#### Abstract

Let denote a generalized Wiener space, the space of real-valued continuous functions on the interval and define a stochastic process by for and , where with a.e. and is continuous on . Let random vectors and be given by and , where is a partition of . In this paper we derive a translation theorem for a generalized Wiener integral and then prove that is a generalized Brownian motion process with drift . Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on with the drift and the conditioning functions and . As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on .

#### 1. Introduction

Let denote the Wiener space, the space of real-valued continuous functions on with . On the space, Yeh [1] introduced an inversion formula that a conditional expectation can be found by a Fourier-transform. But Yeh’s inversion formula is very complicated in its application when the conditioning function is vector-valued. In [2], Park and Skoug derived a simple formula for conditional Wiener integrals on with a vector-valued conditioning function given by , where is a partition of the interval . In their simple formula, they expressed the conditional Wiener integral directly in terms of an ordinary Wiener integral. Using the simple formula in [2], Chang and Skoug [3] investigated the effect that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them.

On the other hand, let denote the space of real-valued continuous functions on the interval . Im and Ryu [4] introduced a probability measure on , where is a probability measure on the Borel class of . When , the Dirac measure concentrated at , is exactly the Wiener measure on . On the space , the author [5, 6] derived two simple formulas for the conditional Wiener -integral of functions on with the vector-valued conditioning functions and given by and which generalize the Park and Skoug’s formula in [2]. Using these formulas with the conditioning functions and , he evaluated the conditional Wiener -integral of function of the form for any positive integer .

Let with a.e. on , and let be a continuous function on . Define a stochastic process by for and . Let and be given by In this paper, we derive a translation theorem for a generalized Wiener -integral, and then prove that is a generalized Brownian motion process with drift and variance for . Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on with the drift and the conditioning functions and . As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of functions of the forms for , where and is a complex Borel measure on .

#### 2. A Generalized Brownian Motion Process with Drift

In this section, we introduce a generalized Brownian motion process with drift which generalizes the generalized Wiener space as given in [7].

Let and denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let be the analogue of Wiener space associated with a probability measure on the Borel class of , where denotes the Borel class of [4]. Let , and let be a measure on such that for . For in and in , let denote the Paley-Wiener-Zygmund integral of according to [4]. Let with a.e. on , and let be a continuous function on . Define stochastic processes by and for and . Let for .

By (3.1) in [4, Theorem 3.1], we have the following translation theorem for a generalized Wiener integral on the analogue of Wiener space.

Theorem 1. *Let be of bounded variation on , let for , and let be a measurable function on . Then, is also measurable function of on and
**
where means that if one side exists then both sides exist, and they are equal.*

Letting , we have the following corollary by Theorem 1.

Corollary 2. *Let be of bounded variation on and a measurable function on . Then, for is also measurable on and
*

Letting in Theorem 1 and Corollary 2, we have the following corollary.

Corollary 3. *Under the assumptions as given in Theorem 1 and Corollary 2**
In particular for *

Using the same method as used in the proof of Corollary 3.3 in [4], we have the following corollary.

Corollary 4. *Under the assumptions as given in Theorem 1, one has for **
In particular for *

Letting in the equations of Corollary 4, we have the following corollary by Fourier transforms on .

Corollary 5. *Under the assumptions as given in Theorem 1, let . Then, is normally distributed with mean zero and variance on if . Moreover, for is normally distributed with mean and variance on . In particular, is normally distributed on with mean and variance .*

Using the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem.

Theorem 6. *Let be an orthonormal subset of such that each is of bounded variation. For , let . Then, are independent, and each has the standard normal distribution. Moreover if, is Borel measurable, then
*

Theorem 7. *Let be of bounded variation on , and let . Then, the random vector has a joint density function with respect to given by
**
where , , and .*

*Proof. *For , let , where denotes the indicator function on . Let . Then, are orthonormal, and . Let be a Borel subset of . By Theorem 6,
For , let . Then, by the change of variable theorem,
which completes the proof.

*Remark 8. *The condition in Theorem 7 does not mean that the mean function is continuous on with . In fact, it is only needed to express formally the exponential function in .

By Theorem 7 and the change of variable theorem, we have the following corollary.

Corollary 9. * is a Gaussian process with respect to , and its covariance function is given by
**
so that
*

Using Fourier transforms on , we have the following corollary by Theorem 7.

Corollary 10. *Let . Then, is normally distributed on with mean and variance .*

By Corollary 10, we can prove the following corollary.

Corollary 11. *Let . Then and are independent on .*

Theorem 12. * is a generalized Brownian motion process on determined by the mean function and variance function .*

*Remark 13. *(1) A generalized Brownian motion process determined by the mean function and variance function is a Gaussian process whose mean and covariance functions are given by and , respectively, for [1].

(2) In [8], the author proved that is a generalized Brownian motion process using Theorem 6. Comparing with [8], we proved the results in this paper using only the translation theorem [4, Theorem 3.1] which is another approach to prove being a generalized Brownian motion process.

#### 3. Simple Formulas for Generalized Conditional Wiener Integrals

In this section, we derive two simple formulas for generalized conditional Wiener integrals on . For this purpose, we start with this section defining a conditional -integral.

