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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 671909, 12 pages
A Rotation of Admixable Operators on Abstract Wiener Space with Applications
Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea
Received 17 December 2012; Accepted 24 June 2013
Academic Editor: Henryk Hudzik
Copyright © 2013 Jae Gil Choi and Seung Jun Chang. 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.
We investigate certain rotation properties of the abstract Wiener measure. To determine our rotation property for the Wiener measure, we introduce the concept of an admixable operator via an algebraic structure on abstract Wiener space. As for applications, we define the analytic Fourier-Feynman transform and the convolution product associated with the admixable operators and proceed to establish the relationships between this transform and the corresponding convolution product.
1. Introduction and Preliminaries
Let denote one-parameter Wiener space. Bearman's rotation theorem  for Wiener measure has played an important role in various research areas in mathematics and physics involving Wiener integration theory. Bearman's theorem was further developed by Cameron and Storvick  and by Johnson and Skoug  in their studies of Wiener integral equations. Recently, in , using results in , Chang et al. obtained results involving a very general multiple Fourier-Feynman transform on Wiener space.
Let be a real separable infinite-dimensional Hilbert space with inner product and norm . Let be a measurable norm on with respect to the Gaussian cylinder set measure on . Let denote the completion of with respect to . Let denote the natural injection from to . The adjoint operator of is one to one and maps continuously onto a dense subset of , where and are topological duals of and , respectively. By identifying with and with , we have a triple and for all in and in , where denotes the natural dual pairing between and . By the well-known result of Gross , provides a unique countably additive extension, , to the Borel -algebra of . is a probability measure on the Borel -algebra of which satisfies The triple is called an abstract Wiener space. For more details, see [6–9].
Let be a complete orthonormal set in such that 's are in . For each and , we define the stochastic inner product by For every (≠0) in , exists for -a.e. , and it is a Gaussian random variable on with mean zero and variance ; that is, (1) holds with replaced with . In fact, the stochastic inner product is essentially independent of the choice of the complete orthonormal set used in its definition. Also, if both and are in , then Parseval's identity gives . Furthermore, for any , , and . We also see that, if is an orthonormal set in , then the random variables ’s are independent.
Let be the class of -measurable subsets of . A subset of is said to be scale-invariant measurable [3, 7] provided is -measurable for every , and a scale-invariant measurable subset of is said to be scale-invariant null provided for every . A property that holds except on a scale-invariant null set is said to hold scale invariant almost everywhere (-a.e.). A functional on is said to be scale-invariant measurable provided is defined on a scale-invariant measurable set and is -measurable for every . If two functionals and on are equal -a.e., that is, for each , , then we write .
Next, we introduce the concept of an admixable operator on .
Definition 1. Let be an operation between and which satisfies the conditions: (1). (2). (3) If for and , then or . (4) For every and every ,
(5) For every and every ,
(6) For every , there exists such that
where . In this case, we write . (7) For every and every ,
Given , let be a linear operator associated with . The operator is said to be -admixable provided for all .
For a finite subset of , let be the random vector given by A functional is called a cylinder-type functional on if there exists a linearly independent subset of such that where is a complex-valued Lebesgue measurable function on . It is easy to show that, for the given cylinder-type functional of the form (8), there exists an orthogonal subset of such that is expressed as where is a complex-valued Lebesgue measurable function on . Thus, we lose no generality in assuming that every cylinder-type functional on is of the form (9).
Lemma 2 (Chung, ). Let be an abstract Wiener space, and let be an orthogonal set in . Let be a complex-valued function defined on . Then, for the cylinder-type functional given by (9) on ,(i) is -measurable if and only if is Borel measurable on , (ii) is -measurable if and only if is Lebesgue measurable on .
For , let be the -admixable operator on . In this case, for any orthogonal subset of ,
The main purpose of this paper is to establish a rotation property for the abstract Wiener integral, where is given by (9) and is determined by and .
2. A Typical Example of an Abstract Wiener Space
The classical Wiener space , which is one of the most important examples of abstract Wiener spaces (see ), is a triple , where (i) is a Banach space consisting of real-valued continuous functions with defined on the compact interval endowed with uniform norm , (ii) is a real separable infinite dimensional Hilbert space consisting of absolutely continuous functions with , such that endowed with the inner product (iii) is the Wiener measure on the Borel -algebra of with
Let be the unitary operator from , onto given by for and let For any and , let the operation between and be defined by where denotes the pointwise multiplication of the functions and .
