Translation Theorem for Conditional Function Space Integrals and Applications

1. Introduction

In [1], Nobert Wiener introduced the concept of “integration in function space”. Nowadays, the space of real-valued continuous functions $C_0[0,T]$ equipped with a Gaussian measure is called the Wiener space. In [2], Yeh introduced a generalized Wiener space $C_{a,b}[0,T]$ related to a generalized Brownian motion process (henceforth, GBMP). This theory for the function space $C_{a,b}[0,T]$ was developed further by Chang and Chung in [3] for appropriate functions $a\left(t\right)$ and $b\left(t\right)$ on $[0,T]$, and was used extensively in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] with various related results. The function $a\left(t\right)$ is often interpreted as the “drift” of the associated stochastic process.

Let $(\Omega ,\mathcal{F},\mathrm{P})$ be a probability space. A real-valued stochastic process Y on $(\Omega ,\mathcal{F},\mathrm{P})$ and a time interval $[0,T]$ is called a GBMP provided $Y(0,\omega )=0$ a.e. $\omega$ and for $0=t_0<t_1<\dots <t_n\le T$, the random vector $(Y(t_1,\omega ),\dots ,Y(t_n,\omega ))$ has a normal distribution with density function

$${\left({\left(2\pi \right)}^n\prod _{j=1}^n(b\left(t_j\right)-b\left(t_{j-1}\right))\right)}^{-1/2}exp\left\{-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{\displaystyle \frac{{[(u_j-a\left(t_j\right))-(u_{j-1}-a\left(t_{j-1}\right))]}^2}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\right\},$$

with $u_0=0$, and where $a\left(t\right)$ and $b\left(t\right)$ are suitable continuous real-valued functions. We note that the GBMP Y determined by the functions $a(·)$ and $b(·)$ is Gaussian with mean $a\left(t\right)$ and covariance $r(s,t)=min\left\{b\right(s),b(t\left)\right\}$. Let $(C_{a,b}[0,T]$, $\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$ denote the complete generalized Wiener space where $C_{a,b}[0,T]$ is the continuous sample paths of the GBMP Y. Then the function space $C_{a,b}[0,T]$ reduces to the Wiener space $C_0[0,T]$ precisely when $a\left(t\right)=0$ and $b\left(t\right)=t$ for all $t\in [0,T]$. In Section 2 below, we will provide a more detailed construction of the function space $C_{a,b}[0,T]$.

Let $(C_0[0,T],\mathcal{W},m_w)$ denote the one parameter Wiener space. The Cameron–Martin translation theorem [20] and several analogies [21,22,23,24] describe how and when the Wiener measure $m_w$ changes under translation by a specific elements of the Wiener space $C_0[0,T]$. This translation theorem was developed for the Yeh–Wiener integral [25], the abstract Wiener integral [26,27,28], the conditional Wiener integral [29,30,31], the analytic Feynman integral [32,33,34,35], and the conditional analytic Feynman integral [36]. Furthermore, a translation theorem on the generalized Wiener space $C_{a,b}[0,T]$ was first established by Chang and Chung in [3]. In [15], those results was improved for functionals on the function space $C_{a,b}[0,T]$, and in [7,10,11], the translation theorem combined with integral processes was established with related topics.

The aim of this paper is to provide a translation theorem for conditional function space integrals on the function space $C_{a,b}[0,T]$. As an application, we also derive a translation theorem for conditional generalized analytic Feynman integrals of functionals F on $C_{a,b}[0,T]$. In order to establish the translation theorem for conditional generalized analytic Feynman integral on the function space, we assumed the existence of the conditional Feynman integral appeared in the theorem, because the drift term $a\left(t\right)$ makes establishing the existence of the conditional Feynman integral very difficult. Thus, in Section 6, we provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the translation theorems can be applied. The formulas and the results in this paper are more complicated than the corresponding formulas and the results in previous researches illustrated above, because the generalized Wiener process used in this paper is nonstationary in time and is subject to the drift $a\left(t\right)$. However, choosing $a\left(t\right)=0$ and $b\left(t\right)=t$ on $[0,T]$, the function space $C_{a,b}[0,T]$ reduces to the Wiener space $C_0[0,T]$, and so the expected results on $C_0[0,T]$ are immediate corollaries of the results in this paper.

2. Definitions and Preliminaries

2.1. Backgrounds

Two functions $a(·)$ and $b(·)$ be given as in Section 1 above. Furthermore, we assume that the function $a(·)$ is continuous and of bounded variation on $[0,T]$ with $a\left(0\right)=0$, and the function $b(·)$ is continuous, monotone increasing and of bounded variation on $[0,T]$ with $b\left(0\right)=0$. Then, by [37], there exists a probability space $(\Omega ,\mathcal{F},\mathrm{P})$ and a continuous additive process Y on $(\Omega ,\mathcal{F},\mathrm{P})$ and a time interval $[0,T]$, where $\mathrm{P}$ is a Gaussian measure such that the probability distribution of $Y(t,·)-Y(s,·)$, $s<t$, is normally distributed with mean $a\left(t\right)-a\left(s\right)$ and variance $b\left(t\right)-b\left(s\right)$. The stochastic process Y on $(\Omega ,\mathcal{F},\mathrm{P})$ and $[0,T]$ is called a GBMP. The GBMP Y determined by $a(·)$ and $b(·)$ is a Gaussian process with mean function $a\left(t\right)$ and covariance function $r(s,t)=min\left\{b\right(s),b(t\left)\right\}$.

Let $C_{a,b}[0,T]$ be the space of continuous sample paths of the GBMP Y determined by $a(·)$ and $b(·)$. The function space $C_{a,b}[0,T]$ is equivalent to the Banach space of continuous functions x on $[0,T]$ with $x\left(0\right)=0$ under the supremum norm. Let $\mathcal{B}\left(C_{a,b}[0,T]\right)$ be the Borel $\sigma$-field on $C_{a,b}[0,T]$. Then, as explained in [37], Y induces a probability measure $\mu$ on the measurable space $(C_{a,b}[0,T],\mathcal{B}\left(C_{a,b}[0,T]\right))$. Hence $(C_{a,b}[0,T],\mathcal{B}\left(C_{a,b}[0,T]\right),\mu )$ is the function space induced by Y. We then complete this function space to obtain $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$ where $\mathcal{W}\left(C_{a,b}[0,T]\right)$ is the set of all $\mu$-Carathéodory measurable subsets of $C_{a,b}[0,T]$. For more details, see [2,37]. Note that choosing $a\left(t\right)=0$ and $b\left(t\right)=t$ on $[0,T]$, one can see that the GBMP reduces a standard Brownian motion (or Wiener process).

In this paper, we assume that

(i)

the mean function $a\left(t\right)$ of the GBMP Y is absolutely continuous on $[0,T]$,

(ii)

the derivative $a^{\prime }\left(t\right)$ of $a\left(t\right)$ is of class $L_2[0,T]$, and ${\int }_0^T\left|a^{\prime }\left(t\right){|}^{2}d\right|a|\left(t\right)<+\infty$ where $\left|a\right|\left(t\right)$ denotes the total variation function of $a\left(t\right)$ on $[0,T]$,

(iii)

the variance function $b\left(t\right)$ of the GBMP Y is continuously differentiable on $[0,T]$,

(iv)

for each $t\in [0,T]$, $b^{\prime }\left(t\right)>0$.

Then it follows that for any cylinder set $I_{t_1,\dots ,t_n,U}$ having the form

$$I_{t_1,\dots ,t_n,U}=\{x\in C_{a,b}[0,T]:(x\left(t_1\right),\dots ,x\left(t_n\right))\in U\}$$

with a set of time moments $0=t_0<t_1<\dots <t_n\le T$ and a Borel set $U\subset {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill \mu \left(I_{t_1,\dots ,t_n,U}\right)& ={\left({\left(2\pi \right)}^n\prod _{j=1}^n(b\left(t_j\right)-b\left(t_{j-1}\right))\right)}^{-1/2}\hfill \\ & \times {\int }_{B}exp\left\{-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{\displaystyle \frac{{[(u_j-a\left(t_j\right))-(u_{j-1}-a\left(t_{j-1}\right))]}^2}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\right\}d{u}_1\dots d{u}_n\hfill \end{array}$$

where $u_0=0$.

Let $C_{a,b}^{\prime }[0,T]$ be the linear space of (equivalence classes of) Lebesgue measurable functions w on $[0,T]$ which satisfy the conditions

$${\int }_0^T{\left(Dw\left(t\right)\right)}^{2}d|a|\left(t\right)<+\infty \mathrm{and}{\int }_0^T{\left(Dw\left(t\right)\right)}^{2}db\left(t\right)<+\infty ,$$

where $Dw={\displaystyle \frac{dw}{dt}}/{\displaystyle \frac{db}{dt}}=w^{\prime }/b^{\prime }$.

