About this Journal Submit a Manuscript Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 578174, 13 pages
http://dx.doi.org/10.5402/2012/578174
Research Article

A Rotation on Wiener Space with Applications

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Received 25 April 2012; Accepted 26 June 2012

Academic Editors: M. Brünig, M. Langthjem, H. T. Yau, and C. Zhang

Copyright © 2012 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.

Abstract

We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

1. Introduction

Let 𝐶0[0,𝑇] denote one-parameter Wiener space, that is, the space of all real-valued continuous functions 𝑥 on [0,𝑇] with 𝑥(0)=0. Let denote the class of all Wiener measurable subsets of 𝐶0[0,𝑇], and let 𝑚 denote Wiener measure. Then (𝐶0[0,𝑇],,𝑚) is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional 𝐹 by 𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥).(1.1)

In [1], Bearman gave a significant theorem for Wiener integral on product Wiener space. It can be summarized as follows.

Theorem 1.1 (Bearman's Rotation Theorem). Let 𝐺(𝑤,𝑧) be an 𝑚×𝑚-integrable functional on 𝐶20[0,𝑇], the product of 2 copies of 𝐶0[0,𝑇], and let 𝜃 be a function of bounded variation on [0,𝑇]. Let 𝑇𝜃𝐶20[0,𝑇]𝐶20[0,𝑇] be the transformation defined by 𝑇𝜃(𝑤,𝑧)=(𝑤,𝑧) with 𝑤(𝑡)=𝑡0cos𝜃(𝑠)𝑑𝑤(𝑠)𝑡0𝑧sin𝜃(𝑠)𝑑𝑧(𝑠),(𝑡)=𝑡0sin𝜃(𝑠)𝑑𝑤(𝑠)+𝑡0cos𝜃(𝑠)𝑑𝑧(𝑠).(1.2) Then the transform 𝑇𝜃 is measure preserving and 𝐶20[0,𝑇]𝐺(𝑤,𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶20[0,𝑇]𝐺𝑇𝜃(𝑤,𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧).(1.3)

As a special case of Theorem 1.1, one can obtain the following corollary.

Corollary 1.2. Let 𝐹 be Wiener integrable on 𝐶0[0,𝑇]. Then for any 𝜃, 𝐹(𝑤sin𝜃+𝑧cos𝜃) is integrable on 𝐶20[0,𝑇] and 𝐶20[0,𝑇]𝐹(𝑤sin𝜃+𝑧cos𝜃)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥).(1.4)

The following more general case of Corollary 1.2 is due to Cameron and Storvick [2]. But we state the theorem with some assumption for our research.

Theorem 1.3. Let 𝐹 be Wiener measurable on 𝐶0[0,𝑇]. Assume that for any 𝜌>0, 𝐹(𝜌) is Winer integrable. Then for any 𝑎,𝑏, 𝐹(𝑎𝑤+𝑏𝑧) is integrable on 𝐶20[0,𝑇] and 𝐶20[0,𝑇]𝐹(𝑎𝑤+𝑏𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝑎2+𝑏2𝑥𝑑𝑚(𝑥).(1.5)

In many papers, Theorem 1.3 is used to study relationships between analytic Fourier-Feynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance [36]. In this paper, we will extend the result in Theorem 1.3 to a more general case for functionals of Gaussian processes given by (2.2) below. We then apply our rotation property of Wiener measure to establish a fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product.

2. A Rotation on Wiener Space

The most important concepts we will employ in the statements and proofs of our results are the concepts of the scale-invariant measurability and the Paley-Wiener-Zygmund stochastic integral [7].

A subset 𝐵 of 𝐶0[0,𝑇] is said to be scale-invariant measurable [8] provided 𝜌𝐵 for all 𝜌>0, and a scale-invariant measurable set 𝑁 is said to be scale-invariant null provided 𝑚(𝜌𝑁)=0 for all 𝜌>0. A property that holds except on a scale-invariant null set is said to be hold scale-invariant almost everywhere (s-a.e.). If two functionals 𝐹 and 𝐺 are equal s-a.e., we write 𝐹𝐺.

