Translation Theorem for Conditional Function Space Integrals and Applications

Abstract

The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–Martin space elements in abstract Wiener space. Translation theorems for analytic Feynman integrals have been established in many research articles. This study aims to present a translation theorem for the conditional function space integral of functionals on the generalized Wiener space $C_{a,b}[0,T]$ induced via a generalized Brownian motion process determined using continuous functions $a\left(t\right)$ and $b\left(t\right)$. As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on $C_{a,b}[0,T]$. We then provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the conditional translation theorem on $C_{a,b}[0,T]$ can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research on the Wiener space $C_0[0,T]$ because the generalized Brownian motion process used in this study is neither stationary in time nor centered. In this study, the stochastic process used is subject to a drift function.

1. Introduction

In [1], Nobert Wiener introduced the concept of “integration in function space”. At present, 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) associated with continuous functions $a\left(t\right)$ and $b\left(t\right)$. This theory for the function space $C_{a,b}[0,T]$ was developed further by Chang and Chung in [3,4] and was used extensively in [5,6,7,8,9] with various related results. The authors in [5] derived simple formulas for conditional function space integrals of functionals on the generalized Wiener space $C_{a,b}[0,T]$ and its applications. In [6], the authors studied relationships among the integral transforms, convolution products, first variation, and inverse transforms of functionals on $C_{a,b}[0,T]$. Chang [7] proposed the conditional generalized Fourier–Feynman transform of functionals in Fresnel-type classes. In [8,9], the authors investigated generalized integral transforms of functionals on $C_{a,b}[0,T]$, and extended generalized Fourier–Feynman transforms with a first variation (the Gâteaux derivative) on the function space $C_{a,b}[0,T]$.

Let $(\mathrm{\Omega },\mathcal{F},\mathrm{P})$ be a probability space. The stochastic process Y on $(\mathrm{\Omega },\mathcal{F},\mathrm{P})$ and an interval $[0,T]\subset \mathbb{R}$ is called a GBMP, provided the following apply:

(i)

$Y(0,\omega )=0$ $\mathrm{P}$-a.e.;

(ii)

For $0=t_0<t_1<\cdots <t_n\le T$, the random vector $(Y(t_1,\omega ),\dots ,Y(t_n,\omega ))$ has a normal distribution with the following density function:

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

where $(u_0,u_1,\dots ,u_n)\in \left\{0\right\}\times {\mathbb{R}}^n$, $a\left(t\right)$, and $b\left(t\right)$ are suitable continuous real-valued functions on $[0,T]$.

We note that the GBMP Y determined by the functions $a\left(t\right)$ and $b\left(t\right)$ is Gaussian with mean $a\left(t\right)$ and covariance $r(s,t)=min\left\{b\right(s),b(t\left)\right\}$. Let $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$ be the complete generalized Wiener space, where $C_{a,b}[0,T]$ is the continuous sample paths of the GBMP Y. In the case of $a\left(t\right)=0$ and $b\left(t\right)=t$, the function space $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$ reduces to the classical Wiener space $(C_0[0,T],\mathcal{W},m_w)$. In Section 2, we will provide a more detailed construction of the function space $C_{a,b}[0,T]$.

Many physical problems can be represented 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 following 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 from [10,11] 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 to 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 [3,4].

On the other hand, the Cameron–Martin translation theorem [12] and several analogies [13,14] describe how and when the Wiener measure $m_w$ changes under translation with specific elements in the Wiener space $C_0[0,T]$. This translation theorem was developed for the Yeh–Wiener [15], abstract Wiener [16], conditional Wiener [17,18], analytic Feynman [19,20], and conditional analytic Feynman [21] integrals. Furthermore, a translation theorem on the generalized Wiener space $C_{a,b}[0,T]$ was first established by Chang and Chung in [3] and improved in [9].