For $w_1,w_2\in C_{a,b}^{\prime }[0,T]$, let

$${(w_1,w_2)}_{C_{a,b}^{\prime }}={\int }_0^{T}D{w}_1\left(t\right)D{w}_2\left(t\right)db\left(t\right).$$

Then ${(·,·)}_{C_{a,b}^{\prime }}$ is an inner product on $C_{a,b}^{\prime }[0,T]$ and ${\parallel w\parallel }_{C_{a,b}^{\prime }}=\sqrt{{(w,w)}_{C_{a,b}^{\prime }}}$ is a norm on $C_{a,b}^{\prime }[0,T]$. In particular note that ${\parallel w\parallel }_{C_{a,b}^{\prime }}=0$ if and only if $Dw\left(t\right)=0$$m_L$-a.e. on $[0,T]$, where $m_L$ denotes the Lebesgue measure on $[0,T]$. Furthermore, $(C_{a,b}^{\prime }[0,T],\parallel ·{\parallel }_{C_{a,b}^{\prime }})$ is a separable Hilbert space. Using the assumptions on the functions $a(·)$ and $b(·)$, one can see that the functions $a(·)$ and $b(·)$ are element of the Hilbert space $C_{a,b}^{\prime }[0,T]$. For more details, see [16,17].

Let ${\left\{e_n\right\}}_{n=1}^{\infty }$ be a complete orthonormal set of functions in $(C_{a,b}^{\prime }[0,T]$, ${\parallel ·\parallel }_{C_{a,b}^{\prime }})$ such that the $D{e}_n$’s are of bounded variation on $[0,T]$. Then for $w\in C_{a,b}^{\prime }[0,T]$ and $x\in C_{a,b}[0,T]$, we define the Paley–Wiener–Zygmund (PWZ) stochastic integral ${(w,x)}^{\sim }$ as follows:

$${(w,x)}^{\sim }=\underset{n\to \infty }{lim}{\int }_0^T\sum _{j=1}^n{(w,e_j)}_{C_{a,b}^{\prime }}D{e}_j\left(t\right)dx\left(t\right)$$

if the limit exists. For each $w\in C_{a,b}^{\prime }[0,T]$, the PWZ stochastic integral ${(w,x)}^{\sim }$ exists for $\mu$-a.e. $x\in C_{a,b}[0,T]$. For each $w\in C_{a,b}^{\prime }[0,T]\setminus \left\{0\right\}$, the PWZ stochastic integral ${(w,x)}^{\sim }$ is a non-degenerate Gaussian random variable with mean ${(w,a)}_{C_{a,b}^{\prime }}$ and variance ${\parallel w\parallel }_{C_{a,b}^{\prime }}^2$. If $\{g_1,\dots ,g_n\}$ is an orthogonal set of functions in $C_{a,b}[0,T]$, then the random variables, ${(g_j,x)}^{\sim }$’s are independent. Furthermore, if $Dw$ is of bounded variation on $[0,T]$, then the PWZ stochastic integral ${(w,x)}^{\sim }$ equals the Riemann–Stieltjes integral ${\int }_0^{T}Dw\left(t\right)dx\left(t\right)$. Also we note that for $w,x\in C_{a,b}^{\prime }[0,T]$,

$${(w,x)}^{\sim }={(w,x)}_{C_{a,b}^{\prime }}.$$

(2.1)

In particular, for each $w\in C_{a,b}^{\prime }[0,T]$ and $x\in C_{a,b}[0,T]$, it follows that ${(w,b)}^{\sim }={(w,b)}_{C_{a,b}^{\prime }}=w\left(T\right)$ and ${(b,x)}^{\sim }={\int }_0^{T}dx\left(t\right)=x\left(T\right)$. For a more detailed study of the PWZ stochastic integral on $C_{a,b}[0,T]$, see [9,16].

2.2. Generalized Analytic Feynman Integral

We denote the function space integral of a $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable functional F by

$$E\left[F\right]\equiv E_x\left[F\left(x\right)\right]={\int }_{C_{a,b}[0,T]}F\left(x\right)d\mu \left(x\right)$$

whenever the integral exists.

A subset B of $C_{a,b}[0,T]$ is said to be scale-invariant measurable provided $\rho B$ is $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable for all $\rho >0$, and a scale-invariant measurable set N is said to be scale-invariant null provided $\mu \left(\rho N\right)=0$ for all $\rho >0$. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional F is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and $F(\rho ,·)$ is $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable for every $\rho >0$.

Throughout this paper, we will always assume that each functional $F:C_{a,b}[0,T]\to \mathbb{C}$ that we consider satisfies the conditions:

$$F:C_{a,b}[0,T]\to \mathbb{C}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\text{-}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\text{-}\mathrm{a}.\mathrm{e}.\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$$

(2.2)

and

$$E_x\left[|F(\rho x)|\right]={\int }_{C_{a,b}[0,T]}|F\left(\rho x\right)|d\mu \left(x\right)<+\infty \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\rho >0.$$

(2.3)

Also, let $\mathbb{C}$, ${\mathbb{C}}_{+}$ and ${\tilde{\mathbb{C}}}_{+}$ denote the set of complex numbers, complex numbers with positive real part, and non-zero complex numbers with nonnegative real part, respectively. Furthermore, for each $\lambda \in \mathbb{C}$, ${\lambda }^{1/2}$ denotes the principal square root of $\lambda$, i.e., ${\lambda }^{1/2}$ is always chosen to have nonnegative real part.

We are now ready to state the definition of the generalized analytic Feynman integral.

Definition 2.1.

Let a functional F on $C_{a,b}[0,T]$ satisfy conditions (2.2) and (2.3). If there exists a function $J^{*}\left(\lambda \right)$ analytic in ${\mathbb{C}}_{+}$ such that

$$J^{*}\left(\lambda \right)=E_x\left[F\left({\lambda }^{-1/2}x\right)\right]={\int }_{C_{a,b}[0,T]}F\left({\lambda }^{-1/2}x\right)d\mu \left(x\right)$$

for all $\lambda >0$, then $J^{*}\left(\lambda \right)$ is defined to be the analytic function space integral of F over $C_{a,b}[0,T]$ with parameter λ, and for $\lambda \in {\mathbb{C}}_{+}$ we write

$$E^{{\mathrm{an}}_{\lambda }}\left[F\right]\equiv E_x^{{\mathrm{an}}_{\lambda }}\left[F\left(x\right)\right]=J^{*}\left(\lambda \right).$$

Let $q\ne 0$ be a real number and let F be a functional on $C_{a,b}[0,T]$ such that the analytic function space integral, $E^{{\mathrm{an}}_{\lambda }}\left[F\right]$, exists for all $\lambda \in {\mathbb{C}}_{+}$. If the following limit exists, we call it the generalized analytic Feynman integral of F with parameter q and we write

$$E^{{\mathrm{anf}}_q}\left[F\right]\equiv E_x^{{\mathrm{anf}}_q}\left[F\left(x\right)\right]=\underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}_x^{{\mathrm{an}}_{\lambda }}\left[F\left(x\right)\right].$$

2.3. Conditional Function Space Integrals

We now state the definitions of the conditional function space integral and the conditional generalized analytic Feynman integral.

Definition 2.2.

Let $X:C_{a,b}[0,T]\to {\mathbb{R}}^n$ be a $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable function on $C_{a,b}[0,T]$ whose probability distribution ${\mu }_X=\mu \circ X^{-1}$ is absolutely continuous with respect to Lebesgue measure on ${\mathbb{R}}^n$. Let F be a $\mathbb{C}$-valued μ-integrable functional on $C_{a,b}[0,T]$. Then the conditional integral of F given X, denoted by $E\left(F\right|X=\overrightarrow{\eta })$$\equiv E_x\left(F\left(x\right)\right|X\left(x\right)=\overrightarrow{\eta })$, is a Lebesgue measurable function of $\overrightarrow{\eta }$, unique up to null sets in ${\mathbb{R}}^n$, satisfying the equation

$${\int }_{X^{-1}\left(B\right)}F\left(x\right)d\mu \left(x\right)={\int }_{B}E\left(F\right|X=\overrightarrow{\eta })d{\mu }_X\left(\overrightarrow{\eta }\right)$$

for all Borel sets B in ${\mathbb{R}}^n$.