Let {𝜙𝑛} be a complete orthonormal set in 𝐿2[0,𝑇], each of whose elements is of bounded variation on [0,𝑇]. Then for each 𝑣𝐿2[0,𝑇], the Paley-Wiener-Zygmund (PWZ) stochastic integral 𝑣,𝑥 is defined by the formula 𝑣,𝑥=lim𝑛𝑇0𝑛𝑗=1𝑣,𝜙𝑗2𝜙𝑗(𝑡)𝑑𝑥(𝑡)(2.1) for all 𝑥𝐶0[0,𝑇] for which the limit exists, where (,)2 denotes the 𝐿2-inner product.

It was shown in [7] that for each 𝑣𝐿2[0,𝑇], the limit defining the PWZ integral 𝑣,𝑥 exists for 𝑚-a.e. 𝑥𝐶0[0,𝑇] and that this limit is essentially independent of the choice of the complete orthonormal set {𝜙𝑛}. It was also shown in [7] that if 𝑣 is of bounded variation on [0,𝑇], then the PWZ integral 𝑣,𝑥 equals the Riemann-Stieltjes integral 𝑇0𝑣(𝑡)𝑑𝑥(𝑡) for 𝑚-a.e. 𝑥𝐶0[0,𝑇]. In fact, the integrals are equal for s-a.e. 𝑥𝐶0[0,𝑇] and that for all 𝑣𝐿2[0,𝑇], 𝑣,𝑥 is a Gaussian random variable with mean 0 and variance 𝑣22.

For any 𝐿2[0,𝑇] with 2>0, let 𝒵 be the Gaussian process 𝒵(𝑥,𝑡)=𝑡0(𝑠)𝑑𝑥(𝑠)=𝑣,𝑥(2.2) introduced by Park and Skoug in [9] and used extensively since; see for example [5, 6, 10, 11]. Of course if (𝑡)1 on [0,𝑇], then 𝒵(𝑥,𝑡)=𝑥(𝑡).

It is easy to see that 𝒵 is a Gaussian process with mean zero and covariance function 𝐶0[0,𝑇]𝒵(𝑥,𝑠)𝒵(𝑥,𝑡)𝑑𝑚(𝑥)=0min{𝑠,𝑡}2(𝑢)𝑑𝑢.(2.3) In addition, 𝒵(,𝑡) is stochastically continuous in 𝑡 on [0,𝑇], and for any 1,2𝐿2[0,𝑇], 𝐶0[0,𝑇]𝒵1(𝑥,𝑠)𝒵2(𝑥,𝑡)𝑑𝑚(𝑥)=0min{𝑠,𝑡}1(𝑢)2(𝑢)𝑑𝑢.(2.4)

For any complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇] and for any 𝑛, define the projection map 𝒫𝑛 from 𝐿2[0,𝑇] into span{𝜙1,,𝜙𝑛} by 𝒫𝑛(𝑡)=𝑛𝑗=1,𝜙𝑗2𝜙𝑗(𝑡).(2.5) Then for 𝐿2[0,𝑇] and 𝑥𝐶0[0,𝑇], we see that 𝒵(𝑥,𝑡)=lim𝑛𝑡0𝒫𝑛(𝑠)𝑑𝑥(𝑠)=lim𝑛𝒵𝒫𝑛(𝑥,𝑡),(2.6) that is, 𝒵𝒫𝑛(𝑥,𝑡) converges in 𝐿2(𝐶0[0,𝑇])-mean to 𝒵(𝑥,𝑡).

Throughout this paper, we will assume that each functional 𝐹𝐶0[0,𝑇] we consider is scale-invariant measurable and that 𝐶0[0,𝑇]||𝐹𝒵(||𝑥,)𝑑𝑚(𝑥)<+(2.7) for all 𝐿2[0,𝑇].

We are now ready to state the main theorem of this paper.

Theorem 2.1. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any 1,2𝐿2[0,𝑇], 𝐶20[0,𝑇]𝐹𝒵1(𝑤,)+𝒵2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,)𝑑𝑚(𝑥),(2.8) where 1, 2, and 𝑘 are related by 𝑘(𝑡)=𝑛=11,𝜙𝑛22+2,𝜙𝑛22𝜙𝑛(𝑡)(2.9) for some complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇], each of those elements is of bounded variation on [0,𝑇].

