Journal of Probability and Statistics
Volume 2017 (2017), Article ID 8510782, 13 pages
https://doi.org/10.1155/2017/8510782

Research Article

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Dong Hyun Cho

Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea

Correspondence should be addressed to Dong Hyun Cho; j94385@kyonggi.ac.kr

Received 29 July 2016; Accepted 24 November 2016; Published 28 February 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Dong Hyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of . We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

1. Introduction

Let denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [1]. The conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products on are introduced by the author [2] using a conditioning function which contains present positions of the paths. According to the author’s paper, he investigated the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur among them. In particular, he derived several change of scale formulas for the conditional transforms and the conditional convolution products, which simplify evaluating the conditional expectations, because the probability measure used on cannot be scale-invariant [3, 4].

Let be absolutely continuous on and let be of bounded variation with a.e. on . Define a stochastic process by

for and , where denotes the Paley-Wiener-Zygmund stochastic integral [1]. For a partition of , define random vectors and by

With the conditioning function which contains the present position of the path , the author [5] evaluated the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by

where is a complex Borel measure of bounded variation on , with , and is an orthonormal subset of . He then investigated relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3).

In this paper, using a simple formula for conditional expectations over continuous paths [6], we evaluate conditional expectations of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. We then investigate their relationships. Particularly, we prove that the conditional transform of the convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish change of scale formulas for the conditional expectations. In these evaluation formulas and change of scale formulas we use multivariate normal distributions so that the conditioning function does not contain the present positions of the paths. In fact, with the conditioning function which does not contain the present position of the path , we evaluate conditional expectations, that is, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3). We then show that the -analytic conditional Fourier-Feynman transform of the conditional convolution product for the functions and which are described in (3) can be expressed by the formula

for a nonzero real , almost surely , and almost surely , where and is the probability distribution of on the Borel class of . Compared with the results in [5, 7], the conditioning function in this paper does not contain the present position of and the effects of drift depend on the polygonal function of so that we can generalize the theorems in [5, 7] and the results of this paper do not depend on a particular choice of the initial distribution of the paths.

2. Preliminaries on an Analogue of Wiener Space

We begin this section with introducing a concrete form of an analogue of Wiener space which is our underlying space.

For a positive real let denote the space of real-valued continuous functions on the time interval with the supremum norm. For with let be the function given by . For   () in the Borel class of , the subset of is called an interval and let be the set of all such intervals. For a probability measure on , let

where for . , the Borel -algebra of , coincides with the smallest -algebra generated by and there exists a unique probability measure on such that for all . This measure is called an analogue of Wiener measure associated with the probability measure [1].

Let and denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let be integrable and let be a random vector on assuming that the value space of is a normed space with the Borel -algebra. Then we have the conditional expectation of given from a well-known probability theory [8]. Furthermore, there exists a integrable complex-valued function on the value space of such that

where is the probability distribution of . The function is called the conditional -integral of given and it is also denoted by .

For an extended real number with , suppose that and are related by (possibly if ). Let . For let be a measurable function on such that

Then we write

Let be a partition of , where is a fixed nonnegative integer. Let be of bounded variation with a.e. on . Let be absolutely continuous on and define stochastic processes by

for and , where denotes an indicator function. Define a random vector by

for . For , let and for any function on , define a polygonal function of by

for . For , define a polygonal function of by (11), where is replaced by . If , is interpreted as with . For and let

For , , , and any nonsingular positive matrix on , let

where denotes the dot product on . Let be the identity matrix on . For a measurable function , let

let , and let for . Suppose that exists, where the expectation is taken over the first variable. By Theorem in [2] and Theorem in [6]

for a.e. , where , is the probability distribution of on and the expectation is taken over the variable . Let be the right-hand side of (15). If, for a.e. and a.e. , has an analytic extension on , then it is called a generalized analytic conditional Fourier-Wiener transform of given with the parameter and is denoted by

Moreover if has a limit as approaches through , then it is called a generalized -analytic conditional Fourier-Feynman transform of given with the parameter and is denoted by

For , define a generalized -analytic conditional Fourier-Feynman transform of given by the formula

Let be defined on . For and , redefine and suppose that exists. By Theorem in [2] and Theorem in [6]

Let be the right-hand side of (19). If has an analytic extension on , then it is called a generalized conditional convolution product of and given with the parameter and denoted by

Moreover if has a limit as approaches through , then it is called a generalized conditional convolution product of and given with the parameter and denoted by

For , let

let be the subspace of generated by , and let be the orthogonal complement of . Let be the orthogonal projection given by

and let be the orthogonal projection. For , , and any nonsingular positive matrix on , let

The following lemmas are useful to prove the results in the next sections [5].