Let n be a positive integer and let $\{g_1,\dots ,g_n\}$ be an orthonormal set of functions in the Hilbert space $(C_{a,b}^{\prime }[0,T],{(·,·)}_{C_{a,b}^{\prime }})$. Throughout this paper, we will always condition by the function $X:C_{a,b}[0,T]\to {\mathbb{R}}^n$ defined by

$$X\left(x\right)=({(g_1,x)}^{\sim },\dots ,{(g_n,x)}^{\sim }).$$

(2.4)

We now define the conditional generalized Feynman integral $E^{{\mathrm{anf}}_q}\left(F\right|X=\overrightarrow{\eta })$ of functionals F on $C_{a,b}[0,T]$ using the conditioning function X given by (2.4) above.

Definition 2.3.

Let a functional $F:C_{a,b}[0,T]\to \mathbb{C}$ satisfy conditions (2.2) and (2.3) and let $X:C_{a,b}[0,T]\to {\mathbb{R}}^n$ be given by equation (2.4). For $\lambda >0$ and $\overrightarrow{\eta }\in {\mathbb{R}}^n$, let

$$\begin{array}{cc}\hfill J(\lambda ;\overrightarrow{\eta })& =E_x\left(F\left({\lambda }^{-1/2}x\right)\right|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\eta })\hfill \\ & =E_x\left(F\left({\lambda }^{-1/2}x\right)\right|X\left(x\right)={\lambda }^{1/2}\overrightarrow{\eta })\hfill \end{array}$$

denote the conditional function space integral of $F\left({\lambda }^{-1/2}x\right)$ given $X\left({\lambda }^{-1/2}x\right)={\lambda }^{-1/2}({(g_1,x)}^{\sim },\dots ,{(g_n,x)}^{\sim })$. If for a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$, there exists a function $J^{*}(\lambda ;\overrightarrow{\eta })$, analytic in λ on ${\mathbb{C}}_{+}$ such that $J^{*}(\lambda ;\overrightarrow{\eta })=J(\lambda ;\overrightarrow{\eta })$ for all $\lambda >0$, then $J^{*}(\lambda ;\overrightarrow{\eta })$ is defined to be the conditional analytic function space integral of F given $X\left(x\right)=({(g_1,x)}^{\sim },\dots ,{(g_n,x)}^{\sim })$ with parameter λ and for $\lambda \in {\mathbb{C}}_{+}$ we write

$$E^{{\mathrm{an}}_{\lambda }}\left(F\right|X=\overrightarrow{\eta })\equiv E_x^{{\mathrm{an}}_{\lambda }}\left(F\left(x\right)\right|X\left(x\right)=\overrightarrow{\eta })=J^{*}(\lambda ;\overrightarrow{\eta }).$$

If for fixed real $q\ne 0$, the limit

$$\underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}^{{\mathrm{an}}_{\lambda }}\left(F\right|X=\overrightarrow{\eta })$$

exists for a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$, we will denote the value of this limit by $E^{{\mathrm{anf}}_q}\left(F\right|X=\overrightarrow{\eta })\equiv E^{{\mathrm{anf}}_q}\left(F\left(x\right)\right|X\left(x\right)=\overrightarrow{\eta })$ and we call it the conditional generalized analytic Feynman integral of F given X with parameter q.

Next we define $[·]:{\mathbb{R}}^n\to C_{a,b},[0,T]$ by

$$\left[\overrightarrow{\eta }\right]\equiv \sum _{j=1}^n{\eta }_j{g}_j$$

for $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$, and we write

$$\left[x\right]\equiv \left[X\left(x\right)\right]=\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j$$

for $x\in C_{a,b}[0,T]$.

In [12], Chang, Choi and Skoug gave a formula for expressing conditional function space integrals in terms of ordinary function space integrals. We provide a modified result from [12] which plays an important role in this paper. The proof given in [12] with the current hypotheses on $a\left(t\right)$ and $b\left(t\right)$ and the definition of the PWZ stochastic integral also works here.

Theorem 2.4.

Let X be given by equation (2.4) and let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then

$$\begin{array}{cc}\hfill E\left(F\right|X=\overrightarrow{\eta })& =E_x\left[F(x-\left[x\right]+\left[\overrightarrow{\eta }\right])\right]\hfill \\ \hfill & \equiv E_x\left[F\left(x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n{\eta }_j{g}_j\right)\right].\hfill \end{array}$$

(2.5)

Let F be a functional on $C_{a,b}[0,T]$ which satisfies conditions (2.2) and (2.3). Then, one can easily see from (2.5) that for all $\lambda >0$,

$$\begin{array}{cc}& E_x\left(F\left({\lambda }^{-1/2}x\right)|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\eta }\right)\hfill \\ & =E_x\left[F\left({\lambda }^{-1/2}x-{\lambda }^{-1/2}\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n{\eta }_j{g}_j\right)\right]\hfill \end{array}$$

for a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$. Thus we have that

$$E_x^{{\mathrm{an}}_{\lambda }}\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\eta }\right)=E_x^{{\mathrm{an}}_{\lambda }}\left[F\left(x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n{\eta }_j{g}_j\right)\right]$$

(2.6)

and

$$E_x^{{\mathrm{anf}}_q}\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\eta }\right)=E_x^{{\mathrm{anf}}_q}\left[F\left(x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n{\eta }_j{g}_j\right)\right]$$

(2.7)

where in (2.6) and (2.7), the existence of either side implies the existence of the other side and their equality.

3. Translation Theorems for Conditional Function Space Integrals

We start this section with translation theorems on the function space $C_{a,b}[0,T]$, see [3], Theorem 5.4 and Chang and Skoug [15], Theorem 3.2. We then use this translation theorem to obtain conditional function space integration formulas.

Theorem 3.1.

Let $X\left(x\right)$ be given by (2.4) and let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then it follows that for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$$E_x\left[F\left(x\right)\right]=exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}\right\}{E}_x[F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}]$$

and

$$E_x\left[F(x+x_0)\right]=exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}\right\}{E}_x[F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}].$$

(3.1)

With the current hypotheses on $a\left(t\right)$ and $b\left(t\right)$ and the definition of the PWZ stochastic integral, we have the following lemma.

Lemma 3.2

([12]). The processes $\left\{x\right(t)-[x\left]\right(t),t\in [0,T\left]\right\}$ and $\left\{\right[x\left]\right(t),t\in [0,T\left]\right\}$ are independent.

Applying the change of variables theorem, we have the following lemma.

Lemma 3.3.

Give an orthonormal set $\{g_1,\dots ,g_n\}$ of functions in $C_{a,b}^{\prime }[0,T]$ and a function $x_0$ in $C_{a,b}^{\prime }[0,T]$, it follows that

$$E_x\left[exp\left\{\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}{(g_j,x)}^{\sim }\right\}\right]=exp\left\{{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}{(g_j,a)}_{C_{a,b}^{\prime }}\right\}.$$

(3.2)

Theorem 3.4.

Let $X\left(x\right)$ be given by (2.4) and let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then it follows that for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$$\begin{array}{cc}\hfill & E_x\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right),\hfill \end{array}$$

(3.3)

where ${(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}=({(x_0,g_1)}_{C_{a,b}^{\prime }},\dots ,{(x_0,g_n)}_{C_{a,b}^{\prime }})$.

Proof. 

Using equations (2.5), (3.1), and (2.1), and applying Lemma 3.2, it follows that

$$\begin{array}{cc}\hfill & E_x\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =E_x\left[F\left(\left\{x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n{\xi }_j{g}_j\right\}+x_0\right)\right]\hfill \\ \hfill & =E_x\left[F\left(x+x_0-\sum _{j=1}^n{(g_j,x+x_0)}^{\sim }{g}_j+\sum _{j=1}^n\left({\xi }_j+{(g_j,x_0)}^{\sim }\right)g_j\right)\right]\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left[F\left(x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n\left({\xi }_j+{(g_j,x_0)}_{C_{a,b}^{\prime }}\right)g_j\right)exp\left\{{(x_0,x)}^{\sim }\right\}\right]\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n{\left(x_0,\left({\xi }_j+{(x_0,g_j)}_{C_{a,b}^{\prime }}\right)g_j\right)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x[F\left(x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n\left({\xi }_j+{(x_0,g_j)}_{C_{a,b}^{\prime }}\right)g_j\right)\hfill \\ \hfill & \times exp\left\{{\left(x_0,x-\sum _{j=1}^n{(g_j,x)}^{\sim }{g}_j+\sum _{j=1}^n\left({\xi }_j+{(x_0,g_j)}_{C_{a,b}^{\prime }}\right)g_j\right)}^{\sim }\right\}]\hfill \\ \hfill & \times E_x\left[exp\left\{\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}{(g_j,x)}^{\sim }\right\}\right].\hfill \end{array}$$

(3.4)

By (2.5) and (3.2), equation (3.4) can be rewritten by

$$\begin{array}{cc}& E_x\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n\left({\xi }_j+{(x_0,g_j)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & \times E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & \times exp\left\{{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}{(g_j,a)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & =exp\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n{\xi }_j{(x_0,g_j)}_{C_{a,b}^{\prime }}\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n{(g_j,a)}_{C_{a,b}^{\prime }}{(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ & \times E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right),\hfill \end{array}$$

as desired. □

Corollary 3.5.