3. Proof of the Main Theorem

We begin this section with three lemmas in order to establish (2.8).

Lemma 3.1. Let 𝐹 be a functional on 𝐶0[0,𝑇], and let 𝜙 be a function of bounded variation on [0,𝑇]. Then for all 𝑎,𝑏, 𝐶20[0,𝑇]𝐹𝑎𝒵𝜙(𝑤,)+𝑏𝒵𝜙(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝑎2+𝑏2𝒵𝜙(𝑥,)𝑑𝑚(𝑥).(3.1)

Proof. We first note that for each 𝑡[0,𝑇], 𝑎𝒵𝜙(𝑤,𝑡)+𝑏𝒵𝜙(𝑧,𝑡)=𝑡0𝜙(𝑠)𝑑(𝑎𝑤(𝑠)+𝑏𝑧(𝑠))=𝒵𝜙(𝑎𝑤+𝑏𝑧,𝑡).(3.2) We also note that 𝐹(𝒵(𝑥,)) is Wiener integrable as a functional of 𝑥. Hence, by (1.5), we obtain that for all 𝑎,𝑏, 𝐶20[0,𝑇]𝐹𝑎𝒵𝜙(𝑤,)+𝑏𝒵𝜙(=𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝒵𝜙=(𝑎𝑤+𝑏𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]𝐹𝒵𝜙𝑎2+𝑏2=𝑥,𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐹𝑎2+𝑏2𝒵𝜙(𝑥,)𝑑𝑚(𝑥).(3.3) Thus (3.1) is established.

Lemma 3.2. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any 1,2𝐿2[0,𝑇] and each 𝑛, 𝐶20[0,𝑇]𝐹𝒵𝒫𝑛1(𝑤,)+𝒵𝒫𝑛2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝒫𝑛𝑘(𝑥,)𝑑𝑚(𝑥),(3.4) where 1, 2, and 𝑘 are related by (2.9).

Proof. Since the addition is continuous in the uniform topology on 𝐶0[0,𝑇], we can apply (3.1) to the functional 𝐹(𝑛𝑗=1𝒵𝜙𝑗(𝑥,)). Thus using (2.5) and (3.1), we have 𝐶20[0,𝑇]𝐹𝒵𝒫𝑛1(𝑤,)+𝒵𝒫𝑛2(=𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝑛𝑗=11,𝜙𝑗2𝒵𝜙𝑗(𝑤,)+2,𝜙𝑗2𝒵𝜙𝑗=(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]𝐹𝑛𝑗=11,𝜙𝑗22+2,𝜙𝑗22𝒵𝜙𝑗=(𝑥,)𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐹𝒵𝒫𝑛𝑘(𝑥,)𝑑𝑚(𝑥).(3.5) Thus (3.4) is established.

Lemma 3.3. Let 𝐹 be bounded and continuous on 𝐶0[0,𝑇]. Then for any 1,2𝐿2[0,𝑇], 𝐶20[0,𝑇]𝐹𝒵1(𝑤,)+𝒵2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,)𝑑𝑚(𝑥),(3.6) where 1, 2, and 𝑘 are related by (2.9) above.

Proof. We clearly see that 𝐹 is Wiener integrable. We also note that {𝒫𝑛} is a sequence of functions of bounded variation on [0,𝑇] such that 𝒫𝑛 converges to in the space 𝐿2[0,𝑇] as 𝑛. For each 𝑛 and 𝐿2[0,𝑇], let 𝐹𝑛(𝒵(𝑥,))=𝐹(𝒵𝒫𝑛(𝑥,)). Since 𝒵𝒫𝑛 converges to 𝒵 uniformly and 𝐹 is continuous in the uniform topology, by (2.6), 𝐹𝒵(𝑥,)=𝐹lim𝑛𝒵𝒫𝑛(𝑥,)=lim𝑛𝐹𝑛𝒵(𝑥,).(3.7) Since 𝐹 is bounded, by using the dominated convergence theorem and (3.4), we have 𝐶20[0,𝑇]𝐹𝒵1(𝑤,)+𝒵2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=lim𝑛𝐶20[0,𝑇]𝐹𝑛𝒵1(𝑤,)+𝒵2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=lim𝑛𝐶0[0,𝑇]𝐹𝑛𝒵𝑘=(𝑥,)𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,)𝑑𝑚(𝑥),(3.8) which concludes the proof of Lemma 3.3.