Lemma 1. Let . Then for a.e.

where is the multiplication operator defined by

Lemma 2. Let and . Then

where .

Lemma 3. Let be a subset of such that is an independent set. Then the random vector has the multivariate normal distribution with mean vector and covariance matrix . Moreover, for any Borel measurable function one has

where means that if either side exists, then both sides exist and they are equal.

Remark 4. Because the Borel sets of are always scale-invariant measurable and we use the Borel class of on which is defined [1], the scale-invariant measurability is not essential in (7).

3. Conditional Fourier-Wiener and Fourier-Feynman Transforms

Let , let be any fixed positive integer, and let be an orthonormal subset of such that both and are independent sets. Let be the space of cylinder functions having the form

for a.e. , where and . Without loss of generality we can take to be Borel measurable.

Theorem 5. Let and let be given by (29). Then for

for a.e. and a.e. , where , , and

Moreover .

Proof. For let , where . For and a.e. we have by Lemmas 1 and 2

where and are given by (12) and (13), respectively. Using the same method as used in the proof of Theorem in [7]

We note that if , then by the change of variable theorem

and for we have by Schwarz’s inequality

Now, by Morera’s theorem with aids of Hölder’s inequality and the dominated convergence theorem, we have (30) for . Since and by (35), the final result follows by the change of variable theorem and Young’s inequality [9].

Corollary 6. Let and let be given by (29).

(1) If , then one has for , and

 where for

(2) If , then one has

 where for

 and for

(3) If and , then one has

 where for

 and for

By the dominated convergence theorem and Theorem 5 we have the following theorem.

Theorem 7. Let be given by (29). Then for a nonzero real , a.e. , and a.e. , exists and it is given by the right-hand side of (30), where is replaced by . Furthermore .

Using Lemma in [10] we can prove the following theorem.

Theorem 8. Let be given by (29) with , let be a nonzero real number, and let .

(1) Suppose that for . Then for a.e. and a.e. , is given by

 where is given by (31). Furthermore .(2) If , then for a.e. and a.e. , , exists and it is given by the right-hand sides of (38), where is replaced by . Furthermore .(3) If and , then for , is given by the right-hand sides of (41), where is replaced by .

Using the same method as used in the proof of Theorem in [7], we can prove the following theorem.

Theorem 9. Let be given by (29) with , and let .

(1)Suppose that for . For a.e. and a.e. , let . Then

 for , and for

 as approaches through .(2)Suppose that . Let . Then one has (45) if and has (46) if , where is replaced by .(3)Suppose that and . Let . Then one has (45) if and has (46) if , where is replaced by .

4. Relationships between Conditional Fourier-Feynman Transforms and Convolution Products

In this section we evaluate the conditional convolution products and investigate their relationships.

Now, we have the following two theorems by (35) and Theorems and in [7].

Theorem 10. Let , , and and be related by (29), respectively, where . Furthermore let and . Then for , a.e. , and a.e. , exists and is given by

Moreover one has if either or , if , and if .

Theorem 11. Let be a nonzero real number. Then for or and a.e. , one has the following:

(1)if , then ,(2)if , then ,(3)if and , then ,(4)if and , then ,(5)if and , then .

Theorem 12. Let and let . Then for , a.e. , and a.e. , one has

Proof. We note that is well-defined by Theorems 5 and 10. By those theorems as stated above we have for , a.e. , and a.e.

Let and . Then we have by the change of variable theorem and Lemma 2

which completes the proof.

We now have the following theorem from Theorems 5, 10, 11, and 12.