Let be integrable on and a random vector on assuming that the value space of is a normed space equipped with the Borel -algebra. Then, we have the conditional expectation of given from a well-known probability theory. Furthermore, there exists a -integrable -valued function on the value space of such that for a.e. , where is the probability distribution of . The function is called the conditional -integral of given , and it is also denoted by . Let be a partition of . For , let , and , where . Define random vectors and by and for . For any in , define a polygonal function of by for , where and denote indicator functions. For and , define polygonal functions and of and by (15), where is replaced by and , respectively, for ( and formally). Moreover, define a polygonal function of by (15), where is replaced by . For , the symbol is understood as on .

By Theorems 2.4 and 2.9 in [8], we can easily prove the following theorem.

Theorem 14. *Let for some . Then, is normally distributed on with mean and variance .*

Theorem 15. *The process and are stochastically independent on .*

*Proof. *Let for some . Then, for ,
by Corollary 2.4 and Theorems 2.9 and 2.10 in [8]. By Corollary 5 and Theorem 14 we have
Since and are normally distributed on , they are independent. The proofs of remainder cases follow easily.

Theorem 16. *The processes , where , are stochastically independent on .*

*Proof. *Let and with . If or , we can prove easily the independence of and . Suppose that . By Theorems 3.1 and 2.11 in [8], we have
which completes the proof.

Applying the same method as used in the proof of Theorem 2 in [2, page 383] with Problem 4 of [9, page 216], we have the following theorem by Theorem 15.

Theorem 17. *Let be a function, and let be -integrable over the variable . Then, for a Borel subset of **
where the expectation is taken over and is the probability distribution of on . Moreover, for a.e. (hence for a.e. )
*

Using the same method as used in the proof of Theorem 2.5 in [6], we can prove the following theorem.

Theorem 18. *Let be a function and -integrable over the variable . Moreover, let be a probability distribution of on . Then, for a.e. (hence for a.e. )
**
where and the expectation is taken over .*

Note that the conditioning functions and describe the positions of paths at the times (the present time). contains the present position of the path for , while does not. Moreover if we let a.e., , and , the Dirac measure concentrated at , then we can obtain Theorems 1 and 2 in [10] by the translation theorem (Theorem 1). If , , , and , then we can obtain the space in [7] by Theorem 12. Furthermore, if is replaced by the generalized Brownian motion process on , and we let , then we can also obtain Theorem 3.4 in [7] by Theorem 17. If we let and , then we can obtain Theorem 3 in [11] by Theorem 17. If we let , then we can obtain Theorem 2.12 in [8] by Theorem 17. If we let and , then we can obtain Remark 2.2 in [3] by Theorem 17. Finally, if we let a.e., and , then we can obtain Theorem 2 in [2] by Theorem 17 which is among the first result expressing the conditional Wiener integrals of functions on as ordinary Wiener integrals.

*Remark 19. *Note that Theorems 17 and 18 are not generalizations of Theorem 2.9 in [5] and Theorem 2.5 in [6]. In Theorem 2.9 of [5] and Theorem 2.5 of [6], the conditioning functions have initial distributions , while and in Theorems 17 and 18 have no initial distributions.

#### 4. Evaluation Formulas Using the Simple Formulas

In this section, we derive evaluation formulas for the generalized conditional Wiener integrals of various functions which are of interest in Feynman integration theories and quantum mechanics. For a function let for .

Lemma 20. *Let for . Then, is -integrable and
**
where denotes the greatest integer function.*

*Proof. *By Corollary 5, the change of variable theorem and the binomial expansion,
where denote the gamma function, so that is integrable over . Furthermore, we have
which completes the proof.

Theorem 21. *Let the assumptions be as given in Lemma 20. Then, for a.e. (hence, a.e. )
**
where the expectation is taken over .*

*Proof. *For a.e. , we have by Theorems 14 and 17
by the same method as used in the proof of Lemma 20.

Theorem 22. *Let the assumptions be as given in Theorem 21, and for let
**
Then, for a.e. (hence a.e. )
**
where the expectation is taken over .*

*Proof. *For , let . By the binomial expansion, we have for
so that by Theorems 18 and 21, we have for a.e.
by the change of variable theorem. Using the same method as used in the proof of Lemma 20, we have the desired result.

From now on, we assume that every expectation is taken over unless otherwise specified. By Corollary 9, Theorems 14 and 17, and Theorem 3.4 in [8], we have the following theorem.

Theorem 23. *Let , and . For , let . *(1)*If , then for a.e. (hence, a.e. )
*(2)*If , then for a.e. (hence, a.e. )
*

Theorem 24. *Let the assumptions be as given in Theorem 23. *(1)*If , and , then for a.e. (hence, a.e. )
*(2)*If , and , then for a.e. (hence a.e. )
*(3)*If and , then for a.e. (hence a.e. )
*(4)*If and , then for a.e. (hence a.e. )
*(5)*If and , then for a.e. (hence a.e. )
*

*Proof. *For , let . (1) and (2) follow immediately from Theorem 23. If , then