In [3,4], Chang and Chung used the n-dimensional conditioning function

$$X_{\overrightarrow{t}}\left(x\right)=(x\left(t_1\right),\dots ,x\left(t_n\right)),x\in C_{a,b}[0,T],0=t_0<t_1<\cdots <t_n=T$$

to study a generalized heat equation and a translation theorem for the conditional function space integral of functionals on the function space $C_{a,b}[0,T]$.

This study aims to present a translation theorem for conditional function space integrals on the function space $C_{a,b}[0,T]$. To achieve this, we use the conditioning function with the following form:

$$X\left(x\right)=({(g_1,x)}^{\sim },\dots ,{(g_n,x)}^{\sim }),x\in C_{a,b}[0,T],$$

where $\{g_1,\dots ,g_n\}$ is an orthonormal subset of the Cameron–Martin space in $C_{a,b}[0,T]$, and ${(g,x)}^{\sim }$ denotes the Paley–Wiener–Zygmund (henceforth, PWZ) stochastic integral. We also derive a translation theorem for conditional generalized analytic Feynman integrals of functionals F on $C_{a,b}[0,T]$. To establish the translation theorem for the conditional generalized analytic Feynman integral on the function space, we assume that the conditional generalized analytic Feynman integral that appears in the theorem exists because the drift term $a\left(t\right)$ of the GBMP makes establishing the existence of the conditional Feynman integral very difficult. The function $a\left(t\right)$ defining the GBMP is interpreted as the “drift” of the GBMP. Thus, in Section 6, we provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the translation theorems can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research because the generalized Wiener process used in this study is nonstationary in time and subject to drift $a\left(t\right)$, which can be used to explain the position of the Ornstein–Uhlenbeck process in an external force field [22]. However, by 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 $C_0[0,T]$; thus, the expected results on $C_0[0,T]$ are immediate corollaries of our results.

The generalized Wiener space $C_{a,b}[0,T]$ is fundamentally different from the classical Wiener space $C_0[0,T]$ as it is induced by a GBMP that incorporates a non-trivial drift function. The presence of this drift means that the underlying process is neither stationary nor centered and introduces significant mathematical complexity that prior work on $C_0[0,T]$ does not address.

The authors in [23] introduced the octonion linear canonical transform to expand transformation theory to higher algebraic structures. In [24,25], the authors proposed a piecewise scheme for Chebyshev finite-difference time-domain methods in computational mathematics and worked on homogeneity pursuit in functional-coefficient quantile regression models with censored panel data. The researcher in [26] applied neural ordinary differential equations for robust parameter estimation with physical priors. The authors in [27] developed quantum stochastic differential equations related to annihilation and creation operators. In [28,29], the researcher focused on quantum stochastic processes, including boson field implementations, Girsanov transforms, and covariant semigroups. The researchers in [30,31] developed white noise differential equations and the quantum Lévy Laplacian with heat equation connections. Our findings could be useful in the field of infinite dimensional analysis and further advance generalized Fourier transform [23], difference equations [24], linear data models [25], ordinary differential equations [26], stochastic differential equations [27], quantum field theory [28,29], white noise differential equations [30,31], etc. If we construct rather rigorous mathematical interpretations associated with elaborate stochastic processes such as the GBMP, then various mathematical theories and related results based on the research [23,24,25,26,27,28,29,30,31] will be more relevant to real-world problems.

The theoretical application in this article is also used to derive a corresponding translation theorem for the conditional generalized analytic Feynman integral, which is non-trivial and provides a toolkit for performing rigorous calculations in functional integration within this generalized setting.