Remark 3. Let be the set of all -measurable subsets of . Then, is a complete measure space. It is well known that coincides with , the completion of .
Let with . Then, the stochastic integral which was introduced by Park and Skoug in , is a Gaussian process with mean zero and covariance function In addition, is stochastically continuous in on . For more detailed studies of this process, see [14–17]. Furthermore, if is an element of , then, for all , is continuous in , and so is in .
From [14, Lemma 1], we note that, for each and each with , for -a.e. .
Given , define an operator by Then, for all , Thus, is -admixable in view of Definition 1.
3. A Rotation of Admixable Operators
In this section, we establish a rotation property for the abstract Wiener integral involving admixable operators. We first introduce an integration formula which plays a key role.
Lemma 4. Let be an orthogonal set in and let be given by (7). Let be a Lebesgue measurable function. Then where by one means that if either side exists, both sides exist and equality holds.
The following integration formula is also used several times in this paper: for complex numbers and with .
Remark 5. Let and be elements in with . Then, the random variables and will have the same distribution .
Let and be orthogonal sets in with for each . Using the aforementioned facts and applying (23), we see that, for any Lebesgue measurable function on ,
To simplify the expressions in our results, we use the following notations: for and given by (7).
Proposition 6. Let and be orthogonal sets in . Then, for any Lebesgue measurable function on , where is an orthogonal set in which satisfies the condition for each . Also, both of the expressions in (27) are given by the last expression in (29).
Proof. First, using (23), the Fubini theorem, and (24), we have that
Next, let be any orthonormal set in . For each , let . Then is an orthogonal set in and satisfies (28) above. In this case, using (23), the Fubini theorem, and (24), we see that is given by the last expression of (29). In view of Remark 5, we obtain the desired result.
The following corollary follows by the use of mathematical induction.
Corollary 7. For each , let be an orthogonal set in . Then for any Lebesgue measurable function on , where is an orthogonal set in which satisfies the condition for each . Also, both of the expressions in (30) are given by the expression
The next corollary follows directly from Proposition 6.
Throughout the rest of this section, for convenience, we use the following notation: for a finite sequence in , let
For our rotation property presented, namely, Theorem 11, we will consider the pair of finite subsets of and of (this allows ) such that (c1) is orthogonal in , (c2) is orthogonal in for all .
Let us return to the classical Wiener space . See Section 2. For each , let on . Then, is an orthogonal sequence in . Additionally, is a complete orthonormal set in . In fact, for all . We can take different pair from , which satisfies conditions (c1) and (c2) above. For instance, let and let , where .
Remark 9. Let and satisfy the conditions (c1) and (c2). Then, the stochastic inner products form a set of independent Gaussian random variables on with mean and variance where is given by (33).
Remark 10. Given an orthogonal subset , let be the space of every element in such that is orthogonal in . Then, for any finite subset of , the pair satisfies conditions (c1) and (c2) above.
Given an orthogonal set in , let be the class of all cylinder-type functionals, , given by (9) for -a.e. , where the corresponding function of satisfies the condition for all .
We are now ready to present our rotation property of abstract Wiener measure associated with admixable operators.
Proof. For each and each , let . For each , is orthogonal in by condition (c2). Hence, for each , the stochastic inner products
form a set of independent Gaussian random variables with mean and variance .
We observe that Thus, using (9), (39), and (30), we obtain where is an orthogonal set in , such that for each .
Next, we note that, for each with , Hence, from (35) and (42), we see that is an orthogonal set in with for all , and so we can choose to be the orthogonal set . In this case, we see that Equation (37) follows from (40) and (43).
Example 12. Let us return to the classical Wiener space again. We introduce the family of functions from :
These functions have the reproducing property
for all . In fact, .
For , we consider the admixable operator given by (21). We note that, for each , From this, we see that, for and , and so, using (37), we find that, for any and every , Using (16), we now have on . Thus, by (48), (49), and (47), we have
Proposition 13 (Cameron and Storvick, ). Let be Wiener integrable on for . As a result, is integrable on , and
Proof. Let be any positive integer, and let be any partition of . It suffices to show that (51) holds for any tame function
Let be defined by (16) between and . For , let the -admixable operator be given by (21), and, for each , let on . For any , (c1) is an orthogonal set in , and (c2) and are orthogonal sets in .
Also, for any , , and, for each , Thus, the left side of (51) with given by (52) is rewritten by Thus, from Theorem 11, we obtain
Using standard methods, similar to those in , we can obtain the result for general functionals on .
