theorem 6.1. Let $f(t)$ be continuous on $[0,T]$ , differentiable in $(0,T)$ , and $satisp|f’(t)|\leq$$(0<p< \frac{1}{2})$for some constant C. Then for $b\geq-f(O)$,(6.3)$m_{w}( \{x\in C_{0}[0,T]|\sup_{0\leq t\leq T}(x(t)-f(t))\geq b\})$$=2 \int_{(f(T)+b)/\sqrt{T}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-z^{-}}u^{2}du-4\int_{0}^{T}M(T,t)[\int_{(f(T)+b)/\sqrt{T}}+_{u^{2}}\infty\frac{1}{\sqrt{2\pi}}e^{-T}du]dt$$+ \sum_{n=1}^{\infty}4^{n}\int_{0}^{T}K_{n}(T,t)[2\int_{(\int(t)+b)/\sqrt{t}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-T}duu^{2}$$-4 \int_{0}^{t}M(t,s)\int_{(f(s)+b)/\sqrt{s}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-T}u^{2}duds]dt$

,

where

$M(t,s)=\{\begin{array}{ll}\frac{\partial}{\partial s}\int_{-\infty}^{(f(t)-f(s))/\sqrt{t-s}}\frac{1}{\sqrt{2\pi}}e^{-T}duu^{2} (0\leq s<t\leq T),0 (0\leq t<s\leq T),\end{array}$$K_{1}(T,t)= \int^{T}M(T,s)M(s,t)ds$,and$K_{n+1}(T,t)= \int_{t}^{T}K_{n}(T,s)K_{1}(s,t)ds$.

of
The main purpose of this section is to find the analogue of Wiener measurecontinuously differentiable function $f$ on $[0, T]$ , which is a$C[0,T]| \sup_{0\leq t\leq T}(x(t)-f(t))\geq 0\}$ for$m_{\varphi}$$\{x\in$generalization of Theorem 6.1.
Throughout in this section, $\int_{a}^{b}f(u)du$ means the Henstock integral of $f$.

Let $f:[0,T]arrow \mathbb{R}$ be continuously differentiable and $f(s)=0$ if $s\leq 0$ . For $t\in[0,T]$ , the$f(t)-f(s)$exists and equals to .
limit
$0$

$\lim_{sarrow t^{-}}\overline{\sqrt{t-s}}$be the first hitting time of the curve $f$ from below by , that is, $x(\tau(x))=$
For $x\in C[0, T]$ , let$f(\tau(x))$ . If
never reaches the curve $f$, let $\tau(x)=+\infty$ .
For $t\in[0, T]$ , let$\tau(x)$

$x$

$x$

(6.4)

$A_{t}=\{x\in C[0,$ $T]|x(0)<f(0)$and for some $s0\in[0,t],$ $x(s_{0})\geq f(so)\}$ .

71
Let

$G:\mathbb{R}arrow \mathbb{R}$be a function with

$G(t)=\{\begin{array}{ll}0 (t<0),m_{\varphi}(A_{t}) (0\leq t\leq T),m_{\varphi}(A_{T}) (T<t).\end{array}$

Lemma 6.2.

$G$is increasing and continuous with $G(O)=0$ .

Lemma 6.3. If $0\leq s<t\leq T$ then $\tau(x)=s$ and $x(t)-x(s)$ are independent.
The following theorem is one of main theorems in these notes.

Theorem 6.4. For $0<t\leq T,$ $G(t)$ satisfies the following Volterm’s integml equation of thesecond kind(6.5) $G(t)=2 \int_{-\infty}^{f(0)}[\int_{f(t)}^{+\infty}\frac{1}{\sqrt{2\pi t}}\exp\{-\frac{(u_{1}-u_{0})^{2})}{2t}\}du_{1}]d\varphi(u_{0})-2\int_{0}^{t}G(s)M(t,s)ds$

where$M(t,s)=\{\begin{array}{ll}\frac{\partial}{\partial s}\int_{-\infty}^{(f(t)-f(s))/\sqrt{t-s}}\frac{1}{\sqrt{2\pi}}e^{-\frac{u^{2}}{2}}du (0\leq s<t\leq T),0 (0\leq t\leq s\leq T).\end{array}$

The equality (6.5) and the change of order of integration gives(6.6)$G(t)=2 \int_{-\infty}^{f(0)}[\int_{f(t)}^{+\infty}\frac{1}{\sqrt{2\pi t}}\exp\{-\frac{(u_{1}-uo)^{2})}{2t}\}du_{1}]d\varphi(u_{0})$$-4 \int_{0}^{t}[l_{-\infty}^{f(0)}[\int_{f(s)}^{+\infty}\frac{1}{\sqrt{2\pi s}}\exp\{-\frac{(u_{1}-u_{0})^{2})}{2s}\}du_{1}]d\varphi(u_{0})]M(t,s)ds$$+4 \int_{0}^{t}[l^{t}M(s,z)M(t,s)ds]G(z)dz$,

if $M(s,z)M(t,s)G(z)$ is integrable on $\{(s,z)|0\leq z<s\leq t\}$ .
By [47], we obtain the main theorem in these notes.

Theorem 6.5. If $l^{t}M(s,z)M(t,s)ds$ is square integmble on $\{(z,t)|0\leq z<t\leq T\}$ then theequation (6.5) has one and essentially only one solution in the classby the formula(6.7)$G(t)=2 \int_{-\infty}^{f(0)}[\int_{f(t)}^{+\infty}\frac{1}{\sqrt{2\pi t}}\exp\{-\frac{(u_{1}-u_{0})^{2})}{2t}\}du_{1}]d\varphi(u_{0})$$L^{2}$. This solution is given

72,$+ \sum_{n=1}^{\infty}(-1)^{n}2^{n+1}\int_{0}^{t}[\int_{-\infty}^{f(0)}[\int_{f(s)}^{+\infty}\frac{1}{\sqrt{2\pi s}}\exp\{-\frac{(u_{1}-uo)^{2})}{2s}\}du\iota]d\varphi(u_{0})]H_{n}(t,s)ds$

where $H_{1}(t,s)=M(t,s)$ and $H_{n+1}(t,s)= \int^{t}H_{n}(t,z)H_{1}(z,s)dz$ .