2. Definitions and Preliminaries

2.1. Backgrounds

Two functions, $a(·)$ and $b(·)$, were given in Section 1. 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, considering Theorems 12.2 and 14.2 from [32], there exists a probability space $(\mathrm{\Omega },\mathcal{F},\mathrm{P})$ and an additive process Y on $(\mathrm{\Omega },\mathcal{F},\mathrm{P})$ and the interval $[0,T]$ where $\mathrm{P}$ is a Gaussian measure such that the probability distribution of $Y(t,·)-Y(s,·)$ with $s<t$ 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 $(\mathrm{\Omega },\mathcal{F},\mathrm{P})$ and $[0,T]$ is called a GBMP. The GBMP Y is determined using $a(·)$, 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 [32], pp. 18–21, 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 $C_{a,b}[0,T]$. It is worth noting that, by choosing $a\left(t\right)=0$ and $b\left(t\right)=t$ on $[0,T]$, one can see that the GBMP reduces to a standard Brownian motion.

In this study, we assume the following:

(i)

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

$${\int }_0^T|a^{\prime }\left(t\right){|}^{2}d|a|\left(t\right)<+\infty$$

(1)

where $\left|a\right|\left(t\right)$ denotes the total variation function of $a\left(t\right)$ on $[0,T]$;

(ii)

The derivative $a^{\prime }\left(t\right)$ of $a\left(t\right)$ is of class $L_2[0,T]$;

(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 $I_{t_1,\dots ,t_n,U}$ of 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 subdivision $0=t_0<t_1<\cdots <t_n\le T$ of $[0,T]$ and a Borel set $U\subset {\mathbb{R}}^n$,

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

where $u_0=0$.

Let $C_{a,b}^{\prime }[0,T]$ be the linear space (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, it is worth noting 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 $a\left(t\right)$ and $b\left(t\right)$, one can see that the functions $a\left(t\right)$ and $b\left(t\right)$ are elements of the Hilbert space $C_{a,b}^{\prime }[0,T]$. It is worth noting that the function $a\left(t\right)=t^{2/3}$, $0\le t\le T$ does not satisfy requirement (1), even though its derivative is an element of $L_2[0,T]$.

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 PWZ stochastic integral ${(w,x)}^{\sim }$ as

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

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, ${(g_j,x)}^{\sim }$, 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)

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

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

A subset S of $C_{a,b}[0,T]$ is called a scale-invariant measurable set provided $\rho S$ is $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable for all $\rho >0$, and a scale-invariant measurable set N is called a scale-invariant null set 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 invariance almost everywhere (s-a.e.). A functional F is said to be scale-invariant measurable provided that 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 $\rho >0$.

It was pointed out in [33] that the concept of scale-invariant measurability, rather than Borel or Wiener measurability, is accurate for the analytic Feynman integration theory. Hence, throughout this study, we always assume that each functional $F:C_{a,b}[0,T]\to \mathbb{C}$ that we consider satisfies the following conditions:

$$F:C_{a,b}[0,T]\to \mathbb{C}\mathrm{is} \mathrm{scale-invariant} \mathrm{measurable} \mathrm{and} \mathrm{s-a.e.} \mathrm{defined}$$

(3)

and

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

(4)

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

Definition 1.

Let a functional F on $C_{a,b}[0,T]$ satisfy conditions (3) and (4). If there exists a function $J^{\ast }\left(\lambda \right)$ analytic in ${\mathbb{C}}_{+}$ such that

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

for all $\lambda >0$, then $J^{\ast }\left(\lambda \right)$ 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

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

Let $q\ne 0$ be a real number and 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 write the following:

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

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.

Let $X:C_{a,b}[0,T]\to {\mathbb{R}}^n$ be a $\mathcal{W}\left(C_{a,b}[0,T]\right)$-measurable function with a probability distribution ${\mu }_X=\mu \circ X^{-1}$ that is absolutely continuous with respect to the 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 following equation: for all Borel sets B in ${\mathbb{R}}^n$

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

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

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

(5)

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

Definition 3.