Theorem 13. Let be a nonzero real number. Then one has the following:

(1)if , then one has for a.e. and a.e.

(2)if and , then

5. Evaluation Formulas for the Functions in a Banach Algebra

Let be the function defined by

where is a complex-valued Borel measure of bounded variation over . For a.e. , let be given by

By Theorem in [11], we can prove the following theorem.

Theorem 14. Let and let be given by (54). Then for , a.e. , and a.e. , is given by

For a nonzero real , exists and it is given by the right-hand side of (55), where is replaced by . Furthermore and .

Theorem 15. Let and let be given by (54). Then one has that for a.e.

converges to and for a.e.

as approaches through .

Theorem 16. Let and and and be related by (53), respectively. Let and for a.e. . Then for , a.e. , and a.e. , exists and it is given by

For a nonzero real , is given by the above equation, where is replaced by . Furthermore, .

Theorem 17. Let be a nonzero real number and let . Furthermore let and be as given in Theorem 16. Then one has for a.e. and a.e.

Let be the class of all complex-valued Borel measures of bounded variation over and let be the space of all functions which for have the form

for a.e. . Note that is a Banach algebra [1].

Theorem 18. Let and let be given by (60). Then for , a.e. , and a.e. , is given by

For a nonzero real , is given by the right-hand side of the above equality, where is replaced by . Furthermore, .

Proof. For let , where . Then for , a.e. , and a.e. , we have by Lemmas 1 and 2

By Morera’s theorem and the dominated convergence theorem we have the theorem.

Theorem 19. Under the assumptions as given in Theorem 18, one has that for a.e.

converges to and for a.e.

as approaches through .

Proof. By Theorem 18, is well-defined so that we have for

Applying a similar method as used in the proof of Theorem in [11] with minor modifications we can obtain the remainder part of the proof.

Theorem 20. Let and and and be related by (60), respectively. Then for , a.e. , and a.e.

For a nonzero real , is given by the above equation, where is replaced by . Furthermore, .

Proof. For let , where . Then for , a.e. , and a.e. , we have by Lemmas 1 and 2

By Morera’s theorem and the dominated convergence theorem we have the theorem.

Theorem 21. Let be a nonzero real number and let . Furthermore let and be as given in Theorem 20. Then one has for a.e. and a.e.

Proof. By Theorems 18 and 20 we have for , a.e. , and a.e.

Let be given by the right-hand side of the above equality, where is replaced by . The existence of follows from the dominated convergence theorem. Now let and . Then we have by the dominated convergence theorem

as approaches through , which shows the existence of . The equality in the theorem follows from Theorem 18 with simple calculations.

Remark 22. Theorems 15 and 17 can be obtained from Theorems 9 and 12, respectively. They also can be proved directly with aids of Theorems 14 and 16.
Comparing Theorems 17 and 21 with Theorem 13, the results in Theorems 17 and 21 hold for .

6. Change of Scale Formulas for the Transforms and Convolutions

In this section, we derive two types of change of scale formulas for the conditional expectations as described in the previous sections.

For , and let

For a nonzero real , let be any sequence in converging to .

By simple calculations, we have the following two theorems.

Theorem 23. Let and let . Then for , a.e. , and a.e.

where and is given by (72). If , then

Theorem 24. Let and with . Then for , a.e. , and a.e. , is given by the right-hand side of (73) with replacing by . Moreover if , then is given by the right-hand side of (74) with replacing by .

By Theorems 14 and 23 and the dominated convergence theorem, we have the following theorem.

Theorem 25. Let and let be given by (54). Then, for , a.e. , and a.e. , and are given by the right-hand sides of (73) and (74), respectively, with replacing by .

By Theorems 16 and 24 and the dominated convergence theorem, we have the following theorem.

Theorem 26. Let the assumptions be as given in Theorem 16. Then, for , a.e. , and a.e. , and are given by the right-hand sides of (73) and (74), respectively, with replacing by .

For , and let

Theorem 27. Let and let be given by (29). Then for , a.e. , and a.e.

where is given by (75). If , then

Proof. For , a.e. , and a.e. , we have by Lemma 3

By the analytic continuation, the dominated convergence theorem, and Theorem 5, we have the theorem.