Remark If$\varphi=\delta_{0}$then the equation (6.3) and the equation (6.7) are exactly same.

and $f(t)=b$ a constant function with $b\geq 0$ . The $M(t,s)=0$ for$s<t\leq T$ , so we have the equation (6.1), that is,
Remark Let$\varphi=\delta_{0}$$G(t)=2 \int_{b}^{+\infty}\frac{1}{\sqrt{2\pi}}\exp\{-\frac{u^{2}}{2t}\}du$$0\leq$.

\S 7. The Relationship Between Conditional Expectation and Bartle Integral
with Respect to a Vector Measure$V_{\varphi}$

In this section, we will show that the Bartle integral with respect to can be written as theiterated integrals with respect to complex-valued measure. From this, we recognize the relation
between the Bartle integral and the conditional expectation on $(C[a,b],\omega_{\varphi})[41]$ .
. Let be a non-negative integer. Let be a
Let be a probability measure on. We write
for a measure on-valued measurable function on.
determined by , that is, $P_{X}(E)=\omega_{\varphi}(X^{-1}(E))$ for, let. Then is a measureFor
andvalued measure on $(C[a,b],\mathcal{B}(C[a,b]))$ in the total variation norm sense.
$V_{\varphi}$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$

$X$

$n$

$\varphi$$\mathbb{R}^{n+1}$$(C[a,b],\mathcal{B}(C[a,b]),\omega_{\varphi})$$P_{X}$$E\in \mathcal{B}(\mathbb{R}^{n+1})$$X$$(\mathbb{R}^{n+1},\mathcal{B}(\mathbb{R}^{n+1}))$$[V_{\varphi}(B)](E)=\omega_{\varphi}(B\cap X^{-1}(E))$$B\in \mathcal{B}(C[a,b])$$\varphi\in \mathcal{M}(\mathbb{R})$

Theorem 7.1. Let be a probability measure on$(C[a,b],\mathcal{B}(C[a,b]))$ . Then$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\varphi$$V_{\varphi}$

and $f$ bounded measumble on$[( Ba)-\int_{C[a,b]}f(x)dV_{\varphi}(x)](E)=\int_{E}E(f|X)(\xi)dP_{X}(\xi)$for.

$E\in \mathcal{B}(\mathbb{R}^{n+1})$

For a non-negative finite real valued measure in$\varphi$, that is,$\varphi^{N}(E)=\frac{\varphi(E)}{|\varphi|(\mathbb{R})}$is a zero measure. For
and for $j=1,2,3,4,$$\varphi=\sum_{j=1}^{4}\alpha_{j}\varphi_{j}$$\varphi$

for

in

$E$

in

$\mathcal{M}(\mathbb{R})$$\mathcal{B}(\mathbb{R})$$\mathcal{M}(\mathbb{R})$, let$\varphi^{N}$

if is a non-zero measure and$\varphi$with the Jordan decomposition. Hence, for$\omega_{\varphi_{j}}=|\varphi_{j}|(\mathbb{R})\varphi_{j}^{N}$, for $B\in \mathcal{B}(C[a,b])$ and for

be a normalized measure of$E\in \mathcal{B}(\mathbb{R})$$\varphi\in \mathcal{M}(\mathbb{R})$$\varphi^{N}$is a zero measure if$\varphi=\sum_{j=1}^{4}\alpha_{j}\varphi_{j},$$\varphi$$\omega_{\varphi}=\sum_{j=1}^{4}\alpha_{j}\omega_{\varphi_{j}}$with the Jordan decomposition,$[V_{\varphi}(B)](E)=[ \sum_{j=1}^{4}\alpha_{j}|\varphi_{j}|(\mathbb{R})V_{\varphi_{j}^{N}}(B)](E)$,73
so we have
$V_{\varphi}= \sum_{j=1}^{4}\alpha_{j}|\varphi_{j}|(\mathbb{R})V_{\varphi_{j}^{N}}$.

Theorem 7.2. Let $\varphi\in M(R)$ . For a bounded measumblefunction $f$ on $(C[a,b], \mathcal{B}(C[a,b]))$and $X(x)=x(b)$,$[( Ba)-\int_{C[a,b]}f(x)dV_{\varphi}(x)](E)=\frac{1}{2\pi}\int_{E}\int_{\mathbb{R}}e^{-i\zeta u}\int_{C[a,b]}e^{iux(t)}f(x)d\omega_{\varphi}(x)dm_{L}(u)dm_{L}(\xi)$for$E\in \mathcal{B}(\mathbb{R})$.

Remark By putting$\varphi=\delta_{0},$$\omega_{\varphi}=\omega$

and $X(x)=x(b)$ , the classical Wiener measure and.

$[( Ba)-\int_{C[a,b]}f(x)dV_{\varphi}(x)](E)=\int_{x-1(E)}f(x)d\omega(x)$Here $f$ is a bounded measurable function and$E\in \mathcal{B}(\mathbb{R})$

Theorem 7.3 (The Wiener Integration Formula for.

$V_{\varphi}$). Suppose for $k=1,2,$ $\ldots,n,$$i_{k}$

is anonnegative integer such that $m=n+ \sum_{j=1}^{n}i_{j}+1$ and $a\equiv t_{0}\equiv t_{0,0}<t_{0,1}<t_{0,2}<\cdots<t_{0,i_{1}}<$and for $j=1,2,$.
is a Borel measumble fiunction then the following$t_{1}\equiv t_{0,i_{1}+1}\equiv t_{1,0}<t_{1,1}<t_{1,2}<\cdots<t_{n-1,i_{n}}<t_{n}\equiv t_{n-1,i_{n}+1}\equiv b$$X(x)=(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$. If$f:B^{m}arrow \mathbb{R}$equality holds:(7.1)$\ldots,n$$I_{\mathscr{J}}t$$[( Ba)-\int_{C[a,b]}f(y(t_{0,0}),y(t_{0,1}), \ldots,y(t_{n-1,i_{n}+1}))dV_{\varphi}^{J_{\vec{t}}}(y)](E)$$=* \int_{\mathbb{R}}[\int_{\mathbb{R}^{m-1}}f(u_{0,0},u_{0,1}, \ldots,u_{n-1,i_{n}+1})W_{m+1}\prod_{g=0}^{n}\chi_{E^{[g]}}(u_{g,0})$$d( \prod_{i=1}^{m-1}\omega)(u0,0,u0,1, \ldots,u_{n-1,i_{n}+1})]dm_{L}(u0,0)$

where$E^{[g]}$is the$g^{th}$,-section of .
$E$

\S 8. The Simple Formula for Conditional Expectation
on Analogue of Wiener Measure SpaceIn this section, we prove the simple formula for conditional expectation on analogue ofWiener measure. Throughout in this section, let $a=t_{0}<t_{1}<\cdots<t_{n}=b$ be given, let$[y](s)= \sum_{j=1}^{n}\chi[t_{j-1},t_{j})(s)[y(t_{j-1})+\frac{s-t_{j-1}}{t_{j}-t_{j-1}}(y(t_{j})-y(t_{j-1}))]+y(b)\xi_{\{b\}}(s)$74
for $y\in C[a,b]$ and$[u](s)= \sum_{j=1}^{n}x[t_{j-1},t_{j})(s)[u_{j-1}+\frac{s-t_{j-1}}{t_{j}-t_{j-1}}(u_{j}-u_{j-1})]+u_{n}\xi_{\{b\}}(s)$for$(u_{0},u_{1}, \ldots,u_{n})\in \mathbb{R}^{n+1}$.