We are now ready to prove our main theorem.

Proof of Theorem 2.1. Let 𝐹 be Wiener integrable. Suppose that the left-hand side of (2.8) exists. By usual arguments of integration theory, there exists a sequence {𝐹𝑛} of bounded and continuous functionals such that 𝐹𝑛 converges to 𝐹. By Lemma 3.3 and the dominated convergence theorem, we can obtain the desired result.

Corollary 3.4. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for all 𝐿2[0,𝑇] and all 𝑎,𝑏, 𝐶20[0,𝑇]𝐹𝑎𝒵(𝑤,)+𝑏𝒵(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝑎2+𝑏2𝒵(𝑥,)𝑑𝑚(𝑥).(3.9)

Proof. Simply choose 1=𝑎 and 2=𝑏 in (2.8) and use the linearity property of the PWZ stochastic integral.

Using similar arguments as in the proofs of Lemmas 3.1, 3.2, and 3.3 and Theorem 2.1 above, we can obtain the following theorems.

Theorem 3.5. Let 𝐹 be a functional on 𝐶0[0,𝑇], and let {1,,𝜈} be any subset of 𝐿2[0,𝑇]. Then 𝐶𝜈0[0,𝑇]𝐹𝜈𝑗=1𝒵𝑗𝑥𝑗,𝑑𝑚𝜈𝑥=𝐶0[0,𝑇]𝐹𝒵𝑘𝜈(𝑥,)𝑑𝑚(𝑥),(3.10) where 𝑚𝜈 is the product Wiener measure on 𝐶𝜈0[0,𝑇], the product of 𝜈 copies of 𝐶0[0,𝑇], and 𝑘𝜈(𝑡)=𝑛=1𝜈𝑗=1𝑗,𝜙𝑛22𝜙𝑛(𝑡)(3.11) for some complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇].

Theorem 3.6. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any 1 and 2 in 𝐿2[0,𝑇], 𝐶20[0,𝑇]𝐹𝒵2(𝑤,)𝒵1(𝑧,),𝒵1(𝑤,)+𝒵2(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶20[0,𝑇]𝐹𝒵𝑘(𝑤,),𝒵𝑘(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧),(3.12) where 1, 2, and 𝑘 are related by (2.9).

Remark 3.7. Equations (2.8) and (3.12) are indeed very general formulas.(1)For any 𝜃, choosing 1(𝑡)sin𝜃 and 2(𝑡)cos𝜃 in (2.8) yields (1.4).(2)For any 𝑎,𝑏, choosing 1(𝑡)𝑎 and 2(𝑡)𝑏 in (2.8) or choosing (𝑡)1 in (3.9) yields (1.5). (3)For any function of bounded variation 𝜃(), choosing 1(𝑡)=sin𝜃(𝑡) and 2(𝑡)=cos𝜃(𝑡) on [0,𝑇] in (3.12) yields (1.3).

4. Generalized Fourier-Feynman Transform and Generalized Convolution Product

In this section, we will apply our main theorem to the generalized analytic Fourier-Feynman transform and the convolution product theories.

In defining various analytic Feynman integrals, one usually starts, for 𝜆>0, with the Wiener integral 𝐶0[0,𝑇]𝐹𝜆1/2𝑥𝑑𝑚(𝑥)(4.1) and then extends analytically in 𝜆 to the right-half complex plane. Here we start with the (generalized) Wiener integral 𝐶0[0,𝑇]𝐹𝜆1/2𝒵(𝑥,)𝑑𝑚(𝑥)=𝐽(;𝜆),(4.2) where 𝒵 is the Gaussian process given by (2.2) above.

Throughout this section, let + and + denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.