Let X and F be as in Theorem 3.4. Then it follows that for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$$\begin{array}{cc}\hfill & E_x\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n\left({\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }-{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right).\hfill \end{array}$$

(3.5)

Proof. 

Let $G\left(x\right)=F\left(x\right)exp\{-{(x_0,x)}^{\sim }\}$. Then using (3.3) with F replaced with G, it follows that

$$\begin{array}{cc}& E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\left\{\parallel x_0{\parallel }_{C_{a,b}^{\prime }}^2\right\}{E}_x\left(F(x+x_0)exp\{-{(x_0,x+x_0)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\left\{\parallel x_0{\parallel }_{C_{a,b}^{\prime }}^2\right\}{E}_x\left(G(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & \times E_x\left(G\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & \times E_x\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right).\hfill \end{array}$$

From this, we obtain that

$$\begin{array}{cc}\hfill & E_x\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}+{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n\left({\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }\right).\hfill \end{array}$$

(3.6)

Replacing $\overrightarrow{\xi }$ by $\overrightarrow{\xi }-{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}$ in (3.6), we have equation (3.5) as desired. □

Remark 3.6.

Using the techniques similar to those used in the proof of Theorem 3.4, we can establish equation (3.5) without use of equation (3.3). Also, equation (3.3) can be established by use of equation (3.5).

4. Conditional Function Space Integration Formulas

In [3], Chang and Chung extended the results of [38,39] to the function space $C_{a,b}[0,T]$ using the vector-valued conditioning function $X_{\overrightarrow{t}}:C_{a,b}[0,T]\to {\mathbb{R}}^n$ given by

$$X_{\overrightarrow{t}}\left(x\right)=(x\left(t_1\right),\dots ,x\left(t_n\right)),0=t_0<t_1<\dots <t_n=T.$$

(4.1)

The conditioning function $X_{\overrightarrow{t}}\left(x\right)$ given by (4.1) can be represented by the conditioning function $X\left(x\right)$ given by (2.4) with the special choice of $g_j$’s.

Let $0=t_0<t_1<\dots <t_n=T$ be a partition of $[0,T]$. For each $j\in \{1,\dots ,n\}$, let

$$g_{\overrightarrow{t},j}\left(t\right)={\int }_0^t{\displaystyle \frac{1}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}}{\chi }_{[t_{j-1},t_j]}\left(s\right)db\left(s\right).$$

(4.2)

Then ${\mathcal{G}}_{\overrightarrow{t}}=\{g_{\overrightarrow{t},1},\dots ,g_{\overrightarrow{t},n}\}$ is an orthonormal set of functions in $C_{a,b}[0,T]$, and thus the conditioning function $X_{{\mathcal{G}}_{\overrightarrow{t}}}:C_{a,b}[0,T]\to {\mathbb{R}}^n$ given by

$$X_{{\mathcal{G}}_{\overrightarrow{t}}}=({(g_{\overrightarrow{t},1},x)}^{\sim },\dots ,{(g_{\overrightarrow{t},n},x)}^{\sim })$$

(4.3)

is under our consideration. Given a vector $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)\in {\mathbb{R}}^n$, let

$$k_{\overrightarrow{t},j}=\sum _{l=1}^j\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}{g}_{\overrightarrow{t},l}\in C_{a,b}^{\prime }[0,T]$$

for each $j\in \{1,\dots ,n\}$. Then it follows that

$$X_{\overrightarrow{t}}\left(x\right)=({(k_{\overrightarrow{t},1},x)}^{\sim },\dots ,{(k_{\overrightarrow{t},n},x)}^{\sim }).$$

From this, it also follows that for any $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)\in {\mathbb{R}}^n$,

$$\begin{array}{cc}& X_{{\mathcal{G}}_{\overrightarrow{t}}}\left(x\right)=({\xi }_1,\dots ,{\xi }_n)⟺\hfill \\ & X_{\overrightarrow{t}}\left(x\right)=\left(\sqrt{b\left(t_1\right)-b\left(t_0\right)}{\xi }_1,\sum _{l=1}^2\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}{\xi }_l,\dots ,\sum _{l=1}^n\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}{\xi }_l\right)\hfill \end{array}$$

and for any $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill & X_{\overrightarrow{t}}\left(x\right)=({\eta }_1,\dots ,{\eta }_n)⟺\hfill \\ \hfill & X_{{\mathcal{G}}_{\overrightarrow{t}}}\left(x\right)=\left({\displaystyle \frac{{\eta }_1-{\eta }_0}{\sqrt{b\left(t_1\right)-b\left(t_0\right)}}},\dots ,{\displaystyle \frac{{\eta }_n-{\eta }_{n-1}}{\sqrt{b\left(t_n\right)-b\left(t_{n-1}\right)}}}\right)\hfill \end{array}$$

(4.4)

with ${\eta }_0=0$.

Given a vector $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$, let ${\xi }_l={\displaystyle \frac{{\eta }_l-{\eta }_{l-1}}{\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}}}$ for each $l\in \{1,\dots ,n\}$. Then it follows that for each $t\in [t_{j-1},t_j]$,

$$\begin{array}{cc}\hfill [\overrightarrow{\eta }{]}_{\overrightarrow{t}}\left(t\right)\equiv [\overrightarrow{\xi }]\left(t\right)& =\sum _{l=1}^n{\xi }_l{g}_{\overrightarrow{t},l}\left(t\right)\hfill \\ \hfill & =\sum _{l=1}^n{\displaystyle \frac{{\eta }_l-{\eta }_{l-1}}{\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}}}{\int }_0^t{\displaystyle \frac{1}{\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}}}{\chi }_{[t_{l-1},t_l]}\left(s\right)db\left(s\right)\hfill \\ \hfill & ={\eta }_{j-1}+{\displaystyle \frac{b\left(t\right)-b\left(t_{j-1}\right)}{b\left(t_j\right)-b\left(t_{j-1}\right)}}({\eta }_j-{\eta }_{j-1})\hfill \end{array}$$

(4.5)

where ${\eta }_0=0$, and

$$\begin{array}{cc}\hfill [x{]}_{\overrightarrow{t}}\left(t\right)\equiv [x]\left(t\right)& =\sum _{l=1}^n{(g_{\overrightarrow{t},l},x)}^{\sim }{g}_{\overrightarrow{t},l}\left(t\right)\hfill \\ \hfill & =\sum _{l=1}^n{\displaystyle \frac{x\left(t_l\right)-x\left(t_{l-1}\right)}{\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}}}{\int }_0^t{\displaystyle \frac{1}{\sqrt{b\left(t_l\right)-b\left(t_{l-1}\right)}}}{\chi }_{[t_{l-1},t_l]}\left(s\right)db\left(s\right)\hfill \\ \hfill & =x\left(t_{j-1}\right)+{\displaystyle \frac{b\left(t\right)-b\left(t_{j-1}\right)}{b\left(t_j\right)-b\left(t_{j-1}\right)}}(x\left(t_j\right)-x\left(t_{j-1}\right)).\hfill \end{array}$$

(4.6)

In view of Theorem 2.4 and with the above setting, we have the following corollary.

Corollary 4.1.

Let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then the conditioning function $X_{{\mathcal{G}}_{\overrightarrow{t}}}$ given by equation (4.3) yields the conditioning function $X_{\overrightarrow{t}}$ given by (4.1), and it follows the conditional function space integration formula:

$$E_x\left(F\right|X_{\overrightarrow{t}})\left(\overrightarrow{\eta }\right)=E_x\left[F\left(x-{\left[x\right]}_{\overrightarrow{t}}+{\left[\overrightarrow{\eta }\right]}_{\overrightarrow{t}}\right)\right]$$

where ${\left[x\right]}_{\overrightarrow{t}}$ and ${\left[\overrightarrow{\eta }\right]}_{\overrightarrow{t}}$ are given by (4.6) and (4.5) respectively.

Lemma 4.2.