Let functional $F:C_{a,b}[0,T]\to \mathbb{C}$ satisfy conditions (3) and (4) and let $X:C_{a,b}[0,T]\to {\mathbb{R}}^n$ be given by Equation (5). 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 $J^{\ast }(\lambda ;\overrightarrow{\eta })$ analytic in λ on ${\mathbb{C}}_{+}$ such that $J^{\ast }(\lambda ;\overrightarrow{\eta })=J(\lambda ;\overrightarrow{\eta })$ for all $\lambda >0$, then $J^{\ast }(\lambda ;\overrightarrow{\eta })$ 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. For $\lambda \in {\mathbb{C}}_{+}$, we write

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

if, for a 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.

We define $[·]:{\mathbb{R}}^n\to C_{a,b},[0,T]$ by $\left[\overrightarrow{\eta }\right]\equiv {\displaystyle \sum _{j=1}^n}{\eta }_j{g}_j$ for $\overrightarrow{\eta }=({\eta }_1,\cdots ,{\eta }_n)\in {\mathbb{R}}^n$, and we write $\left[x\right]\equiv \left[X\left(x\right)\right]={\displaystyle \sum _{j=1}^n}{(g_j,x)}^{\sim }{g}_j$ for $x\in C_{a,b}[0,T]$.

In [5], Chang, Choi, and Skoug derived a formula for expressing conditional function space integrals in terms of ordinary function space integrals. We provide a modified result from [5], Theorem 3.4, which plays an important role in this paper. The proof given in [5] 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 1.

Let X be given by Equation (5) and let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then,

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

(6)

Remark 1.

Equation (6) is indeed a very simple formula equating the conditional function space integral in terms of an ordinary function space integral.

Let F be a functional on $C_{a,b}[0,T]$ which satisfies conditions (3) and (4). Then, one can easily see from (6) that, for all $\lambda >0$,

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

for a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$. Thus, we have the following:

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

(7)

and

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

(8)

where, in (7) and (8), the existence of either side implies the existence of the other side and their equality.

3. Translation Theorems for Conditional Function Space Integrals

Due to their nature, Gaussian measures on function spaces do not have the property of translation invariance. However, as first shown by Cameron and Martin in [12], they enable the computation of the Radon–Nikodym derivatives of measures resulting from certain translations. In particular, there is a specific class of functions which, along with translation results, yields an equivalent Gaussian measure. In the case of ordinary Wiener space $(C_0[0,T],\mathcal{W},m_w)$, this collection of allowable translates coincides with the Sobolev space $H_0^2(0,T)$ of functions vanishing at 0 with square-integrable weak derivatives on $(0,T)$. For the generalized Wiener space $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$, similar but more complicated results hold.

We start this section with translation theorems on the function space $C_{a,b}[0,T]$. We then use this translation theorem to obtain conditional function space integration formulas.

Theorem 2

 ([3,9]). (Translation theorems for function space integral). Let X be given by (5) and let F be a μ-integrable functional on $C_{a,b}[0,T]$. Then, for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

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

and

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

(9)

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 1

 ([5]). 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.

Remark 2.

Lemma 1 makes an interesting observation about the process $x\left(t\right)-\left[x\right]\left(t\right)$. The process $\left[x\right]\left(t\right)$ has been widely used to approximate the generalized Brownian motion $x\left(t\right)$, while $x\left(t\right)-\left[x\right]\left(t\right)$ has been applied to a Brownian bridge process.

We are obliged to point out the facts that, for each $w\in C_{a,b}^{\prime }[0,T]$, the PWZ stochastic integral ${(w,x)}^{\sim }$ is a Gaussian random variable with mean ${(w,a)}_{C_{a,b}^{\prime }}$ and variance ${\parallel w\parallel }_{C_{a,b}^{\prime }}^2$ and that, 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. Thus, applying the change of variables theorem, the Fubini theorem, and the integration formula,

$${\int }_{\mathbb{R}}\exp \{-\alpha u^2+\beta u\}=\sqrt{{\displaystyle \frac{\pi }{\alpha }}}\exp \left\{{\displaystyle \frac{{\beta }^2}{4\alpha }}\right\}\alpha ,\beta \in \mathbb{C} \mathrm{with} \mathrm{Re}\alpha >0,$$

(10)

and we have the following lemma:

Lemma 2.