Let 𝐹 be a complex-valued scale-invariant measurable functional on 𝐶0[0,𝑇] such that 𝐽(;𝜆) given by (4.2) exists and is finite for all 𝜆>0. If there exists a function 𝐽(;𝜆) analytic on + such that 𝐽(;𝜆)=𝐽(;𝜆) for all 𝜆>0, then 𝐽(;𝜆) is defined to be the generalized analytic Wiener integral (with respect to the process 𝒵) of 𝐹 over 𝐶0[0,𝑇] with parameter 𝜆, and for 𝜆+ we write anw𝜆𝐶0[0,𝑇]𝐹𝒵(𝑥,)𝑑𝑚(𝑥)=𝐽(;𝜆).(4.3) Let 𝑞 be a nonzero real number and let 𝐹 be a functional such that anw𝜆𝐶0[0,𝑇]𝐹(𝒵(𝑥,))𝑑𝑚(𝑥) exists for all 𝜆+. If the following limit exists, we call it the generalized analytic Feynman integral of 𝐹 with parameter 𝑞 and we write anf𝑞𝐶0[]0,𝑇𝐹𝒵(𝑥,)𝑑𝑚(𝑥)=lim𝜆𝑖𝑞anw𝜆𝐶0[]0,𝑇𝐹𝒵(𝑥,)𝑑𝑚(𝑥),(4.4) where 𝜆 approaches 𝑖𝑞 through values in +.

Note that if 1 on [0,𝑇], then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 1214].

Next (see [5, 6, 15]) we state the definition of the generalized Fourier-Feynman transform (GFFT).

Definition 4.1. For 𝜆+ and 𝑦𝐶0[0,𝑇], let 𝑇𝜆,(𝐹)(𝑦)=anw𝜆𝐶0[]0,𝑇𝐹𝑦+𝒵(𝑥,)𝑑𝑚(𝑥).(4.5) Let 𝑞 be a non-zero real number. For 𝑝(1,2], we define the 𝐿𝑝 analytic GFFT with respect to 𝒵, 𝑇(𝑝)𝑞,(𝐹) of 𝐹, by the formula (𝜆+), 𝑇(𝑝)𝑞,(𝐹)(𝑦)=l.i.m.𝜆𝑖𝑞𝑇𝜆,(𝐹)(𝑦)(4.6) if it exists; that is, for each 𝜌>0, lim𝜆𝑖𝑞𝐶0[0,𝑇]|||𝑇𝜆,(𝐹)(𝜌𝑦)𝑇(𝑝)𝑞,(|||𝐹)(𝜌𝑦)𝑝𝑑𝑚(𝑦)=0,(4.7) where 1/𝑝+1/𝑝=1. We define the 𝐿1 analytic GFFT, 𝑇(1)𝑞,(𝐹) of 𝐹, by the formula (𝜆+)𝑇(1)𝑞,(𝐹)(𝑦)=lim𝜆𝑖𝑞𝑇𝜆,(𝐹)(𝑦)(4.8) if it exists.

We note that for 𝑝[1,2], 𝑇(𝑝)𝑞,(𝐹) is defined only s-a.e. We also note that if 𝑇(𝑝)𝑞,(𝐹) exists and if 𝐹𝐺, then 𝑇(𝑝)𝑞,(𝐺) exists and 𝑇(𝑝)𝑞,(𝐺)𝑇(𝑝)𝑞,(𝐹). One can see that for each 𝐿2[0,𝑇], 𝑇(𝑝)𝑞,(𝐹)𝑇(𝑝)𝑞,(𝐹) since 𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥)=𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥).(4.9)

Next we give the definition of the generalized convolution product (GCP).

Definition 4.2. Let 𝐹 and 𝐺 be scale-invariant measurable functionals on 𝐶0[0,𝑇]. For 𝜆+ and 1,2𝐿2[0,𝑇], we define their GCP with respect to {𝒵1,𝒵2} (if it exists) by (𝐹𝐺)(1,2)𝜆=(𝑦)anw𝜆𝐶0[0,𝑇]𝐹𝑦+𝒵1(𝑥,)2𝐺𝑦𝒵2(𝑥,)2𝑑𝑚(𝑥),𝜆+,anf𝑞𝐶0[]0,𝑇𝐹𝑦+𝒵1(𝑥,)2𝐺𝑦𝒵2(𝑥,)2𝑑𝑚(𝑥),𝜆=𝑖𝑞,𝑞,𝑞0.(4.10) When 𝜆=𝑖𝑞, we denote (𝐹𝐺)(1,2)𝜆 by (𝐹𝐺)(1,2)𝑞.