For each $j\in \{1,\dots ,n\}$, let $g_{\overrightarrow{t},j}$ be given by (4.2). Then it follows that

$${(g_{\overrightarrow{t},j},a)}_{C_{a,b}^{\prime }}={\displaystyle \frac{a\left(t_j\right)-a\left(t_{j-1}\right)}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}},$$

(4.7)

and for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$${(x_0,g_{\overrightarrow{t},j})}_{C_{a,b}^{\prime }}={\displaystyle \frac{x_0\left(t_j\right)-x_0\left(t_{j-1}\right)}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}}.$$

(4.8)

Corollary 4.3

([3]). Let F be a μ-integrable function on $C_{a,b}[0,T]$ and given a partition $0=t_0<t_1<\dots <t_n=T$ and a vector $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$, let $X_{\overrightarrow{t}}$, $g_j$$(j\in \{1,\dots ,n\left\}\right)$, ${\left[\overrightarrow{\eta }\right]}_{\overrightarrow{t}}$, and ${\left[x\right]}_{\overrightarrow{t}}$ be as above. Then it follows that for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$$\begin{array}{cc}\hfill & E_x\left(F\left(x\right)|x\left(t_j\right)={\eta }_j,j=1,\dots ,n\right)\hfill \\ \hfill & =E_x\left(F\left(x\right)|{(g_{\overrightarrow{t},j},x)}^{\sim }={\displaystyle \frac{{\eta }_j-{\eta }_{j-1}}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}},j=1,\dots ,n\right)\hfill \\ \hfill & =exp\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{\displaystyle \frac{{(x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}^2}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\hfill \\ \hfill & +\sum _{j=1}^n{\displaystyle \frac{[({\eta }_j-{\eta }_{j-1})-(a\left(t_j\right)-a\left(t_{j-1}\right))](x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\}\hfill \\ \hfill & \times E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|x\left(t_j\right)={\eta }_j-x_0\left(t_j\right),j=1,\dots ,n\right)\hfill \end{array}$$

(4.9)

where ${\eta }_0=0$.

Proof. 

Using (4.1), (3.5) together with (4.4), (4.7), and (4.8), one can derive equation (4.9). □

Corollary 4.4.

Let F be a μ-integrable function on $C_{a,b}[0,T]$ and given a partition $0=t_0<t_1<\dots <t_n=T$ and a vector $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$, let $X_{\overrightarrow{t}}$, $g_j$$(j\in \{1,\dots ,n\left\}\right)$, ${\left[\overrightarrow{\eta }\right]}_{\overrightarrow{t}}$, and ${\left[x\right]}_{\overrightarrow{t}}$ be as above. Then it follows that for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

$$\begin{array}{cc}\hfill & E_x\left(F(x+x_0)|x\left(t_j\right)={\eta }_j,j=1,\dots ,n\right)\hfill \\ \hfill & =exp\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{\displaystyle \frac{{(x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}^2}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\hfill \\ \hfill & -\sum _{j=1}^n{\displaystyle \frac{[({\eta }_j-{\eta }_{j-1})-(a\left(t_j\right)-a\left(t_{j-1}\right))](x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\}\hfill \\ \hfill & \times E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|x\left(t_j\right)={\eta }_j+x_0\left(t_j\right),j=1,\dots ,n\right)\hfill \end{array}$$

(4.10)

where ${\eta }_0=0$.

Proof. 

Using (4.1), (3.3) together with (4.4), (4.7), and (4.8), one can also derive equation (4.10). □

Example 4.5.

Let $S:C_{a,b}^{\prime }[0,T]\to C_{a,b}^{\prime }[0,T]$ be the linear operator defined by

$$Sw\left(t\right)={\int }_0^{t}w\left(s\right)db\left(s\right).$$

Then, we see that the adjoint operator $S^{*}$ of S is given by

$$S^{*}w\left(t\right)=w\left(T\right)b\left(t\right)-{\int }_0^{t}w\left(s\right)db\left(s\right)={\int }_0^t[w\left(T\right)-w\left(s\right)]db\left(s\right).$$

Using an integration by parts formula, it follows that

$${(S^{*}b,x)}^{\sim }={\int }_0^{T}x\left(t\right)db\left(t\right).$$

Let $\{g_1,\dots ,g_n\}$ be an orthonormal set of functions in $C_{a,b}[0,T]$ and let $x_0\in C_{a,b}^{\prime }[0,T]$. Also for $\overrightarrow{\eta }\in {\mathbb{R}}^n$, let $\overrightarrow{\xi }=\overrightarrow{\eta }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}$. Then, using equation (3.5) with $F\left(x\right)\equiv 1$ on $C_{a,b}[0,T]$, we obtain that

$$\begin{array}{cc}\hfill 1& \equiv E_x\left(F\left(x\right)|X\left(x\right)=\overrightarrow{\eta }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ \hfill & =exp\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n\left({\eta }_j+{(x_0,g_j)}_{C_{a,b}^{\prime }}-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ \hfill & \times E_x\left(exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\eta }\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}+{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+\sum _{j=1}^n\left({\eta }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\eta }\right).\hfill \end{array}$$

(4.11)

Using equation (4.11), we immediately obtain the conditional function space integration formula

$$\begin{array}{cc}\hfill & E_x\left(exp\{-{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\eta }\right)\hfill \\ \hfill & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\eta }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}.\hfill \end{array}$$

(4.12)

In particular, using (4.12) with $x_0$ replaced with $S^{*}b$, and an integration by parts formula, we obtain

$$\begin{array}{cc}& E_x\left(exp\left\{-{\int }_0^{T}x\left(t\right)db\left(t\right)\right\}|x\left(t_j\right)={\eta }_j,j=1,\dots ,n\right)\hfill \\ & =E_x\left(exp\left\{-{(S^{*}b,x)}^{\sim }\right\}|{(g_{\overrightarrow{t},j},x)}^{\sim }={\displaystyle \frac{{\eta }_j-{\eta }_{j-1}}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}},j=1,\dots ,n\right)\hfill \\ & =exp\{{\displaystyle \frac{1}{2}}{\parallel S^{*}b\parallel }_{C_{a,b}^{\prime }}-{(S^{*}b,a)}_{C_{a,b}^{\prime }}\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(S^{*}b,g_{\overrightarrow{t},j})}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\displaystyle \frac{{\eta }_j-{\eta }_{j-1}}{\sqrt{b\left(t_j\right)-b\left(t_{j-1}\right)}}}-{(g_{\overrightarrow{t},j},a)}_{C_{a,b}^{\prime }}\right){(S^{*}b,g_{\overrightarrow{t},j})}_{C_{a,b}^{\prime }}\}\hfill \\ & =exp\{{\displaystyle \frac{1}{6}}{b}^3\left(T\right)-{\int }_0^{T}a\left(t\right)db\left(t\right)\hfill \\ & -{\displaystyle \frac{1}{8}}\sum _{j=1}^n(b\left(t_j\right)-b\left(t_{j-1}\right)){(2b\left(T\right)-b\left(t_j\right)-b\left(t_{j-1}\right))}^2\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{j=1}^n[({\eta }_j-{\eta }_{j-1})-(a\left(t_j\right)-a\left(t_{j-1}\right))](2b\left(T\right)-b\left(t_j\right)-b\left(t_{j-1}\right))\}\hfill \end{array}$$

where ${\eta }_0=0$.

Letting $b\left(t\right)=t$ and $a\left(t\right)=0$, the function space $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$ reduces the one-parameter Wiener space $(C_0[0,T],\mathcal{W},m_{\mathrm{w}})$. Many physical problems can be formulated by the conditional Wiener integral $E\left(F\right|X_t)$ of the Wiener integrable functionals F on $C_0[0,T]$ which have the form

$$F\left(x\right)=exp\left\{-{\int }_0^t\theta (s,x\left(s\right))ds\right\}$$

where $X_t\left(x\right)=x\left(t\right)$ and $\theta (·,·)$ is a sufficiently smooth function on $[0,T]\times \mathbb{R}$. It is known [40,41] that the function $U(·,·)$ on $[0,T]\times \mathbb{R}$ defined by

$$U(t,\eta )={\left(2\pi t\right)}^{-1/2}exp\left\{-{\displaystyle \frac{{(\eta -{\eta }_0)}^2}{2t}}\right\}E\left(F(x\left(t\right)+{\eta }_0)\right|x\left(t\right)=\eta -{\eta }_0)$$

forms a solution of the partial differential equation

$${\displaystyle \frac{\partial U}{\partial t}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}U}{\partial {\eta }^2}}-\theta U$$

under an appropriate initial condition at $t=0$. The Kac’s result described above was extended by Chang and Chung in [3,18]. The conditioning function $X_T$ given by (4.13) below was used in [3,18].

In this view point, the formulas with the 1-dimensional conditioning function

$$X_T\left(x\right)=x\left(T\right)$$

(4.13)

is thus more relevant in the heat equation theory and other applications.

Consider the conditioning function $X_b:C_{a,b}[0,T]\to \mathbb{R}$ given by

$$X_b\left(x\right)={(b/\sqrt{b\left(T\right)},x)}^{\sim }.$$

This conditioning function $X_b$ will play a good role between the previous researches and the current research for conditional function space integrals, because

$$X_T\left(x\right)=\eta ⟺X_b\left(x\right)=\eta /\sqrt{b\left(T\right)}.$$

Notice that ${\mathcal{G}}_T=\{b/\sqrt{b\left(T\right)}\}$ is an orthonormal set in $(C_{a,b}^{\prime }[0,T],\parallel ·{\parallel }_{C_{a,b}^{\prime }})$.

Example 4.6.

Let F be a μ-integrable functional on $C_{a,b}[0,T]$ and let $x_0$ be a function in $C_{a,b}^{\prime }[0,T]$. Then equation (4.9) reduces the formula: for $\eta \in \mathbb{R}$,

$$\begin{array}{cc}\hfill & E_x\left(F\left(x\right)\left|x\right(T)=\eta \right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}+{\displaystyle \frac{(\eta -a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}\hfill \\ \hfill & \times E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|x\left(T\right)=\eta -x_0\left(T\right)\right).\hfill \end{array}$$

(4.14)

Replacing η with $\eta +x_0\left(T\right)$ in equation (4.14), it also follows that

$$\begin{array}{cc}& E_x\left(F(x+x_0)exp\{-{(x_0,x)}^{\sim }\}|x\left(T\right)=\eta \right)\hfill \\ & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}-{\displaystyle \frac{(\eta -a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}\hfill \\ & \times E_x\left(F\left(x\right)|x\left(T\right)=\eta +x_0\left(T\right)\right).\hfill \end{array}$$

In particular, setting $F\equiv 1$, we have the conditional function space integration formula

$$\begin{array}{cc}\hfill & E_x\left(exp\{-{(x_0,x)}^{\sim }\}|x\left(T\right)=\eta \right)\hfill \\ \hfill & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}-{\displaystyle \frac{(\eta -a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}.\hfill \end{array}$$

(4.15)

Example 4.7.

Let F be a μ-integrable functional on $C_{a,b}[0,T]$ and let $x_0$ be a function in $C_{a,b}^{\prime }[0,T]$. Then equation (4.10) reduces the formula

$$\begin{array}{cc}\hfill & E_x\left(F(x+x_0)|x\left(T\right)=\eta \right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}-{\displaystyle \frac{({\eta }_j-a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}\hfill \\ \hfill & \times E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|x\left(T\right)=\eta +x_0\left(T\right)\right).\hfill \end{array}$$

(4.16)

Replacing η with $\eta -x_0\left(T\right)$ in equation (4.16), it also follows that

$$\begin{array}{cc}& E_x\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|x\left(T\right)=\eta \right)\hfill \\ & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}+{\displaystyle \frac{({\eta }_j-a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}\hfill \\ & \times E_x\left(F(x+x_0)|x\left(T\right)=\eta -x_0\left(T\right)\right).\hfill \end{array}$$

Also, setting $F\equiv 1$, we have

$$\begin{array}{cc}\hfill & E_x\left(exp\left\{{(x_0,x)}^{\sim }\right\}|x\left(T\right)=\eta \right)\hfill \\ \hfill & =exp\left\{{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{x_0^2\left(T\right)}{2b\left(T\right)}}+{\displaystyle \frac{({\eta }_j-a\left(T\right))x_0\left(T\right)}{b\left(T\right)}}\right\}.\hfill \end{array}$$

(4.17)

One can easily see that equation (4.17) with $x_0$ replaced with $-x_0$ coincides equation (4.15) above.

5. Conditional Generalized Analytic Feynman Integrals

In this section, we will extend the results for conditional function space integrals obtained in previous section to the conditional generalized analytic Feynman integral of functionals F on $C_{a,b}[0,T]$. For some related work involving the conditional analytic Feynman integral on classical and abstract Wiener spaces, see [29,30,36,38].

Lemma 5.1.

Let X be given by (2.4) and let F be a $\mathbb{C}$-valued functional on $C_{a,b}[0,T]$ which satisfies conditions (2.2) and (2.3) above. Then for all $x_0\in C_{a,b},[0,T]$ and any $\rho >0$, it follows that

$$\begin{array}{cc}\hfill & E_x\left(F(\rho x+\rho x_0)|\rho X\left(s\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-{\displaystyle \frac{1}{\rho }}\sum _{j=1}^n\left({\xi }_j-\rho {(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(F\left(\rho x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|\rho X\left(x\right)=\overrightarrow{\xi }+\rho {(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right).\hfill \end{array}$$

(5.1)

Proof. 

Let $G\left(x\right)=F\left(\rho x\right)$. Then $G(x+x_0)=F(\rho x+\rho x_0)$. Hence by equation (3.3) with F replaced with G, we have

$$\begin{array}{cc}& E_x\left(F(\rho x+\rho x_0)|\rho X\left(s\right)=\overrightarrow{\xi }\right)=E_x\left(G(x+x_0)|X\left(x\right)={\rho }^{-1}\overrightarrow{\xi }\right)\hfill \\ & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n\left({\displaystyle \frac{1}{\rho }}{\xi }_j-{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & \times E_x\left(G\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|X\left(x\right)={\rho }^{-1}\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}-{(x_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-{\displaystyle \frac{1}{\rho }}\sum _{j=1}^n\left({\xi }_j-\rho {(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ & \times E_x\left(F\left(\rho x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|\rho X\left(x\right)=\overrightarrow{\xi }+\rho {(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \end{array}$$

as desired. □

Theorem 5.2.

Let X and F be as in Lemma 5.1. Assume that given a non-zero real q, the conditional generalized Feynman integral of F, $E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)\right|X\left(x\right)=\overrightarrow{\xi })$ exists. Then it follows that for any $x_0\in C_{a,b}^{\prime }[0,T]$,

$$\begin{array}{cc}\hfill & E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & \stackrel{*}{=}exp\{{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(-iq)}^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ \hfill & +{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+iq\sum _{j=1}^n\left({\xi }_j-{(-iq)}^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ \hfill & \times E_x^{{\mathrm{anf}}_q}\left(F\left(x\right)exp\{-iq{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right),\hfill \end{array}$$

(5.2)

where $\stackrel{*}{=}$ means that if either side of equation (5.2) exists, both side exist and equality holds.

Proof. 

Let $\rho >0$ be given. Since $x_0$ is in the linear space $C_{a,b}^{\prime }[0,T]$, $(1/\rho )x_0$ is also in $C_{a,b}^{\prime }[0,T]$. Let $y_0=(1/\rho )x_0$. Then, using equation (5.1) with $x_0$ replaced with $y_0$, it follows that

$$\begin{array}{cc}\hfill & E_x\left(F(\rho x+x_0)|X\left(\rho x\right)=\overrightarrow{\xi }\right)=E_x\left(F(\rho x+\rho y_0)|\rho X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =exp\left\{-{\displaystyle \frac{1}{2}}{\parallel y_0\parallel }_{C_{a,b}^{\prime }}^2-{(y_0,a)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{1}{2}}\sum _{j=1}^n{(y_0,g_j)}_{C_{a,b}^{\prime }}^2-{\displaystyle \frac{1}{\rho }}\sum _{j=1}^n\left({\xi }_j-\rho {(g_j,a)}_{C_{a,b}^{\prime }}\right){(y_0,g_j)}_{C_{a,b}^{\prime }}\right\}\hfill \\ \hfill & \times E_x\left(F\left(\rho x\right)exp\left\{{(y_0,x)}^{\sim }\right\}|\rho X\left(x\right)=\overrightarrow{\xi }+\rho {(y_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ \hfill & =exp\{-{\displaystyle \frac{1}{2{\rho }^2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{\displaystyle \frac{1}{\rho }}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ \hfill & -{\displaystyle \frac{1}{2{\rho }^2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-{\displaystyle \frac{1}{{\rho }^2}}\sum _{j=1}^n\left({\xi }_j-\rho {(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ \hfill & \times E_x\left(F\left(\rho x\right)exp\left\{{\rho }^{-2}{(x_0,\rho x)}^{\sim }\right\}|\rho X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right).\hfill \end{array}$$

(5.3)

Now let $\rho ={\lambda }^{-1/2}$. Then equation (5.3) becomes

$$\begin{array}{cc}\hfill & E_x\left(F({\lambda }^{-1/2}x+x_0)|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\xi }\right)=E_x\left(F({\lambda }^{-1/2}x+x_0)|{\lambda }^{-1/2}X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ \hfill & =exp\{-{\displaystyle \frac{\lambda }{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{\lambda }^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\lambda \sum _{j=1}^n\left({\xi }_j-{\lambda }^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ \hfill & \times E_x\left(F\left({\lambda }^{-1/2}x\right)exp\left\{\lambda {(x_0,{\lambda }^{-1/2}x)}^{\sim }\right\}|{\lambda }^{-1/2}X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right).\hfill \end{array}$$

(5.4)

Since $\rho >0$ was arbitrary, we have that equation (5.4) holds for all $\lambda >0$. We now use Definition 2.3 to obtain the following conclusion

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =\underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}_x\left(F({\lambda }^{-1/2}x+x_0)|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\xi }\right)\hfill \\ & =\underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}exp\{-{\displaystyle \frac{\lambda }{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{\lambda }^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ & -{\displaystyle \frac{\lambda }{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-\lambda \sum _{j=1}^n\left({\xi }_j-{\lambda }^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ & \times E_x\left(F\left({\lambda }^{-1/2}x\right)exp\left\{\lambda {(x_0,{\lambda }^{-1/2}x)}^{\sim }\right\}|{\lambda }^{-1/2}X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & =exp\{{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(-iq)}^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ & +{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+iq\sum _{j=1}^n\left({\xi }_j-{(-iq)}^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ & \times \underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}_x\left(F\left({\lambda }^{-1/2}x\right)exp\left\{\lambda {(x_0,{\lambda }^{-1/2}x)}^{\sim }\right\}|{\lambda }^{-1/2}X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & =exp\{{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(-iq)}^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ & +{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+iq\sum _{j=1}^n\left({\xi }_j-{(-iq)}^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ & \times \underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}_x\left(F\left({\lambda }^{-1/2}x\right)exp\{-iq{(x_0,{\lambda }^{-1/2}x)}^{\sim }\}|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \\ & =exp\{{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2-{(-iq)}^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ & +{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2+iq\sum _{j=1}^n\left({\xi }_j-{(-iq)}^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}\hfill \\ & \times E_x^{{\mathrm{anf}}_q}\left(F\left(x\right)exp\{-iq{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}\right)\hfill \end{array}$$

as desired □

Corollary 5.3.

Letting $F\equiv 1$ and replacing $\overrightarrow{\xi }$ with $\overrightarrow{\xi }-{(x_0,\overrightarrow{g})}_{C_{a,b}}^{\prime }$ in equation (5.2), it follows the conditional generalized analytic Feynman integration formula

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(exp\{-iq{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\{-{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_{a,b}^{\prime }}^2+{(-iq)}^{1/2}{(x_0,a)}_{C_{a,b}^{\prime }}\hfill \\ & +{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_{a,b}^{\prime }}^2-iq\sum _{j=1}^n\left({\xi }_j+{(-iq)}^{-1/2}{(g_j,a)}_{C_{a,b}^{\prime }}\right){(x_0,g_j)}_{C_{a,b}^{\prime }}\}.\hfill \end{array}$$

6. Explicit Examples

In this section, we provide that the assumption (and hence the conclusion) of Theorem 5.2 is indeed satisfied by several large classes of functionals; we shall very briefly discuss three such classes.

6.1. Banach Algebra $\mathcal{F}\left(C_{a,b}[0,T]\right)$

In our next corollary, we will see that the translation formula (5.2) holds for the conditional generalized analytic Feynman integral of functionals in the Banach algebra $\mathcal{F}\left(C_{a,b}[0,T]\right)$, which is a generalized class of the Banach algebra $\mathcal{S}$ introduced by Cameron and Storvick [42]. The Banach algebra $\mathcal{F}\left(C_{a,b}[0,T]\right)$ consists of functionals expressible in the form

$$F\left(x\right)={\int }_{C_{a,b}^{\prime }[0,T]}exp\left\{i{(w,x)}^{\sim }\right\}df\left(w\right)$$

(6.1)

for s-a.e. $x\in C_{a,b}[0,T]$, where f is an element of $\mathcal{M}\left(C_{a,b}^{\prime }[0,T]\right)$, the space of all $\mathbb{C}$-valued countably additive finite Borel measures on $C_{a,b}^{\prime }[0,T]$. Further work involving the functionals in $\mathcal{F}\left(C_{a,b}[0,T]\right)$ and related topics include [4,5,6,9,10,16].

Corollary 6.1.

Let $X\left(x\right)$ be given by (2.4) and let $F\in \mathcal{F}\left(C_{a,b}[0,T]\right)$ be given by (6.1). Assume that

$${\int }_{C_{a,b}^{\prime }[0,T]}exp\left\{{\displaystyle \frac{(1+n)}{\sqrt{2q_0}}}{\parallel w\parallel }_{C_{a,b}^{\prime }}{\parallel a\parallel }_{C_{a,b}^{\prime }}\right\}d\left|f\right|\left(w\right)<+\infty$$

(6.2)

for some positive real number $q_0>0$. Then the conditional generalized analytic Feynman integrals in both sides of (5.2) exist, and so equation (5.2) holds true for all real q with $\left|q\right|>q_0$.

Proof. 

By [4], Corollary 5.4, the conditional generalized analytic Feynman integral of F, $E^{{\mathrm{anf}}_q}\left(F\right|X=\overrightarrow{\xi })$, exists for all real number q with $\left|q\right|>q_0$. Thus this corollary follows from Theorem 5.2. □

To ensure the existence of the conditional generalized analytic Feynman integral of functionals in the class $\mathcal{F}\left(C_{a,b}[0,T]\right)$, we have to require the condition (6.2). There is an example of a functional F of class $\mathcal{F}\left(C_{a,b}[0,T]\right)$ which is not generalized analytic Feynman integrable (and hence does not exist the conditional generalized analytic Feynman integral of F), see [5].

For $F\in \mathcal{F}\left(C_{a,b}[0,T]\right)$ which satisfies condition (6.2), direct calculations show indeed that

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & ={\int }_{C_{a,b}^{\prime }[0,T]}exp\{i{(w,x_0)}_{C_{a,b}^{\prime }}+i\sum _{j=1}^n{\xi }_j{(w,g_j)}_{C_{a,b}^{\prime }}-{\displaystyle \frac{i}{2q}}\left({\parallel w\parallel }_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n{(g_j,w)}_{C_{a,b}^{\prime }}^2\right)\hfill \\ & +i{(-iq)}^{-1/2}\left({(w,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n{(w,g_j)}_{C_{a,b}^{\prime }}{(g_j,a)}_{C_{a,b}^{\prime }}\right)\}df\left(w\right).\hfill \end{array}$$

We easily see by the Cauchy–Schwarz inequality that

$$\begin{array}{cc}& \left|E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\right|\hfill \\ & \le {\int }_{C_{a,b}^{\prime }[0,T]}exp\left\{{\displaystyle \frac{(1+n)}{\sqrt{2q_0}}}{\parallel w\parallel }_{C_{a,b}^{\prime }}{\parallel a\parallel }_{C_{a,b}^{\prime }}\right\}d\left|f\right|\left(w\right)<+\infty .\hfill \end{array}$$

Thus the assumption (and hence the conclusion) of Theorem 5.2 is indeed satisfied.

6.2. Bounded Cylinder Functionals

Next we want to briefly discuss another class of functionals to which our general translation theorem applied. Given a $\mathbb{C}$-valued Borel measure $\nu$ on ${\mathbb{R}}^m$, the Fourier transform $\widehat{\nu }$ of $\nu$ is a $\mathbb{C}$-valued function on ${\mathbb{R}}^m$ defined by the formula

$$\widehat{\nu }\left(\overrightarrow{u}\right)={\int }_{{\mathbb{R}}^m}exp\left\{i\sum _{k=1}^m{u}_k{v}_k\right\}d\nu \left(\overrightarrow{v}\right),$$

where $\overrightarrow{u}=(u_1,\dots ,u_m)$ and $\overrightarrow{v}=(v_1,\dots ,v_m)$ are in ${\mathbb{R}}^m$.

Given a complex Borel measure $\nu$ on ${\mathbb{R}}^m$ and an orthogonal subset $\mathcal{A}=\{e_1,\dots ,e_m\}$ of non-zero functions in $C_{a,b}^{\prime }[0,T]$, define the functional $F:C_{a,b}[0,T]\to \mathbb{C}$ by

$$F\left(x\right)=\widehat{\nu }({(e_1,x)}^{\sim },\dots ,{(e_m,x)}^{\sim })$$

(6.3)

for s-a.e. $x\in C_{a,b}[0,T]$.