Given 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

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

(11)

In our next theorem, we establish a translation theorem for the conditional function space integral.

Theorem 3.

Let X and F be as in Theorem 2. Then, for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

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

(12)

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 (6), (9), and (2) and applying Lemma 1, it follows that

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

(13)

Using (6) and (11), Equation (13) can be rewritten as

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

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

as desired. □

Corollary 1.

Let X and F be as in Theorem 2. Then, it follows that, for any function $x_0$ in $C_{a,b}^{\prime }[0,T]$,

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

(14)

Proof. 

Let $G\left(x\right)=F\left(x\right)\exp \{-{(x_0,x)}^{\sim }\}$. Then, using (12) with F replaced by G, it follows that

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

From this, we obtain the following:

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

(15)

Replacing $\overrightarrow{\xi }$ with $\overrightarrow{\xi }-{(x_0,\overrightarrow{g})}_{C_{a,b}^{\prime }}$ in (15), we have Equation (14) as desired. □

Remark 3.

Using techniques similar to those used in the proof of Theorem 3, we can establish Equation (14) without using Equation (12). Also, Equation (12) can be established using Equation (14).

4. Conditional Function Space Integration Formulas

In [3], Chang and Chung extended the results of [34,35] 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

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

(16)

The conditioning function $X_{\overrightarrow{t}}$ given by (16) can be represented by the conditioning function X given by (5) with a specific choice of $g_j$s.

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

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

(17)

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

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

(18)

is a specific example of our conditioning function used in the previous section. 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 $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{R}}^n$,

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

(19)

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 $t\in [t_{j-1},t_j]$,

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

(20)

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

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

(21)

In view of Theorem 1 and with the above setting, we have the following corollary:

Corollary 2.

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 (18) yields the conditioning function $X_{\overrightarrow{t}}$ given by (16), and it follows the following conditional function space integration formula:

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

where ${\left[x\right]}_{\overrightarrow{t}}$ and ${\left[\overrightarrow{\eta }\right]}_{\overrightarrow{t}}$ are given by (21) and (20), respectively.

Lemma 3.

For each $j\in \{1,\dots ,n\}$, let $g_{\overrightarrow{t},j}$ be given by (17). Then, it follows that

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

(22)

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

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

(23)

Corollary 3

([3]). Let F be a μ-integrable functional on $C_{a,b}[0,T]$, and, given a partition $0=t_0<t_1<\cdots <t_n=T$ of $[0,T]$ 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 $C_{a,b}^{\prime }[0,T]$,

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

(24)

where ${\eta }_0=0$.

Proof. 

Using (16), (14), (19), (22), and (23), one can derive Equation (24). □

Corollary 4.

Let F be a μ-integrable functional on $C_{a,b}[0,T]$, and, given a partition $0=t_0<t_1<\cdots <t_n=T$ of $[0,T]$ 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 $C_{a,b}^{\prime }[0,T]$,

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

(25)

where ${\eta }_0=0$.

Proof. 

Using (16), (12), (19), (22), and (23), one can also derive Equation (25). □

Example 1.

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 $S^{\ast }$ of S is given by

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

Using an integration with the part formula, it follows that

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

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 (14) with $F\left(x\right)\equiv 1$ on $C_{a,b}[0,T]$, we obtain the following:

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

(26)

Using Equation (26), we immediately obtain the conditional function space integration formula

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

(27)

In particular, using (27) with $x_0$ replaced by $S^{\ast }b$, and integration with the part formula, we obtain the following:

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

where ${\eta }_0=0$.

The functional $\exp \left\{-{\int }_0^{T}x\left(t\right)db\left(t\right)\right\}$ discussed in Example 1 arises naturally in quantum mechanics.

As mentioned in Section 1 above, the formulas with the one-dimensional conditioning function $X_T\left(x\right)=x\left(T\right)$ is more relevant in heat and Schrödinger equation theories 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 and 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 2.

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 (24) reduces the following formula for $\eta \in \mathbb{R}$:

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

(28)

Replacing η with $\eta +x_0\left(T\right)$ in Equation (28), 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

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

(29)

Example 3.

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 (25) reduces the formula

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

(30)

Replacing η with $\eta -x_0\left(T\right)$ in Equation (30), 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

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

(31)

One can easily see that Equation (31) with $x_0$ replaced with $-x_0$ coincides with Equation (29).

5. Conditional Generalized Analytic Feynman Integrals

In this section, we will extend the results for the conditional function space integrals obtained in the previous section to the conditional generalized analytic Feynman integral of functionals F on $C_{a,b}[0,T]$. For related work involving the conditional analytic Feynman integral on classical and abstract Wiener spaces, interested readers are referred to [18,21,34,36].

Lemma 4.

Let X be given by (5) and let F be a $\mathbb{C}$-valued functional on $C_{a,b}[0,T]$, which satisfies conditions (3) and (4) above. Then, for all $x_0\in C_{a,b},[0,T]$ and any $\rho >0$, it follows that

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

(32)

Proof. 

Let $G\left(x\right)=F\left(\rho x\right)$. Then, $G(x+x_0)=F(\rho x+\rho x_0)$. Hence, using Equation (12) with F replaced by G, we have

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

as desired. □

Theorem 4.

Let X and F be as in Lemma 4. Assume that, given a non-zero real q, the conditional generalized analytic Feynman integral $E_x^{{\mathrm{anf}}_q}(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi })$ of F exists. Then, it follows that, for any $x_0\in C_{a,b}^{\prime }[0,T]$,

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

(33)

where $\stackrel{\ast }{=}$ means that, if either side of Equation (33) exists, both sides 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 (32) with $x_0$ replaced with $y_0$, it follows that

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

(34)

Now, let $\rho ={\lambda }^{-1/2}$. Then, Equation (34) becomes

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

(35)

Since $\rho >0$ is arbitrary, we see that Equation (35) holds for all $\lambda >0$. We now use Definition 3 to obtain the following conclusion:

$$\begin{array}{cc}& E_x^{{\mathrm{anf}}_q}(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi })=\underset{\begin{array}{c}\lambda \to -iq\\ \lambda \in {\mathbb{C}}_{+}\end{array}}{lim}{E}_x(F({\lambda }^{-1/2}x+x_0)|X\left({\lambda }^{-1/2}x\right)=\overrightarrow{\xi })\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(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 }})\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(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 }})\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(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 }})\hfill \end{array}$$

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

as desired. □

Corollary 5.

Letting $F\equiv 1$ and replacing $\overrightarrow{\xi }$ with $\overrightarrow{\xi }-{(x_0,\overrightarrow{g})}_{C_{a,b}}^{\prime }$ in Equation (33), the conditional generalized analytic Feynman integration formula is as follows:

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

6. Explicit Examples

Very important classes of functionals in Feynman integration theories and quantum mechanics are functionals on $C_{a,b}[0,T]$ of the form

$$\begin{array}{cc}& \exp \left\{{\int }_0^T\theta (s,x\left(s\right))db\left(s\right)\right\},\hfill \\ & \exp \left\{{\int }_0^T\theta (s,x\left(s\right))db\left(s\right)\right\}\psi \left(x\left(T\right)\right),\mathrm{and}\hfill \\ & \exp \left\{\alpha {\int }_0^{T}x\left(s\right)db\left(s\right)\right\}\hfill \end{array}$$

for some $\alpha \in \mathbb{C}$ and appropriate functions $\theta$ and $\psi$. Such functionals were examined in [3,6,7,20,37,38,39,40,41]. For a more detailed study of these functionals, interested readers are referred to [42].