Remark 4.3. Our definition of the GCP is different than the definition given by Huffman et al. in [5, 6] and used by Chang et al. in [15]. But if we choose 1=2 in (4.10), our GCP (𝐹𝐺)(1,2)𝑞 is the GCP used in [5, 6, 15].

We begin this section with a key lemma for a relationship between the GFFT and the GCP.

Lemma 4.4. Let {𝑔1,𝑔2,𝑔3,𝑔4} be a subset of 𝐿2[0,𝑇], and let 𝑌𝑔1,𝑔2,𝑌𝑔3,𝑔4𝐶20[0,𝑇]×[0,𝑇] be given by 𝑌𝑔1,𝑔2(𝑤,𝑧;𝑡)=𝒵𝑔1(𝑤,𝑡)+𝒵𝑔2𝑌(𝑧,𝑡),𝑔3,𝑔4(𝑤,𝑧;𝑡)=𝒵𝑔3(𝑤,𝑡)𝒵𝑔4(𝑧,𝑡),(4.11) respectively. Then the following assertions are equivalent. (i)𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are independent processes.(ii)𝑔1𝑔3=𝑔2𝑔4.

Proof. Since the processes 𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are Gaussian with mean zero, we know that 𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are independent processes if and only if 𝐶20[0,𝑇]𝑌𝑔1,𝑔2(𝑤,𝑧;𝑠)𝑌𝑔3,𝑔4(𝑤,𝑧;𝑡)𝑑(𝑚×𝑚)(𝑤,𝑧)=0(4.12) for every 𝑠,𝑡[0,𝑇]. But, using (2.4), we have 𝐶20[0,𝑇]𝑌𝑔1,𝑔2(𝑤,𝑧;𝑠)𝑌𝑔3,𝑔4(=𝑤,𝑧;𝑡)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝒵𝑔1(𝑤,𝑠)𝒵𝑔3(𝑤,𝑡)𝒵𝑔1(𝑤,𝑠)𝒵𝑔4(𝑧,𝑡)+𝒵𝑔2(𝑧,𝑠)𝒵𝑔3(𝑤,𝑡)𝒵𝑔2(𝑧,𝑠)𝒵𝑔4=(𝑧,𝑡)×𝑑(𝑚×𝑚)(𝑤,𝑧)0min{𝑠,𝑡}𝑔1(𝑢)𝑔3(𝑢)𝑑𝑢0min{𝑠,𝑡}𝑔2(𝑢)𝑔4(𝑢)𝑑𝑢.(4.13) From this, we can obtain the desired result.

We are now ready to establish fundamental relationships between the GFFT and the GCP.

Lemma 4.5. Let 𝐹 and 𝐺 be functionals on 𝐶0[0,𝑇]. Let {1,2,3} be a subset of 𝐿2[0,𝑇] such that 23=120 almost everywhere on [0,𝑇], and let 𝑘1(𝑡)=21/2𝑛=11,𝜙𝑛22+3,𝜙𝑛22𝜙𝑛(𝑘𝑡),2(𝑡)=21/2𝑛=12,𝜙𝑛22+3,𝜙𝑛22𝜙𝑛(𝑡).(4.14) Furthermore, assume that for all 𝜆+, 𝑇𝜆,3((𝐹𝐺)(1,2)𝜆), 𝑇𝜆,𝑘1(𝐹) and 𝑇𝜆,𝑘2(𝐺) all exist. Then 𝑇𝜆,3(𝐹𝐺)(1,2)𝜆(𝑦)=𝑇𝜆,𝑘1𝑦(𝐹)2𝑇𝜆,𝑘2𝑦(𝐺)2(4.15) for s-a.e. 𝑦𝐶0[0,𝑇].