4. Fourier-Feynman Transforms and Convolutions Associated with Admixable Operators
In this section, to apply our results from the previous section, we first define an analytic Fourier-Feynman transform associated with admixable operators on . Then, we establish the existence theorem and the inverse transform theorem of this transform for some classes of cylinder-type functionals on having the form (9) for -a.e. . Moreover, we present various relationships involving the convolution and the transforms.
Throughout the rest of this paper, let , , and denote, respectively, the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part.
Let , and let be the corresponding admixable operator on . Let be a scale-invariant measurable functional such that exists as a finite number for all . If there exists a function analytic on such that for all , then is defined to be the analytic Wiener integral (associated with the -admixable operator ) of over with parameter . For , we write Let be a real number, and let be a functional such that exists for all . If the following limit exists, we call it the analytic Feynman integral of with parameter , and we write where approaches through values in .
We are now ready to state the definition of the analytic Fourier-Feynman transform associated with admixable operator (admix-FFT).
Definition 14. Let be an abstract Wiener space. For , , and , let where is the -admixable operator on . Let be a nonzero real number. For , we define the analytic -admix-FFT, of , by the formula , if it exists; that is, for each , where . We define the analytic -admix-FFT, of , by the formula , if it exists.
We note that, for , is defined only -a.e.. We also note that if exists and if , then exists and .
Next, we give the definition of the convolution product (CP).
Definition 15. Let and be scale-invariant measurable functionals on . For and , we define their CP with respect to (if it exists) by When , we denote by .
For any scale-invariant measurable functional , we see that, for , if it exists.
Let denote the space of complex-valued, countably additive (and hence finite) Borel measures on , the Borel -algebra of . It is well known that a complex-valued Borel measure necessarily has a finite total variation , and is a Banach algebra under the norm and with convolution as multiplication.
For , the Fourier transform of is a complex-valued function defined on by the formula where and are in .
Let be an orthonormal set in . Define the functional by for -a.e. , where is the Fourier transform of in . Then is a bounded cylinder-type functional because .
Let be the set of all functionals on having the form (67). Note that implies that is scale-invariant measurable on . Throughout this section, we fix the orthogonal set .
We now state the existence theorem for the analytic Feynman integral of the functionals in .
Theorem 16. Let be given by (67). Then, for all and all nonzero real numbers , the analytic Feynman integral of exists and is given by the formula
Proof. By (67), (66), the Fubini theorem, (23), and (24), we see that, for all ,
Now, let for . Then, for all and for all . Thus, applying the dominated convergence theorem, we see that is continuous on . Also, because is analytic on , applying the Fubini theorem, we have for all rectifiable simple closed curve lying in . Thus, by the Morera theorem, is analytic on . Therefore, the analytic Wiener integral exists. Finally, applying the dominated convergence theorem, we know that is given by the right side of (68).
Next, we establish the existence of the admix-FFT for functionals in .
Theorem 17. Let be given by (67). Then, for all and all , the analytic -admix-FFT, , exists for all nonzero real numbers , belongs to , and is given by the formula for -a.e. , where is the complex measure on given by for .
Proof. Proceeding as in the proof of Theorem 16, we see that, for all , all , and for -a.e. ,
is an analytic function of on , and that for any and -a.e. ,
Clearly, the set function given by (72) is a complex measure on , and so the right side of (74) can be rewritten as the right side of (71).
Next, we note that for all and Using these, we see that, for all , all , and all , and so, by the dominated convergence theorem, we see that, for any nonzero real , Hence, exists and is given by the right side of (74) for all desired values of and and all . Thus, the theorem is proved.
Theorem 18. Let be given by (67). Then, for all , all , and all nonzero real , As such, the admix-FFT, , has the inverse transform .
Theorem 19. Let and be elements of with corresponding finite Borel measures and in . Then, for all , the CP, , exists for all nonzero real numbers , belongs to , and is given by the formula for -a.e. , where is a continuous function defined by and is a complex measure on given by for .
Proof. Using (64), the Fubini theorem, (23), and (24), we have that, for all and -a.e. ,
Using the same argument as in the proof of Theorem 17, we can show that the last expression in the previous equation is an analytic function of on and is a bounded continuous function of on because and are finite Borel measures. Hence, exists and is given by
for all and -a.e. .
Consider the set function and the continuous function given by (82) and (81), respectively. Clearly, the set function is a complex measure on . Hence, is an element of , and so the right side of (84) can be rewritten as the right side of (80). Thus, the theorem is proved.
Lemma 20. Let be any orthogonal set in . For every , every and