In this section, we show that the assumption (and hence the conclusion) of Theorem 4 is indeed satisfied by several large classes of functionals. We will very briefly discuss three such classes.

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

In Corollary 6 below, we will see that the translation Formula (33) 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 [38]. The Banach algebra $\mathcal{F}\left(C_{a,b}[0,T]\right)$ consists of functionals expressible in the form

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

(36)

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 Borel measures on $C_{a,b}^{\prime }[0,T]$. Further works involving the functionals in $\mathcal{F}\left(C_{a,b}[0,T]\right)$ and related topics include [7].

Corollary 6.

Let X be given by (5) and let $F\in \mathcal{F}\left(C_{a,b}[0,T]\right)$ be given by (36). Assume that

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

(37)

for some positive real number $q_0>0$. Then, the conditional generalized analytic Feynman integrals on both sides of (33) exist, so Equation (33) holds true for all real q with $|q|>q_0$.

We require condition (37) to ensure the existence of the conditional generalized analytic Feynman integral of functionals in the class $\mathcal{F}\left(C_{a,b}[0,T]\right)$. For a detailed discussion of condition (37), we refer the reader to [9].

For $F\in \mathcal{F}\left(C_{a,b}[0,T]\right)$, which satisfies condition (37), direct calculations, indeed, show that

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

With the Cauchy–Schwarz inequality, we easily obtain the following:

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

Thus, the assumption (and hence the conclusion) of Theorem 4 is satisfied.

Remark 4.

The Banach algebra $\mathcal{F}\left(C_{a,b}[0,T]\right)$ contains several interesting functionals that naturally arise in quantum mechanics. Let $\mathcal{M}\left(\mathbb{R}\right)$ be the class of $\mathbb{C}$-valued countably additive measures on $\mathcal{B}\left(\mathbb{R}\right)$, the Borel class of $\mathbb{R}$. For $\nu \in \mathcal{M}\left(\mathbb{R}\right)$, the Fourier transform $\widehat{\nu }$ of ν is a complex-valued function defined on $\mathbb{R}$ by the following formula:

$$\widehat{\nu }\left(u\right)={\int }_{\mathbb{R}}\exp \left\{iuv\right\}d\nu \left(v\right).$$

Let $\mathfrak{G}$ be the set of all complex-valued functions on $[0,T]\times \mathbb{R}$ of the form $\theta (s,u)={\widehat{\sigma }}_s\left(u\right)$, where $\{{\sigma }_s:0\le s\le T\}$ is a family from $\mathcal{M}\left(\mathbb{R}\right)$ satisfying the following two conditions:

(i) 

For every $E\in \mathcal{B}\left(\mathbb{R}\right)$, ${\sigma }_s\left(E\right)$ is Borel measurable in s;

(ii) 

${\int }_0^T\parallel {\sigma }_s\parallel db\left(s\right)<+\infty$.

Let $\theta \in \mathfrak{G}$ and let H be given by

$$H\left(x\right)=\exp \left\{{\int }_0^T\theta (t,x\left(t\right))dt\right\}$$

for $x\in C_{a,b}[0,T]$. The authors of [43] showed that the function $\theta (t,u)$ is Borel-measurable and that $\theta (t,x(t\left)\right)$, ${\int }_0^T\theta (t,x\left(t\right))dt$, and $H\left(x\right)$ are elements of $\mathcal{F}\left(C_{a,b}[0,T]\right)$. These facts are relevant to quantum mechanics where exponential functions play a prominent role.

6.2. Bounded Cylinder Functionals

Next, we want to briefly discuss another class of functionals to which our general translation theorem can be 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 following formula:

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

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 $C_{a,b}^{\prime }[0,T]$, the functional $F:C_{a,b}[0,T]\to \mathbb{C}$ is defined by

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

(38)

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

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 $C_{a,b}[0,T]$ with the form (38). Note that $F\in {\widehat{\mathfrak{T}}}_{\mathcal{A}}$ implies that F is scale-invariant measurable on $C_{a,b}[0,T]$. As illustrated in Remark 4, the functionals in ${\widehat{\mathfrak{T}}}_{\mathcal{A}}$ arise naturally in quantum mechanics. For a more detailed study of functionals in $F\in {\widehat{\mathfrak{T}}}_{\mathcal{A}}$, interested readers are referred to [37].