Proof. We note that for all 𝜆>0, 𝑇𝜆,3(𝐹𝐺)(1,2)𝜆=(𝑦)𝐶0[0,𝑇](𝐹𝐺)(1,2)𝜆𝑦+𝜆1/2𝒵3=(𝑤,)𝑑𝑚(𝑤)𝐶0[]0,𝑇𝐹𝑦2+1𝒵2𝜆3(𝑤,)+𝒵1𝑦(𝑧,)×𝐺2+1𝒵2𝜆3(𝑤,)𝒵2=(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝑦2+𝜆1/2𝒵3(𝑤,)+𝒵1(𝑧,)2𝑦×𝐺2+𝜆1/2𝒵3(𝑤,)𝒵2(𝑧,)2𝑑(𝑚×𝑚)(𝑤,𝑧).(4.16) But 23=12, and so (𝒵3(𝑤,)+𝒵1(𝑧,))/2 and (𝒵3(𝑤,)𝒵2(𝑧,))/2 are independent processes by Lemma 4.4. Hence by (2.8), we obtain that for all 𝜆>0, 𝑇𝜆,3(𝐹𝐺)(1,2)𝜆=(𝑦)𝐶20[0,𝑇]𝐹𝑦2+1𝒵2𝜆3(𝑤,)+𝒵1×(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐺𝑦2+1𝒵2𝜆3(𝑤,)𝒵2=(𝑧,)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]𝐹𝑦2+𝒵𝑘1(𝑥,)𝜆𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐺𝑦2+𝒵𝑘2(𝑥,)𝜆𝑑𝑚(𝑥)=𝑇𝜆,𝑘1𝑦(𝐹)2𝑇𝜆,𝑘2𝑦(𝐺)2.(4.17) Equation (4.15) holds for all 𝜆+ by analytic continuation.

In next theorem, we show that the GFFT of the GCP is the product of GFFTs.

Theorem 4.6. Let 𝐹, 𝐺, {1,2,3}, 𝑘1, and 𝑘2 be as in Lemma 4.5. Furthermore, assume that for 𝑝[1,2], 𝜆+ and 𝑞{0}, 𝑇𝜆,3((𝐹𝐺)(1,2)𝑞), 𝑇(𝑝)𝑞,3((𝐹𝐺)(1,2)𝑞), 𝑇(𝑝)𝑞,𝑘1(𝐹), and 𝑇(𝑝)𝑞,𝑘2(𝐺) all exist and that 𝑇(𝑝)𝑞,3(𝐹𝐺)(1,2)𝑞=l.i.m.𝜆𝑖𝑞𝑇𝜆,3(𝐹𝐺)(1,2)𝜆],(𝑦),𝑝(1,2lim𝜆𝑖𝑞𝑇𝜆,3(𝐹𝐺)(1,2)𝜆(𝑦),𝑝=1.(4.18) Then 𝑇(𝑝)𝑞,3(𝐹𝐺)(1,2)𝑞(𝑦)=𝑇(𝑝)𝑞,𝑘1𝑦(𝐹)2𝑇(𝑝)𝑞,𝑘2𝑦(𝐺)2(4.19) for s-a.e. 𝑦𝐶0[0,𝑇].

Proof. Equation (4.19) follows from (4.15) by letting 𝜆𝑖𝑞, since all transforms in (4.18) and (4.19) exist.

Remark 4.7. We note that the hypotheses (and hence the conclusions) of Theorem 4.6 above are indeed satisfied by many of the functionals in the following large classes of functionals. These classes of functionals include; (i) the Banach algebra 𝒮 defined by Cameron and Storvick in [16]: also see [3, 5, 14, 15],(ii) various spaces of functionals of the form 𝐹(𝑥)=exp𝑇0𝑓(𝑡,𝑥(𝑡))𝑑𝑡(4.20) for appropriate 𝑓[0,𝑇]× as discussed in [4, 12, 13]; and(iii) various spaces of functionals of the form 𝐹(𝑥)=exp𝑇0𝑓(𝑠,𝑡,𝑥(𝑠),𝑥(𝑡))𝑑𝑠𝑑𝑡(4.21) for appropriate 𝑓[0,𝑇]2×2 as discussed in [3].

Next five corollaries include the results of [36] by Huffman et al. The notations used in [36] are slightly different than ours.

Corollary 4.8. Refer to Theorem 2.1 in [5].