Theorem 28. Let and with . Then for , a.e. , and a.e. , is given by the right-hand side of (76) with replacing by . Moreover if , then is given by the right-hand side of (77) with replacing by .

By Theorems 14, 16, and 27 and the dominated convergence theorem, we have the following two theorems.

Theorem 29. Let and let be given by (54). Then, for , a.e. , and a.e. , and are given by the right-hand sides of (76) and (77), respectively, with replacing by .

Theorem 30. Let the assumptions be as given in Theorem 16. Then, for , a.e. , and a.e. , and are given by the right-hand sides of (76) and (77), respectively, with replacing by .

Let be a complete orthonormal basis of . For , and let

We now have the following relationships among the conditional Fourier-Feynman transform, the conditional convolution products, and the generalized Wiener integrals of the functions in . Their proofs are similar to the proofs of the results in [12] with additional calculations.

Theorem 31. Let and let be given by (60). Then for , a.e. , and a.e.

where for . Moreover, is given by the right-hand side of the above equality with replacing by .

Theorem 32. Let , , and , be related by (60), respectively. Then for , a.e. , and a.e.

Moreover, is given by the right-hand side of the above equality with replacing by .

Remark 33. Letting in the theorems of this section, where , we can obtain the change of scales for and ; that is, the scale in the functions , , and is moved to , , , and in evaluating the conditional expectations as above.

Remark 34. An orthonormal subset of such that both and are independent sets exists [12].
Let and be the orthonormal sets obtained from and , respectively, by the Gram-Schmidt orthonormalization process. For , let

be the linear combinations. Let and be the coefficient matrices of the combinations. Then and can be replaced by and , respectively, in each expression of the theorems.
It does not mean that or in (2). They only satisfy the following equations:

Remark 35. If , then we can obtain the change of scale formulas in [2] with .
If and , then we can obtain the results in [12] with the cylinder functions.
The results of this paper are independent of a particular choice of the initial distribution .

Competing Interests

The author declares that they have no competing interests.

Acknowledgments

This work was supported by Kyonggi University Research Grant 2015.

References

M. K. Im and K. S. Ryu, “An analogue of Wiener measure and its applications,” Journal of the Korean Mathematical Society, vol. 39, no. 5, pp. 801–819, 2002.

D. H. Cho, “Integral transforms on a function space with change of scales using multivariate normal distributions,” Journal of Function Spaces, vol. 2016, Article ID 9235960, 9 pages, 2016.

R. H. Cameron, “The translation pathology of Wiener space,” Duke Mathematical Journal, vol. 21, pp. 623–627, 1954.

R. H. Cameron and W. T. Martin, “The behavior of measure and measurability under change of scale in Wiener space,” Bulletin of the American Mathematical Society, vol. 53, pp. 130–137, 1947.

D. H. Cho, “A conditional Fourier-Feynman transform and conditional convolution product with change of scales on a function space I,” Bulletin of the Korean Mathematical Society, In press.

D. H. Cho, “Analogues of conditional Wiener integrals with drift and initial distribution on a function space,” Abstract and Applied Analysis, vol. 2014, Article ID 916423, 12 pages, 2014.

D. H. Cho, B. J. Kim, and I. Yoo, “Analogues of conditional Wiener integrals and their change of scale transformations on a function space,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 421–438, 2009.

R. G. Laha and V. K. Rohatgi, Probability Theory, John Wiley & Sons, New York, NY, USA, 1979.

G. B. Folland, Real analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, 1984.

G. W. Johnson and D. L. Skoug, “The Cameron-Storvick function space integral: an theory,” Nagoya Mathematical Journal, vol. 60, pp. 93–137, 1976.

D. H. Cho, S. B. Park, and M. Go, “A present position-dependent conditional Fourier-Feynman Transform and convolution product over continuous paths,” International Journal of Mathematical Analysis, vol. 9, no. 48, pp. 2387–2406, 2015.

D. H. Cho and I. Yoo, “Change of scale formulas for a generalized conditional Wiener integral,” Bulletin of the Korean Mathematical Society, vol. 53, no. 5, pp. 1531–1548, 2016.