By [38], we have following theorem from the direct calculations $E(\exp\{i\lambda_{1}X+i\lambda_{2}Y\})=$$E(\exp\{i\lambda_{1}X\})E(\exp\{i\lambda_{2}Y\})$ and $E(\exp\{i\lambda_{1}X+i\lambda_{3}Z\})=E(\exp\{i\lambda_{1}X\})E(\exp\{i\lambda_{3}Z\})$ .

. Let $a=t_{0}<t_{1}<\cdots<s_{1}<$be a probability measure on
and three functions from $C[a,b]$ into with$t_{j-1}<s_{2}<t_{j}<s<\cdots<t_{n}=b$ and $X,$
$X(y)=y(s)-[y](s),$ $Y(y)=y(s_{1})$ and $Z(y)=y(s2)$ , respectively. Then and are stochasticallyindependent andX and are stochastically independent.
Theorem 8.1. Let$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\varphi$

$Y$

$Z$

$\mathbb{R}$

$X$

$Y$

$Z$

In 2008, Professor D. H. Cho [9] proved the next theorem by the quite different and long
method on the analogue of Wiener space over paths in compare with our proof in [38].
$B$

Theorem 8.2 (The Simple Formula for Conditional Expectation). Let
be the fiinction withbility measure on . Let,
Let be -integrable on $C[a,b]$ . Then for$J_{\vec{t}}:C[a,b]arrow \mathbb{R}^{n+1}$$\mathbb{R}$$F$$\varphi$

be a Borel proba-$J_{\overline{t}}(y)=(y(t_{0}),y(t_{1}), \ldots,y(t_{n}))$.

$E\in \mathcal{B}(\mathbb{R}^{n+1})$$m_{\varphi}$(8.1)$[( Ba)-\int_{C(B)}F(y)dV_{h_{t}}^{\varphi}\backslash (y)](E)=\int_{E}E^{\varphi}(F|J_{\vec{t}})dP_{J_{\tilde{t}}}^{\varphi}(u\gamma$,that is,$E^{\varphi}(F|J_{\tilde{t}})=E(F(y-1y]+[u\eta))$.

We know that for any bounded measurable function on $C[a,b]$ and for any probability. What happen if the proba, there is a conditional expectation
measure on
bility measure change?
$F$$\varphi$$E^{\varphi}(F|J_{\vec{t}})$$(\mathbb{R},\mathcal{B}(\mathbb{R}))$$\varphi$

Theorem 8.3 (The Uniqueness Theorem for Giving Distributions). For a bounded measur, independent of theablefunction on $C[a,b]$ , there is a unique conditional expectation
selection of the distribution such that
$F$

$E(F|J_{\vec{t}})$$\varphi$$[( Ba)-\int_{C[a,b]}F(x)dV_{J_{\vec{t}}}^{\varphi}(x)](E)=\int_{E}E(F|J_{\tilde{t}})(u\gamma dP_{J_{\vec{t}}}^{\varphi}(u\gamma$

for any$E\in \mathcal{B}(\mathbb{R}^{n+1})$

and for any Borel probability measure

Remark In Theorem 8.3, if we takebecause $u0=0$ .

$E(F|J_{\vec{t}})$$\varphi=\delta_{0}$

then

$u0$

on .
$\mathbb{R}$$\varphi$does not appear in the representation of

75\S 9. A Measure-Valued Feynman-Kac Formula
Cameron and Storvick [6] introduced an operator-valued function space integral in 1968.
Johnson and Lapidus [18] established the existence theorem of the operator-valued function
space integral as an operator fromto itself for certain functionals involving some Borel
measures, and in 1987, Lapidus [23] proved that the integral satisfies the Schr\"odinger wave
equation. In 1992, Chang and the first author [8] established the existence theorem of theoperator-valued function space integral as an operator fromto $L^{p’}(1<p<2)$ for certain
functionals involving some Borel measures. The first author proved that the integral satisfies
a Volterra-Stieljes integral equation in [36] In this section, we will achieve the measure-valued
Feynman-Kac formula for the integral with respect to a measure-valued measure of suitablefunctional. Throughout in this section and the next sections, we assume $X(x)=x(b)$ and$L^{2}(\mathbb{R}^{N})$$L^{p}$$V_{\varphi}^{X}=$$V_{\varphi}([39])$.

Theorem 9.1. Let
Then$\varphi\in\Lambda 4(\mathbb{R}),$$\eta$

a complex-valued Borel measure on $[a,b]$ and.

$\theta\in L_{\varphi;\infty,1;\eta}$$|\theta(s,x(s))|\leq\Vert\theta(s, \cdot)\Vert_{\varphi;\infty}$

for$|\eta|\cross\omega_{|\varphi|}$-a. . $(s,x)\in[a,b]\cross C[a,b]$ .
$e$Throughout this section let $\eta=\mu+v$ be a complex-valued Borel measure on $[a,b]$ such that$\mu$

is the continuous part of and$\eta$$(p=0,1, \ldots,n)$and$j_{1},$ $\ldots,j_{n}$$v= \sum_{p=0}^{n}c_{p}\delta_{\tau_{p}}$are complex numbers,

where $a=\tau 0<\tau_{1}<\tau_{2}<\cdots<\tau_{n}=b$ and$\varphi\in \mathcal{M}(\mathbb{R})$and$\theta\in L_{\varphi;\infty,1;\eta}$$c_{p}$. For non-negative integers$q$with $q=j_{1}+j_{2}+\cdots+j_{n}$ , let$\triangle_{q;j_{1},j_{2},\ldots,j_{n}}=\{(s_{1,1},s_{1,2}, \ldots,s_{1,j_{1},2,1}s, \ldots,s_{n-1,j_{n-1}},s_{n,1}, \ldots,s_{n,j_{n}})|\tau_{0}=a<s_{1,1}<$$...<s_{1,j_{1}}<\tau_{1}<s_{2,1}<\cdots<\tau_{n-1}<s_{n,1}<\cdots<s_{n,j_{n}}<\tau_{n}=b\}$.

For convenience, we set
for $a\leq s\leq b$ and$\tau_{n}=t=s_{n,j_{n}+1}$ and$\ldots,n-1$$\tau_{k}=s_{k+1,0}=s_{k,j_{k}+1}$ for $k=1,2,$. For non-negative integers $m,q_{0},$with $m=q_{0}+q_{1}+\cdots+q_{n+1}$ and $q_{n+1}=j_{1}+j_{2}+\cdots+j_{n}$ , let$C[a,b]arrow \mathbb{C}$ be afunction defined by$M_{\theta(s,\cdot)}\equiv M_{\theta(s)}$$\tau_{0}=s_{0,0},$$\ldots,q_{n+1},j_{1},$ $\ldots,j_{n}$$K(m,n,q,j):\triangle_{q_{n+1};j_{1},j_{2},\ldots,j_{n}}\cross$$K(m,n,q,j)((s_{1,1}, \ldots,s_{n,j_{n}}),x)=[\prod_{i=0}^{n}\theta(\tau_{i},x(\tau_{i}))^{q_{i}}][\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\theta(s_{i,j},x(s_{i,j}))]$

and$D(m,n,q,j):\triangle_{q_{n+1};j_{1},j_{2},\ldots,j_{n}}arrow \mathbb{R}$a function defined by$D(m,n,q,j)(s_{1,1}, \ldots,s_{n,j_{n}})=[\prod_{i=0}^{n}\Vert\theta(\tau_{i}, \cdot)\Vert_{\varphi^{i};\infty}^{q}][\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\Vert\theta(s_{i,j}, \cdot)\Vert_{\varphi;\infty}]$.

76
Lemma 9.2. (1) $|K(m,n,q,j)|\leq D(m,n,q,j)$ in(2) It follows that$|\mu|\cross\omega_{|\varphi|}$

-a. .
$e$

$| \int_{\triangle_{q_{n+1}.j_{1}}}D(m,n,.q,\oint)(s_{1,1},\ldots,s_{n,j_{n}})d(\prod_{i=1j}^{n}\prod_{=1}^{j_{i}}\mu)(s_{1,1},\ldots,s_{n,j_{n}})|$$\leq\frac{1}{q_{n+1}!}(\prod_{i=0}^{n}\Vert\theta(\tau_{i},\cdot)\Vert_{\varphi^{i};\infty}^{q})(\Vert\theta\Vert_{\varphi;\infty,1;\mu})^{q_{n+1}}$(3) $D(m,n,q,j)$ is

Lemma 9.3.

-Bartle integrable on$( \prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\mu)\cross V_{\varphi}$$\theta(s,x(s))$

is$\mu\cross V_{\varphi}$$\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}|\mu|-a.e$.

.

-Bartle integmble on $[0,t]\cross C[0,t]$ .

Theorem 9.4. (1) $K(m,n,q,j)$ is(2) For$\triangle_{q_{n+1};j_{1},\ldots,j_{n}}\cross C[0,t]$.

$(s_{1,1}, \ldots,s_{n,j_{n}}),$$( \prod_{i=1}^{n}\prod_{j=1}^{j_{i}} \mu)\cross V_{\varphi}$-Banle integmble.

$K(m,n,q,j)((s_{1,1}, \ldots,s_{n,j_{n}}), \cdot)$(3) $( Ba)-\int_{C[0,t]}K(m,n,q,j)((s_{1,1}, \ldots,s_{n,j_{n}}),x)dV_{\varphi}(x)$ is$\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\mu$is$V_{\varphi}$-Banle integmble.

-Bochner integrable.

The proof of the following theorem is pattemed to some extent on earlier work by Johnson
and Lapidus in [18] but the present setting requires a number of new concepts and results in theprevious parts of this section.

Theorem 9.5 (A Measure-Valued Feynman-Kac Formula). $\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}$ isBartle integrable on $C[a,b]$ and for$E\in \mathcal{B}(\mathbb{R})$,$[( Ba)-\int_{C[a,b]}\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)](E)$$\prod c_{p^{p}}^{q}n$$= \sum_{m=0q_{0}+}^{\infty}\ldots\sum_{+q_{n+1}=m}\frac{p=0}{\prod_{p=0}^{n}q_{p}!}\sum_{j_{1}+\cdots+j_{n}=q_{n+1}}$$\int_{\triangle_{q}}[(L_{n},..0,L_{m-1}o\cdots oL_{1})(T(s_{1,1},\varphi,\theta(0, \cdot)^{q_{0}}))](E)d( \prod_{i=1,n+1\cdot j_{1}.j_{n}}^{n}\prod_{j=1}^{j_{i}}\mu)(s_{1,1}, \ldots,s_{n,j_{n}})$.

Moreover,$|( Ba)-\int_{C[a,b]}\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)|(\mathbb{R})\leq 4|\varphi|(\mathbb{R})[\exp\{\Vert\theta\Vert_{\varphi;\infty,1;\eta}\}]$

Here, for $k=2,3,$ $\ldots,n$ ,$L_{k}=M_{\theta(\tau_{k})^{q_{k}}}oS_{\tau_{k}-s_{k,j_{k}}}oM_{\theta(s_{k,j_{k}})}oS_{s_{k,j_{k}}-s_{k,j_{k}-1}}o\cdots oM_{\theta(s_{k,1})}oS_{s_{k,1}-s_{k,0}}$.

$V_{\varphi^{-}}$

77
and$L_{1}=M_{\theta(\tau_{1})^{q}1}oS_{\tau_{1}-s_{1,j_{1}}}oM_{\theta(s_{1,j_{1}})}oS_{s_{1,j1}-s_{1,j_{1}-1}}o\cdots oM_{\theta(s_{1,1})}$.

From Theorem 9.5, directly we deduce the following corollaries.

Corollary 9.6. In Theorem 9.5, we assume that $\eta=\mu$, an arbitmry continuous measure on$[a,b]$ . Then forin$E$$\mathcal{B}(\mathbb{R})$$[( Ba)-\int_{C[a,b]}\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)](E)$,$= \sum_{m=0}^{\infty}\int_{\triangle_{m}}[(S_{t-s_{m}}oM_{\theta(s_{m})}o\cdots oS_{s_{2}-s_{1}}oM_{\theta(s_{1})})(T(s_{1},\varphi,\theta^{0}\equiv 1))](E)d(\prod_{i=1}^{m}\mu)(s_{1},s_{2}, \ldots,s_{m})$where $\triangle_{m}=\{(s_{1},s_{2}, \ldots,s_{m})\in[0,t]^{m}|0<s_{1}<s_{2}<\cdots<s_{m}<t\}$.
Corollary 9.7. In Theorem 9.5, we assume that$[a,b]$ with finite support. Then. for,$\eta=v=\sum_{p=0}^{n}c_{p}\delta_{\tau_{p}}$, a discrete measure on$E\in \mathcal{B}(\mathbb{R})$$[( Ba)-\int_{C[a,b]}\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)](E)$$\prod c_{p}^{q_{p}}n$$= \sum^{\infty}$$\sum$$\frac{p=0}{n}$$m=0q_{0}+ \cdots+q_{n}=m\prod_{p=0}q_{p}!$$[(M_{\theta(\tau_{n})^{q}n}oS_{\tau_{n}-\tau_{n-1}}o\cdots oS_{\tau_{2}-\tau_{1}}oM_{\theta(\tau_{1})^{q_{1}}})(T(\tau_{1},\varphi,\theta(0, \cdot)^{q_{0}}))](E)$Corollary 9.8. In Theorem 9.5, we assume that $c_{n}=0$. Then for

$E$

in

$\mathcal{B}(\mathbb{R})$.

,$[( Ba)-\int_{C[a,b]}\exp\{\int_{[a,b]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)](E)$$\frac{\prod_{p=0}^{n-1}c_{p}^{q_{p}}}{n-1}$$= \sum^{\infty}$$\sum$$\sum$$\int_{\triangle}$$q_{n},j_{1},\ldots,j_{n}$$m=0q_{0}+ \cdots+q_{n}=m\prod_{p=0}q_{p}\downarrow j_{1}+\cdots+j_{n}=q_{n}$$[((S_{t-s_{n}\dot{j}n}oM_{\theta(s_{n,j_{n}})}o\cdots oS_{s_{n,1}-\tau_{n-1}})oL_{n-1}o\cdots oL_{1})$$(T(s_{1,1}, \varphi,\theta(0, \cdot)^{q_{0}}))](E)d(\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\mu)(s_{1,1}, \ldots,s_{n,j_{n}})$\S 10.

.

A Volterra Integral Equation for the Measure-Valued Feynman-Kac Formula

In this section, we prove that the equality in Theorem 9.5, satisfies a suitable Volterra integral
equation.

78
Throughout this section, let $a=0=\tau_{0}<\tau_{1}<\cdots<\tau_{n}=t<\overline{t}=b$ and letmeasure onsuch that $\eta=\mu+v$ where is the continuous part of and$[0,\overline{t}]$further let$\theta\in L_{\varphi;\infty,1;\eta}^{\overline{t}}$be a Borel;$v= \sum_{p=0}^{n}c_{p}\delta_{\tau_{\rho}}$$\eta$$\mu$$\eta$. Let$u(t’)=( Ba)-\int_{C[0,t]}\exp\{\int_{[0,t]}\theta(s,x(s))d\eta(s)\}dV_{\varphi}(x)$for$t<t’\leq\overline{t.}$of the integral
The following theorem is the counterpart for the measure-valued measureequation for the Feynman-Kac formula with Lebesgue-Stieljes measure, obtained by Lapidusin [23, 24, 25] and for the Feynman-Kac formula with an operator-valued measure, obtained by$V_{\varphi}$

Kluvanek in [20].
Theorem 10.1 (The Measure-Valued Feynman-Kac Formula). For
a Volterm integml equation, that is,$t<t’\leq\overline{t,}u(t’)$$u(t’)=S_{t’-t}(u(t))+( Bo\triangleright\int_{(t,t]}(S_{t’-s}\circ M_{\theta(s)})u(s)d\mu(s)$satisfies.

Corollary 10.2. Under the assumptions in Comllary 9.6, for $0<t’\leq\overline{t,}u(t’)$ satisfies a
Volterm integml equation, that is,$u(t’)=S_{t’}( \varphi)+(Bo)-\int_{(0,t’]}(S_{t’-s}\circ M_{\theta(s)})(u(s))d\mu(s)$Corollary 10.3. Under the assumptions in Corollary 9.7, for.

$0<t’\leq\tilde{t,}$$u(t’)= \sum_{m=0q_{0}+}^{\infty}.\sum_{+q_{n}=m}\frac{\prod_{p=0}^{n}c_{p^{p}}^{q}}{\prod_{p=0}^{n}q_{p}!}$$[S_{t’-t}oM\alpha\tau_{n})^{q}n\circ S_{\tau_{n}-\tau_{n-1}}o\cdots oS_{\tau_{2}-\tau_{1}}oM_{\theta(\tau_{1})^{q_{1}}}](T(\tau_{1},\varphi,\theta(0, \cdot)^{q0}))$,$u(t’)=S_{t’-t}(u(t))$ ,and, a zero opemtor.

$( Bo\triangleright\int_{(t,t]}(S_{t’-s}oM_{\theta(s)})(u(s))d\mu(s)=0$\S 11. The Dobrakov integral on the analogue of Wiener space
In this section, we will treat the theoly of Dobrakov integral over $C[a,b]$ . For $B\in \mathcal{B}(C[a,b])$,with $[V(B)](\varphi)=V_{\varphi}(B)$. Then $V(B)$ is a bounded linear operator onlet. From [37], we can check that the following facts.
$V(B):\mathcal{M}(\mathbb{R})arrow \mathcal{M}(\mathbb{R})$$\mathcal{M}(\mathbb{R})$79
Lemma 11.1. For$F$$u0\in \mathbb{R}$Theorem 11.2. Let
is finite if and only$F$, let $P_{u_{0}}=\{x\in C[a,b]|x(a)=uo\}$. Then$\hat{V}(P_{u_{0}})=1$.

be a subset of and let $P(F)=\{x\in C[a,b]|x(0)$ belongs to
is finite and only if $P(F)$ is integrable.
$\mathbb{R}$$\iota f\hat{V}(P(F))$$F\}$. Then$\iota f$

Using Theorem 1.3, we have the following theorem.
Theorem 11.3. is an opemtor-valued measure countably additive in the strong opemtor
topology but is not an opemtor-valued measure countably additive in the uniform opemtor
topology.
$V$

Theorem 11.4.
is not a weakly complete Banach space i. e., there is a subspace of
which is isomorphic to the space .
$\mathcal{M}(\mathbb{R})$$\mathcal{M}(\mathbb{R})$$c_{0}$

Theorem 11.5. If$F$is a finite subset of$\mathbb{R}$, then $V$ is continuous on $P_{0}(F)$.

Lemma 11.6. Let the semivariation be continuous on an integmble set, let be an integmble set and let $f$ be a bounded stmngly function. Then thefu nction
integmble.
$A$$\hat{m}$$f_{\lambda’A}$

Theorem 11.7 (The Wiener Integral Formula for Operator-Valued Measure).
Let $F=$$H$and let $a=t_{0}<t_{1}<--$ $<t_{n}=b$ be given. Suppose
is a function from
$P(F)$ into$H(x)=\delta_{x(a)}h_{x(a)}(x(t_{1}),x(t_{2}),\ldots,x(t_{n}))$such thatand $h_{u_{k}}(k=1,2, \ldots,n)$ arebounded measumble fiunctions on . Then the Dobrakov integml $\int_{P(F)}H(x)dV(x)$ exists andthe following equality holds.
$\{u_{1},u_{2}, \ldots,u_{n}\}$$\lambda 4(\mathbb{R})$$\mathbb{R}^{n}$$[ \int_{P(F)}H(x)dV(x)](E)$$= \frac{1}{\prod_{i=1}^{n}\sqrt{2\pi(t_{i}-t_{i-1})}}\sum_{k=1}^{n}\int_{E}[\int_{\mathbb{R}^{n-1}}h_{u_{k}}(v_{1},v_{2}, \ldots,v_{n})$$\exp\{-\frac{1}{2}\sum_{i=2}^{n}\frac{(v_{i}-v_{i-1})^{2}}{t_{i}-t_{i-1}}\}\exp\{-\frac{1}{2}\frac{(v_{1}-u_{k})^{2}}{t_{1}}\}dm_{L}(v_{1})\cdots dm_{L}(v_{n-1})]dm_{L}(v_{n})$for all Borel subset of .
$E$$\mathbb{R}$

Example 11.8. Let $t=2$ and let $F=\{3,5\}$ . Let $H(x)=\delta_{x(0)}\exp\{-x(1)^{2}\}$ . Then for $E\in$$\mathcal{B}(\mathbb{R})$,$[ \int_{P(E)}H(x)dV(X)](E)=\frac{e^{-3}}{2\sqrt{2\pi}}\int_{E}\exp\{-\frac{3}{8}(u_{2}-1)^{2}\}du_{2}+\frac{e^{-\frac{25}{3}}}{2\sqrt{2\pi}}\int_{E^{\exp\{-\frac{3}{8}(u_{2}-\frac{5}{3})^{2}\}du}}2$.

\S 12. The Operational Calculus for a Measure-Valued Dyson Series
In this section, we investigate Feynman‘s operational calculus for a measure-valued Dyson
series [35]. Throughout this section, let and be two real numbers with $0<t_{1}<t_{2}$ and let$t_{1}$$t_{2}$

80$\varphi\in \mathcal{M}(\mathbb{R})$. Let

be a function with $X^{a,b}(y)=y(b)$ for $a<b$ . For$X^{a,b};C[a,b]arrow \mathbb{R}$and for $B\in \mathcal{B}(C[a,b]),$$V_{\varphi}^{a,b}(B)=\omega_{\varphi}(B\cap X^{a,b}(E))$$E\in \mathcal{B}(\mathbb{R})$.

Theorem 12.1. Let $(so,s_{1},s2, \ldots,s_{m+n})\in \mathbb{R}^{m+n+1}$ with $0<s0<S1<\cdots<s_{m}=t_{1}<s_{m+1}<$
andand be two complex-valued Borel measurablefunctions on$...<s_{m+n}=t_{2}$ . Let, respectively such that$f_{1}$$\mathbb{R}^{m+1}$$f_{2}$$\mathbb{R}^{n}$$f_{1}(u0, u_{1}, \ldots,u_{m})W(m+1;(so,s_{1}, \ldots,s_{m});u0,u1, \ldots,u_{m})$is$| \varphi|\cross\prod_{j=1}^{m}m_{L}$-integmble on$\mathbb{R}^{m+1}$and$f_{1}(u0,u_{1}, \ldots,u_{m})f_{2}(u_{m+1}, \ldots,u_{m+n})W(m+n+1; (so,s_{1}, \ldots,s_{m+n+1});u0,u\iota, \ldots,u_{m+n})$$is| \varphi|\cross\prod_{j=1}^{m+n}m_{L}$-integrable on$\mathbb{R}^{m+n+1}$. Then$F_{1}(x)=f_{1}(x(so),x(s_{1}), \ldots,x(s_{m}))$is-Banle integmble on $C[0,t_{1}]$ ,$V_{\varphi}^{0,t_{1}}$$F(x)=f_{1}(x(so),x(s1), \ldots,x(s_{m}))f_{2}(x(s_{m+1}), \ldots,x(s_{m+n}))$is$V_{\varphi}^{0,t_{2}}$-Bartle integrable on $C[0,t_{2}]$ and$F_{2}(x)=f_{2}(x(s_{m+1}), \ldots,x(s_{m+n}))$is$V_{\tilde{\varphi}}^{t_{1},t_{2}}$-Bartle integrable on$C[t_{1},t_{2}]$, where$\tilde{\varphi}(E)=[(Ba)-\int_{C[0,t_{1}]}F_{1}(x)dV_{\varphi}^{0,t_{1}}(x)](E)$for$E\in \mathcal{B}(\mathbb{R})$. Moreover,$( Ba\triangleright\int_{C[0,t_{2}]}F(x)dV_{\varphi}^{0,t_{2}}(x)=(Ba)-\int_{C[t_{1},t_{2}]}F_{2}(x)dV_{\varphi^{1}}^{t,t_{2}}(x)$.

Remark Let $\varphi\in M(R)$ , let $P_{1};C[0,t_{2}]arrow C[0,t_{1}]$ be a function with $[P_{1}(x)](s)=x(s)$ for.
and let $P_{2}:C[0,t_{2}]arrow C[t_{1},t_{2}]$ be a function with $[P_{2}(x)](s)=x(s)$ for. But it is not
for
where
Then by Theorem 12.1,. Because,
where
for
true always that$t_{1}=1,$ $t_{2}=2$ and $B=\{x\in C[0,t_{2}]|$ either $x(1)\geq 0$ and $x(2)\geq 0$ or $x(1)<0$ andputting$0\leq s\leq t_{1}$$t_{1}\leq s\leq t_{2}$$V_{\varphi}^{0,t_{2}}(D=V_{\tilde{\varphi}}^{t_{1},t_{2}}(P_{2}(I))$$V_{\varphi}^{0,t_{2}}(B)=V_{\tilde{\varphi}}^{t_{1},t_{2}}(P_{2}(B))$$I\in \mathcal{I}$$B\in \mathcal{B}(C[0,t_{2}])$$\tilde{\varphi}=V_{\varphi}^{0,t_{1}}(P_{1}(D)$$\tilde{\varphi}=V_{\varphi}^{0,t_{1}}(P_{1}(B))$$\varphi=\delta_{0},$$x(2)<0$ holds}, $B\in \mathcal{B}(C[0,2]),$$[V_{\varphi}^{0,2}(B)]( \mathbb{R})=\frac{1}{2},$$V_{\varphi}^{0,1}(P_{1}(B))=V_{\varphi}^{0,1}(C[0,1])=S_{1}(\delta_{0})$standard normal distribution, and$[V_{S_{1}(\delta_{0})}^{1,2}(Pb(B))](\mathbb{R})=[S_{1}\circ S_{1}(\delta_{0})](\mathbb{R})=[S_{2}(\delta_{0})](\mathbb{R})=1$

.

, the

81
Here, we establish the operational calculus for a measure-valued Dyson series, the following

theorem in this note.

Theorem 12.2. Under the assumptions in the above theorem, let $g(z)=\exp(z)$ and$\mu+v$ a complex-valued Borel measure onsuch that is a continuous part of andwhere $0=\tau_{0}<\tau 1<\cdots<\tau_{m}=t_{1}<\tau_{m+1}<\cdots<\tau_{m+n}=t_{2}$ and $c_{p}(p=$$\eta=$$[0,t_{2}]$$\mu$$\eta$$v= \sum_{p=0}^{m+n}c_{p}\delta_{\tau_{p}}$$0,1,$ $\ldots,m+n)$

andare complex numbers. Then$\exp(\int_{[0,t_{2}]}\theta(s,x(s))d\eta(s))$$C[t_{1},t_{2}]$are all$V_{\varphi}^{0,t_{1}}-,$$\exp(\int_{[0,t_{1}]}\theta(s,x(s))d\eta(s)),$ $\exp(\int_{(t_{1},t_{2}]}\theta(s,x(s))d\eta(s))$$V_{\tilde{\varphi}}^{t_{1},t_{2}}$-and$V_{\varphi}^{0,t_{2}}$-Bartle integmble of on $C[0,t_{1}]$ ,$x$and $C[0,t_{2}]$ , respectively, where.

$\tilde{\varphi}=(Ba)-\int_{C[0,t_{1}]}\exp(\int_{[0,t_{1}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{1}}(x)$Moreover,$( Ba)-\int_{C[0,t_{2}]}\exp(\int_{[0,t_{2}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{2}}(x)$$=( Ba)-\int_{C[t_{1},t_{2}]}\exp(\int_{(t_{1},t_{2}]}\theta(s,x(s))d\eta(s))dV_{\tilde{\varphi}}^{t_{1},t_{2}}(x)$Remark (a) Let be a constant function on
measure on
and let. Then$\theta$$[0,t_{2}]$$[0,t_{2}]\cross \mathbb{R}$.

, say $\theta(s,u)=c$, let$\eta$

be the Lebesgue$\varphi=\delta_{0}$$\int_{C[0,t_{1}]}\exp(\int_{[0,t_{1}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{1}}(x)=\exp(ct_{1})S_{t_{1}}(\delta_{0})\equiv\tilde{\varphi}$and$\int_{C[t_{1},t_{2}]}\exp(\int_{[t_{1},t_{2}]}\theta(s,x(s))d\eta(s))dV_{\tilde{\varphi}}^{t_{1},t_{2}}(x)$,$= \int_{C[0,t_{2}]}\exp(\int_{[0,t_{2}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{2}}(x)$so, a formula, given in Theorem 12.2, holds.
(b) Taking $g(z)=z$ in (a)$\int_{C[0,t_{1}]}g(\int_{[0,t_{1}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{1}}(x)=ct_{1}S_{t_{1}}(\delta_{0})\equiv\psi$and$\int_{C[t_{1},t_{2}]}g(\int_{[t_{1},t_{2}]}\theta(s,x(s))d\eta(s))dV_{\psi}^{t_{1},t_{2}}(x)=c^{2}t_{1}(t_{2}-t_{1})S_{t_{2}}(\delta_{0})$Since$\int_{C[0,t_{2}]}g(\int_{[0,t_{2}]}\theta(s,x(s))d\eta(s))dV_{\varphi}^{0,t_{2}}(x)=ct_{2}S_{t_{2}}(\delta_{0})$doesn’t hold for the general function .
$g$.

a formula, given in Theorem 12.2,

82\S 13.

Fourier-Feynman Transform on Analogue of Wiener SpaceIn this section, we will develop the theories of fourier-Feynman transform on analogue ofWiener space. First of all, we will establish the existence theorem for our transform. Moreover,
we will find the properties of it. In developing our theories, the rotation theorem is key role, so
we assume that a measure has the Radon-Nikodym derivative with respect to the Lebesgue$\varphi$measure having a form$\frac{d\varphi}{dm_{L}}(x)=Ae^{-ax^{2}}$where $A,$$\alpha$are two positive real numbers.

Lemma 13.1. For a non-zero complex number with $Re/l\geq 0$ and for$\lambda$,$f\in L^{2}(\mathbb{R}^{n+1})$

define$g(v_{0},v_{1}, \ldots,v_{n})=(\frac{\lambda}{2\pi})^{n}+^{1}\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}f(u_{0},u_{1}, \cdots,u_{n})e^{-\Sigma}dm_{L}(uo)\cdots dm_{L}(u_{n})\lambda_{\Sigma_{j=0^{(u_{j}-v_{j})^{2}}}^{n}}$then$g\in L^{2}(\mathbb{R}^{n+1})$

and$\Vert g\Vert_{2}\leq\Vert f\Vert_{2}$.

Lemma 13.2. For $1\geq p\geq 2$ and all non-zero real number, andassure the existence of
of

$q,$

$\iota fF_{1}^{a}\approx^{n}F_{2}$$T_{q}^{(p)}(F_{1})^{a}\approx^{n}T_{q}^{(p)}(F_{2})$$T_{q}^{(p)}(F_{2})$$T_{q}^{(p)}(F_{1})$,, then the existence.

Theorem 13.3. For a partition $a=t_{0}<t_{1}<\cdots<t_{n}=b$ , let $F(x)=f(x(t_{0}),x(t_{1}), \ldots,x(t_{n}))$be in$\mathcal{A}_{n}$. Then for each $y\in C[a,b]$ andfor all complex numbers 1 with$y)d\omega_{\varphi}(x)$(13.1)${\rm Re} A>0,$$\int_{C[a,b]}^{ananw\Lambda}F(x+$exists. Moreover,$\int_{C[a,b]}^{ananw\lambda}F(x+y)d\omega_{\varphi}(x)$$=A \lambda^{n}+^{11}[\prod_{j=1}^{n}2\pi(t_{j}-t_{j-1})]^{-}2\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}f(v_{0},v_{1}, \ldots,v_{n})$$e^{-\Sigma}e^{-a\lambda(v_{0}-y(t_{0}))^{2}}dm_{L}(v_{n})\cdots dm_{L}(v_{1})dm_{L}(vo)r_{\Sigma_{j=1}^{n}\frac{[(v_{j}-v_{j-1})-(y(t_{j}\rangle-y(t_{j-1}))]^{2}}{t_{j^{-1}j-1}}}$.

Moreover, let $h(y(t_{0}),y(t_{1}), \ldots,y(t_{n});\lambda)$ be the right side in (13.1), then$\Vert h(y(t_{0}),y(t_{1}), \ldots,y(t_{n});_{J}t)\Vert_{2}\leq A\sqrt{\frac{\pi}{a}}\Vert f\Vert_{2}$and $h(w0, w_{1}, \ldots,w_{n};\lambda)$ is analytic of .
$\lambda$

Theorem 13.4. Let$f\in L^{2}(\mathbb{R}^{n+1})$$a=t_{0}<t_{1}<\cdots<t_{n}=b$ ,(13.2)and let$q$

be a non-zero real number. For a partition

define$g(v0,v_{1}, \ldots,v_{n})$$= \frac{A}{\sqrt{2\pi}}(-iq)^{\frac{n+1}{2}[\prod_{j=1}^{n}2\pi(t_{j}-t_{j-1})]^{-\Sigma}\int_{-\infty}^{\infty}}1\ldots\int_{-\infty}^{\infty}f(u_{0},u_{1}, \ldots,u_{n})$$ee^{iqa(u_{0}-v_{0})^{2}}dm_{L}(u_{n}) \cdots 1i^{i}\Sigma_{j=1}^{n}\frac{[(u_{j}-u_{j-1})-(v_{j}-v_{j-1})]^{2}}{t_{j}-t_{j-1}})$

.

83
Then

$g$

is in

(13.3)$L^{2}(\mathbb{R}^{n+1})$, andfor all$u0,$ $u_{1},$ $\ldots,u_{n}$$f(u_{0},u_{1}, \ldots,u_{n})$$= \frac{\sqrt{2}a}{A\sqrt{\pi}}(iq)^{\frac{n+1}{2}[\prod_{j=1}^{n}2\pi(t_{j}-t_{j-1})]^{-\frac{1}{2}}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}g(v_{0},v_{1}, \ldots,v_{n})$$e^{-T}iq_{\Sigma_{j=1}^{n}\frac{[(u_{j}-u_{j-1})-(v_{j}-v_{/-1})]^{2}}{t_{j}-t_{/-1}}}e^{-iqa(u_{0}-v_{0})^{2}}dm_{L}(v_{n})\cdots dm_{L}(vl)dm_{L}(vo)$,and(13.4)$lf$112$= \frac{4a^{2}}{A}\Vert g\Vert_{2}$.

Theorem 13.5. For a nonzem real number and a partition $a=t_{0}<t_{1}<\cdots<t_{n}=b,$$F(x)=f(x(t_{0}), \ldots,x(t_{n}))\in \mathcal{A}_{n}$ ,then $G\equiv T_{q}(F)$ exists and
where
$q$

$\iota f$$G(y)\approx g(y(t_{0}), \ldots,y(t_{n}))\in \mathcal{A}_{n}$(13.5)$g(w0,w_{1}, \ldots,w_{n})$$= \frac{A}{\sqrt{2\pi}}(-iq)^{\frac{n+1}{2}[\prod_{j=1}^{n}2\pi(t_{j}-t_{j-1})]^{-\frac{1}{2}}\lim_{Barrow\infty}\int_{D_{B}}\cdots\int_{D_{B}}f(v0,v],\ldots,U_{n})}$$e^{-q_{\Sigma_{j=1}^{n}\frac{[(v_{j}-v_{j-1})-(w_{j}-w_{j-1})]^{2}}{t_{j}-t_{j-1}}}}\iota_{2}e^{iqa(v_{0}-w_{0})^{2}}dm_{L}(v_{n})\cdots dm_{L}(v_{1})dm_{L}(v_{0})$.

Moreover,(13.6)$\Vert T_{q}(F)\Vert_{2}\leq 4\sqrt{a^{3}}\Vert g\Vert_{2}$.

References[1] Bearman J. E., Rotation in the product of two Wiener spaces, Proc. Amer. Math Soc. 3 (1952),

129-137.

[2] Billingsley, P., Convergence of Probability Measures, John Wiley & Sons, New York, 1968.
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[4] Chiang, T. S., Chow, Y and Lee, Y. J., A formula for $E_{W}\exp\{-2^{-1}a^{2}||x+y||_{2}^{2}\}$ , Proc. Amer. Math.

Soc. 100 (1987), 721-724.
[5] Cameron, R. H. and Martin, W. T., Transformations of Wiener integrals under translations, Ann.
Math. 45 (1944), 386-396.
[6] Cameron, R. H. and Storvick, D. A., An operator-valued function space integral and a related integral

equation, J Math Mech. 18 (1968), 517-552.
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Springer-Verlarg (1980), 18-67.
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