Corollary 7.

Let X be given by (5) and let $F\in {\widehat{\mathfrak{T}}}_{\mathcal{A}}$ be given by (38). Given a positive real $q_0$, assume that the complex Borel measure ν corresponding to F given by (38) satisfies the following condition:

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

(39)

Then, the conditional generalized analytic Feynman integrals on both sides of (33) exist, and so Equation (33) holds true for all real q with $\left|q\right|>q_0$.

Proof. 

Using the fact that $\{e_1,\dots ,e_m\}$ is an orthogonal set of functions in $C_{a,b}[0,T]$, (7), the change of variables theorem, the Fubini theorem, and the integration Formula (10), we can guarantee the existence of the conditional generalized analytic Feynman integral $E_x^{{\mathrm{anf}}_q}(F(x+x_0)|X\left(x\right)=\overrightarrow{\xi })$ for functionals F given by (38) under condition (39). Thus, this corollary follows immediately from Theorem 4. □

6.3. Exponential-Type Functionals

An important class of functionals in Feynman integration theory is the class $\mathcal{E}$ of exponential-type functionals, which form a dense set in the Hilbert space $L_2\left(C_{a,b}[0,T]\right)$. Let $\mathcal{E}$ be the class of all functionals which have the form

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

(40)

for each $w\in C_{a,b}^{\prime }[0,T]$ and $x\in C_{a,b}[0,T]$. More precisely, since we identify functionals that coincide with 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 (40). The functionals given by Equation (40) and linear combinations (with complex coefficients) of the ${\mathrm{\Psi }}_w\left(x\right)$s are called (partially) exponential-type functionals on $C_{a,b}[0,T]$.

Remark 5.

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 pointwise multiplication with identity ${\mathrm{\Psi }}_0\equiv 1$. For more details, see [44]. The classes $\mathcal{E}$ and $\mathcal{E}\left(C_{a,b}[0,T]\right)$ are dense in $L_2\left(C_{a,b}[0,T]\right)$.

In view of Definition 3, we see that the conditional generalized analytic Feynman integral of each functional ${\mathrm{\Psi }}_w$ given by (40), $E_x^{{\mathrm{anf}}_q}({\mathrm{\Phi }}_w(x+x_0)|X\left(x\right)=\overrightarrow{\xi })$, exists and is given by

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

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

7. Corollary in Wiener Space ${\mathbf{C}}_{\mathbf{0}}[\mathbf{0},\mathbf{T}]$

From our assertions discussed in this paper, we also have the translation formulas for the conditional analytic Feynman integral defined on the Wiener space $C_0[0,T]$. As illustrated above, 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 $C_0[0,T]$. It also follows that

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

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

Corollary 8.

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

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

and

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

where ${\eta }_0=0$.

For further work on the classical Wiener space $(C_0[0,T],\mathcal{W},m_w)$, interested readers are referred to [21].

8. Conclusions

In the highly lauded monograph [2], Yeh illustrated the concept of the GBMP. This fundamental concept would have been very useful to us (as well as many mathematicians, physicists, and engineers) in establishing various results in research areas involving infinite dimensional analysis, generalized Fourier transform, difference equations, linear data models, ordinary differential equations, stochastic differential equations, quantum field theory, and white noise differential equations. We feel strongly that our results will prove to be very useful in future work for us as well as other researchers in the field. The framework and methods we used to obtain the results in this article are very dependent on the results from the book by Yeh [32] concerning Gaussian processes (i.e., generalized Brownian motion processes) that are nonstationary in time and have a drift function.