Proof. In our Lemma 4.5, simply choose 1=2=3=𝑘1=𝑘2𝐿2[0,𝑇].

Corollary 4.9. Refer to Theorem 1 in [6].

Proof. In our Theorem 4.6, simply choose 1=2=3=𝑘1=𝑘2𝐿2[0,𝑇].

Corollary 4.10. Refer to Theorem 3.3 in [5].

Proof. In our Theorem 4.6, simply choose 1=2=3=𝑘1=𝑘2𝐿[0,𝑇].

Corollary 4.11. Refer to Theorem 3.3 in [3].

Proof. In our Theorem 4.6, simply choose 1=2=3=𝑘1=𝑘21.

Corollary 4.12. Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in [4].

Proof. In our Lemma 4.5 and Theorem 4.6, simply choose 1=2=3=𝑘1=𝑘21.

Acknowledgment

The present research was conducted by the research fund of Dankook University in 2010.

References

  1. J. E. Bearman, “Rotations in the product of two Wiener spaces,” Proceedings of the American Mathematical Society, vol. 3, no. 1, pp. 129–137, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. H. Cameron and D. A. Storvick, “An operator valued Yeh-Wiener integral, and a Wiener integral equation,” Indiana University Mathematics Journal, vol. 25, no. 3, pp. 235–258, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. T. Huffman, C. Park, and D. Skoug, “Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals,” The Michigan Mathematical Journal, vol. 43, no. 2, pp. 247–261, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. T. Huffman, C. Park, and D. Skoug, “Convolution and Fourier-Feynman transforms,” The Rocky Mountain Journal of Mathematics, vol. 27, no. 3, pp. 827–841, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. Huffman, C. Park, and D. Skoug, “Generalized transforms and convolutions,” International Journal of Mathematics and Mathematical Sciences, vol. 20, no. 1, pp. 19–32, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. Park and D. Skoug, “Conditional Fourier-Feynman transforms and conditional convolution products,” Journal of the Korean Mathematical Society, vol. 38, no. 1, pp. 61–76, 2001. View at Zentralblatt MATH
  7. R. E. A. C. Paley, N. Wiener, and A. Zygmund, “Notes on random functions,” Mathematische Zeitschrift, vol. 37, no. 1, pp. 647–668, 1933. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. G. W. Johnson and D. L. Skoug, “Scale-invariant measurability in Wiener space,” Pacific Journal of Mathematics, vol. 83, no. 1, pp. 157–176, 1979. View at Zentralblatt MATH
  9. C. Park and D. Skoug, “A Kac-Feynman integral equation for conditional Wiener integrals,” Journal of Integral Equations and Applications, vol. 3, no. 3, pp. 411–427, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. D. M. Chung, C. Park, and D. Skoug, “Generalized Feynman integrals via conditional Feynman integrals,” The Michigan Mathematical Journal, vol. 40, no. 2, pp. 377–391, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. H. S. Chung and V. K. Tuan, “Generalized integral transforms and convolution products on function space,” Integral Transforms and Special Functions, vol. 22, no. 8, pp. 573–586, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. R. H. Cameron and D. A. Storvick, “An L2 analytic Fourier-Feynman transform,” The Michigan Mathematical Journal, vol. 23, no. 1, pp. 1–30, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. W. Johnson and D. L. Skoug, “An Lp analytic Fourier-Feynman transform,” The Michigan Mathematical Journal, vol. 26, no. 1, pp. 103–127, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. C. Park, D. Skoug, and D. Storvick, “Relationships among the first variation, the convolution product and the Fourier-Feynman transform,” The Rocky Mountain Journal of Mathematics, vol. 28, no. 4, pp. 1447–1468, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song, and I. Yoo, “Relationships involving generalized Fourier-Feynman transform, convolution and first variation,” Integral Transforms and Special Functions, vol. 16, no. 5-6, pp. 391–405, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. H. Cameron and D. A. Storvick, “Some Banach algebras of analytic Feynman integrable functionals,” in Analytic Functions, Kozubnik 1979, A. Dold and B. Eckmann, Eds., vol. 798 of Lecture Notes in Mathematics,, pp. 18–67, Springer, Berlin, Germany, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH