Integral Transforms on a Function Space with Change of Scales Using Multivariate Normal Distributions

Abstract

Using simple formulas for generalized conditional Wiener integrals on a function space which is an analogue of Wiener space, we evaluate two generalized analytic conditional Wiener integrals of a generalized cylinder function which is useful in Feynman integration theories and quantum mechanics. We then establish various integral transforms over continuous paths with change of scales for the generalized analytic conditional Wiener integrals. In these evaluation formulas and integral transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed in terms of the generalized cylinder function itself.

1. Introduction

Let denote the classical Wiener space, the space of continuous real-valued functions on with . It is well known that Wiener measure and Wiener measurability are not invariant under change of scale and under translation [1, 2]. As integral transforms, change of scale formulas for Wiener integrals of various functions was developed on the classical and abstract Wiener spaces [3–7]. Further change of scale formulas for conditional Wiener integrals was introduced by the author and his coauthors [8–10]. In fact change of scale formulas for conditional Wiener integrals was established on , on the infinite dimensional Wiener space, and on , an analogue of Wiener space [11] which is the space of real-valued continuous paths on . Some difficulties in studying the transforms for the conditional Wiener integrals of cylinder functions which play important roles in Feynman integration theories are that they cannot be expressed in terms of the original cylinder functions. To avoid these difficulties, modified cylinder functions expressed by a polygonal function with projection vectors satisfying orthonormality were used to derive the change of scale transforms [8–10]. In this paper we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed by the cylinder function itself and generalize the results in [8–10].

Let and let be of bounded variation with a.e. on . Define a stochastic process by for and , where the integral denotes a generalized Paley-Wiener-Zygmund stochastic integral. For a partition of define random vectors and by Using simple formulas for generalized conditional Wiener integrals on with the conditioning functions and [12] we evaluate generalized analytic conditional Wiener integrals of the following generalized cylinder function: where with and is an orthonormal subset of . We then establish various change of scale transforms for the generalized analytic conditional Wiener integrals of with and . In these evaluation formulas and scale transforms we use multivariate normal distributions so that Gram-Schmidt orthonormalization process of can be removed in the existing change of scale transforms for a suitable orthogonal projection on . In contrast with the existing change of scale transforms in [8–10], the transforms in this paper are expressed in terms of the cylinder function itself and generalize some results in those references.

2. An Analogue of Wiener Space and Preliminary Results

In this section we will introduce the analogue of Wiener space and the preliminary results which are needed in the following sections.

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 subset of is called an interval and let be the set of all such intervals. For a probability measure on let where for and , 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 [11]. Let be a complete orthonormal subset of such that each is of bounded variation. For and in let if the limit exists, where denotes the inner product over . is called the Paley-Wiener-Zygmund integral of associated with .

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. Furthermore there exists a -integrable complex 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 , where is a fixed nonnegative integer. Let be of bounded variation with a.e. on . 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 an orthogonal projection. Let and let it be absolutely continuous. Define stochastic processes by for and . Define random vectors and by for . For let and for any function on define the polygonal function of byfor , where and denote the indicator functions on the interval . For define the polygonal function of by (11) with replaced by for . If , is interpreted as on . For and letFor , , and any nonsingular positive matrix on letwhere denotes the dot product on . For a function let for . By Theorems  6 and 7 in [12], we have the following theorems.

Theorem 1. Let be a complex valued function on and let be integrable over . Then for a.e. where is given by (14), is the probability distribution of on , and the expectation on the right hand side of the equation is taken over the variable .

Theorem 2. Let be integrable over and let be the probability distribution of on . Then for a.e. where and is given by (15) with .

For and let , , and . Suppose that exists. By Theorem 1for a.e. , where and are given by (12) and (13), respectively, and is the probability distribution of on . By Theorem 2 we also have for a.e. where and is the probability distribution of on .

Let and be the right hand sides of (18) and (19), respectively. If has an analytic extension on , then it is called the conditional analytic Wiener -integral of given with the parameter and is denoted by for . Moreover if, for nonzero real , has a limit as approaches to through , then it is called the conditional analytic Feynman -integral of given with the parameter and is denoted by Replacing by , we define and . If exists for and it has an analytic extension on , then we call the analytic Wiener -integral of over with parameter and it is denoted by is also defined by if it exists, where the limit is taken through .

Applying Theorem  2.3 in [13], we can easily prove the following theorem.

Theorem 3. Let be an orthonormal system of . Then are independent and each has the standard normal distribution. Moreover if is Borel measurable, then where is the identity matrix on and means that if either side exists then both sides exist and they are equal.

Since is a probability measure on we have the following corollary.

Corollary 4. Under the assumptions as given in Theorem 3if is Borel measurable.

The following lemmas are useful to prove the results in the next sections and their proofs are simple.

Lemma 5. Let . Then for a.e. where is the multiplication operator defined by

Lemma 6. Let , , and where . Then

Remark 7. (1) The multiplication operator in Lemma 5 is well defined because is of bounded variation which implies the boundedness of . will denote the operator as given in the lemma unless otherwise specified.
(2) For it is possible that if . In this case the symbol does not mean the Paley-Wiener-Zygmund integral of . It is only a formal expression for which is as given in Lemma 6.

3. Multivariate Normal Distributions

In this section we derive a multivariate normal distribution which will be needed in the next section.

Lemma 8. Let be a set of independent vectors in . Then the covariance matrix of the random variables , , exists and is positive definite. Moreover is given byand the determinant of is positive so that is nonsingular and the inverse matrix of is also positive definite.

Proof. By Theorem 3so that the covariance of and is given by which proves (30). We have for Moreover if , then which implies for by the assumption. Thus the covariance matrix is positive definite. Since is symmetric and positive definite, the eigenvalues of are real and positive. Since ,   is invertible. Since implies ; that is, and is positive definite.

For simplicity let for and .

By Lemma 8, Theorem  4 of [14], and the change of variable theorem, we have the following theorem.

Theorem 9. Let the assumptions and notations be as given in Lemma 8. Then for every Borel measurable function where is given by (15), is the identity matrix on , and is the positive definite matrix satisfying .

By the same process as used in Lemma  2.1 of [15], Theorem 9, and the change of variable theorem, we have the following corollary.

Corollary 10. Let be a subset of and suppose 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 , we have

4. Analytic Feynman Integrals and Conditional Analytic Feynman Integrals

We begin this section with introducing the cylinder function on the analogue of Wiener space. Let be an orthonormal subset of , let be any positive integer, let , and let be given byfor a.e. , where . Without loss of generality we can take to be Borel measurable. In the following theorem we evaluate the Wiener and Feynman integrals of .

Theorem 11. Let be given by (37) with and suppose that is an independent subset of . Then for where and is given by (15). Moreover if , then, for a nonzero real , is given by the right hand side of (38) with replacing by .

Proof. Let . Replacing by we have by Corollary 10For any real which is integrable over , so that we have the theorem by the change of variable theorem, Morera’s theorem, the uniqueness of analytic continuation, and the dominated convergence theorem.

Let be the orthonormal set obtained from by the Gram-Schmidt orthonormalization process. For let be the linear combination of ’s and let be the coefficient matrix of the combinations. Sincewe have , where is the transpose of .

We now have the following corollary by (41), Theorem 11, and the change of variable theorem.

Corollary 12. Under the assumptions as given in Theorem 11Moreover if , then is given by the right hand side of (42) with replacing by .

Theorem 13. Let be given by (37) with and suppose that is an independent subset of . Then for for a.e. , where and is given by (12). Moreover if , then, for a nonzero real , is given by the right hand side of (43) with replacing by .

Proof. For and a.e. we have by Lemma 5so that for and we have by Corollary 10where is given by (15). By the Morera and the dominated convergence theorem we have the theorem.

Let be the orthonormal set obtained from by the Gram-Schmidt orthonormalization process. For let be the linear combination of ’s and let be the coefficient matrix of the combinations.

We now have the following corollary by a similar calculation of (41).

Corollary 14. Under the assumptions as given in Theorem 13Moreover if , then is given by the right hand side of (46) with replacing by .

For , , , and any nonsingular positive matrix on let With the above notations, we have the following theorem.

Theorem 15. Let be given by (37) with . Then for for a.e. , where . If , then, for a nonzero real , is given by the right hand side of the above equality with replacing by . Moreover the matrix can be replaced by in the above results.

Proof. For let , where . For we have by Lemma 6, Theorem 13, and the change of variable theorem Using the same method as used in the proof of Theorem  3.2 in [9] by the Schwarz inequality and the Morera theorem. The final results follow from the dominated convergence theorem and by a similar calculation of (41).

Remark 16. (1) An orthonormal subset of such that both and are independent sets exists.
(2) It does not mean that in the equations of Corollary 12 and in the equations of Corollary 14 and of Theorem 15. They satisfy only the following equations:

5. Integral Transforms with Change of Scales

In this section we derive change of scale transforms for the generalized conditional Wiener integrals of the function which is introduced in the previous section. To derive these scale transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors can be removed from the change of scale transforms in [8–10] and the transforms are expressed in terms of itself.

For and let

We now have the following theorem.

Theorem 17. Let and let be given by (37). Then for where is given by (52). If and is a nonzero real number, then for any sequence in converging to as approaches to .

Proof. For we have by (41), Corollary 10, and Theorem 11where is given by (15). If , the final result immediately follows from the dominated convergence theorem.

For , , and let

We now have the following theorem.

Theorem 18. Let and let be given by (37). Then for for a.e. , where is given by (56). If and is a nonzero real number, then for any sequence in converging to as approaches to . Moreover can be replaced by in the above equalities.

Proof. For and a.e. we have by Corollary 10By (41), the analytic continuation, the dominated convergence theorem, and Theorem 13 we have the theorem.

By Theorems 15 and 18 we have the final theorem.

Theorem 19. Let and let be given by (37). Then for for a.e. , where and , are given by (15) and (56), respectively. If and is a nonzero real number, then for any sequence in converging to as approaches to . Moreover can be replaced by in the above equalities.

Remark 20. (1) Letting in the theorems of this section, where , we have the change of scale formulas for , and as integral transforms.
(2) If and a.e., then we can obtain Theorems  5.1 and  5.2 in [9].
(3) If , a.e., and which is the Dirac measure concentrated at , then we can obtain the change of scale transforms in [10].
(4) The results in 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 research was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education (2013R1A1A2058991).