For the orthogonal set $\mathcal{A}=\{e_1,\dots ,e_m\}$, let ${\widehat{\mathfrak{T}}}_{\mathcal{A}}$ be the space of all functionals F on $C_{a,b}[0,T]$ having the form (6.3). Note that $F\in {\widehat{\mathfrak{T}}}_{\mathcal{A}}$ implies that F is scale-invariant measurable on $C_{a,b}[0,T]$. For a more detailed study of functionals in ${\widehat{\mathfrak{T}}}_{\mathcal{A}}$, see [11].

Corollary 6.2.

Let $X\left(x\right)$ be given by (2.4) and let $F\in {\widehat{\mathfrak{T}}}_{\mathcal{A}}$ be given by (6.3). Given a positive real $q_0$, assume that the complex Borel measure ν corresponding to F by (6.3) satisfies the condition

$${\int }_{{\mathbb{R}}^m}exp\left\{{\displaystyle \frac{{(n+1)\parallel a\parallel }_{C_{a,b}^{\prime }}}{\sqrt{2q_0}}}\sum _{k=1}^m{\parallel e_k\parallel }_{C_{a,b}^{\prime }}|v_k|\right\}d|\nu |\left(\overrightarrow{v}\right)<+\infty .$$

(6.4)

Then the conditional generalized analytic Feynman integrals in both sides of (5.2) exist, and so equation (5.2) holds true for all real q with $vertqvert>q_0$.

Proof. 

Using the techniques similar to those used in the proof of [4], Theorem 5.3, we can establish the existence of the conditional generalized analytic Feynman integral $E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)$ for functionals F given by (6.3) under the condition (6.4). Thus this corollary follows immediately from Theorem 5.2. □

6.3. Exponential-Type Functionals

Let $\mathcal{E}$ be the class of all functionals which have the form

$${\Psi }_w\left(x\right)=exp\left\{{(w,x)}^{\sim }\right\}$$

(6.5)

for some $w\in C_{a,b}^{\prime }[0,T]$ and for s-a.e. $x\in C_{a,b}[0,T]$. More precisely, since we shall identify functionals which coincide s-a.e. on $C_{a,b}[0,T]$, the class $\mathcal{E}$ can be regarded as the space of all s-equivalence classes of functionals of the form (6.5). The functionals given by equation (6.5) and linear combinations (with complex coefficients) of the ${\Psi }_w\left(x\right)$’s are called the (partially) exponential-type functionals on $C_{a,b}[0,T]$.

Remark 6.3.

The linear space $\mathcal{E}\left(C_{a,b}[0,T]\right)=\mathrm{Span}\mathcal{E}$ of partially exponential-type functionals is a commutative (complex) algebra under the pointwise multiplication and with identity ${\Psi }_0\equiv 1$. For more details see, [8].

Proceeding formally we see that the conditional generalized analytic Feynman integral of each functional ${\Psi }_w$ given by (6.5), $E_x^{{\mathrm{anf}}_q}\left({\Phi }_w(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)$, exists and is given by the formula

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left({\Phi }_w(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =exp\{{(w,x_0)}_{C_{a,b}^{\prime }}+\sum _{j=1}^n{\xi }_j{(w,g_j)}_{C_{a,b}^{\prime }}+{\displaystyle \frac{i}{2q}}\left({\parallel w\parallel }_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n{(g_j,w)}_{C_{a,b}^{\prime }}^2\right)\hfill \\ & +{(-iq)}^{-1/2}\left({(w,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n{(w,g_j)}_{C_{a,b}^{\prime }}{(g_j,a)}_{C_{a,b}^{\prime }}\right)\}\hfill \end{array}$$

for all real $q\in \mathbb{R}\setminus \left\{0\right\}$. Thus by the linearity of the conditional Feynman integral, one can see that the theorems, corollaries, and formulas established in previous sections hold for the exponential-type functionals in $\mathcal{E}\left(C_{a,b}[0,T]\right)$.

Theorem 6.4.

Let F be a partially exponential-type functional in $\mathcal{E}\left(C_{a,b}[0,T]\right)$. Then F is given by $F\left(x\right)={\sum }_{l=1}^m{c}_l{\Psi }_{w_l}$ for s-a.e. $x\in C_{a,b}[0,T]$, where ${\left\{c_l\right\}}_{l=1}^m$ is a finite sequence in $\mathbb{C}\setminus \left\{0\right\}$ and ${\left\{w_l\right\}}_{l=1}^m$ is a finite sequence of non-zero functions in $C_{a,b}^{\prime }[0,T]$. Then for any non-zero real number q and any function $x_0$ in $C_{a,b}^{\prime }[0,T]$, it follows that

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & =\sum _{l=1}^m{c}_{l}exp\{{(w_l,x_0)}_{C_{a,b}^{\prime }}+\sum _{j=1}^n{\xi }_j{(w_l,g_j)}_{C_{a,b}^{\prime }}+{\displaystyle \frac{i}{2q}}\left(\parallel w_l{\parallel }_{C_{a,b}^{\prime }}^2-\sum _{j=1}^n{(g_j,w_l)}_{C_{a,b}^{\prime }}^2\right)\hfill \\ & +{(-iq)}^{-1/2}\left({(w_l,a)}_{C_{a,b}^{\prime }}-\sum _{j=1}^n{(w_l,g_j)}_{C_{a,b}^{\prime }}{(g_j,a)}_{C_{a,b}^{\prime }}\right)\}.\hfill \end{array}$$

7. Corollary in Wiener Space $C_0[0,T]$.

We can also expect corollaries for the conditional generalized Feynman integrals with the conditioning function $X_{\overrightarrow{t}}$ given by (4.1) (specifically, (4.13)), such as the corollaries and the formulas provided in Section 4 above.

Furthermore, from our assertions discussed in this paper, we also have the translation formulas for the conditional analytic Feynman integral defined on the Wiener space $C_0[0,T]$: letting $a\left(t\right)\equiv 0$ and $b\left(t\right)=t$ on $[0,T]$, the function space $C_{a,b}[0,T]$ reduces to the classical Wiener space $C_0[0,T]$. Also it follows

$$\begin{array}{cc}\hfill C_{0,t}^{\prime }[0,T]& \equiv C_0^{\prime }[0,T]\hfill \\ & =\left\{w\in C_0[0,T]:w\left(t\right)={\int }_0^{t}z\left(s\right)ds\mathrm{for}\mathrm{some}z\in L_2[0,T]\right\}.\hfill \end{array}$$

In this case, we thus have the following translation theorems for the conditional analytic Feynman integral on the one parameter Wiener space $(C_0[0,T],\mathcal{W},m_w)$.

Corollary 7.1.

Setting $a\left(t\right)\equiv 0$ and $b\left(t\right)=t$ yields the formulas: for a scale-invariant measurable functional F on $C_0[0,T]$,

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi }\right)\hfill \\ & \stackrel{*}{=}exp\left\{{\displaystyle \frac{iq}{2}}{\parallel x_0\parallel }_{C_0^{\prime }}^2+{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{(x_0,g_j)}_{C_0^{\prime }}^2+iq\sum _{j=1}^n{\xi }_j{(x_0,g_j)}_{C_0^{\prime }}\right\}\hfill \\ & \times E_x^{{\mathrm{anf}}_q}\left(F\left(x\right)exp\{-iq{(x_0,x)}^{\sim }\}|X\left(x\right)=\overrightarrow{\xi }+{(x_0,\overrightarrow{g})}_{C_0^{\prime }}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}\left(F(x+x_0)|x\left(t_j\right)={\eta }_j,j=1,\dots ,n\right)\hfill \\ & \stackrel{*}{=}exp\{{\displaystyle \frac{iq}{2}}{\int }_0^T{\left(x_0^{\prime }\left(t\right)\right)}^{2}dt+{\displaystyle \frac{iq}{2}}\sum _{j=1}^n{\displaystyle \frac{{(x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}^2}{t_j-t_{j-1}}}\hfill \\ & +iq\sum _{j=1}^n{\displaystyle \frac{({\eta }_j-{\eta }_{j-1})(x_0\left(t_j\right)-x_0\left(t_{j-1}\right))}{t_j-t_{j-1}}}\}\hfill \\ & \times E_x^{{\mathrm{anf}}_q}\left(F\left(x\right)exp\left\{{(x_0,x)}^{\sim }\right\}|x\left(t_j\right)={\eta }_j+x_0\left(t_j\right),j=1,\dots ,n\right)\hfill \end{array}$$

where ${\eta }_0=0$.

For further work on the one parameter Wiener space $(C_0[0,T],\mathcal{W},m_w)$, see [36].