Conditional Fourier-Feynman Transform and Convolution Product Over Wiener Paths In Abstract Wiener Space

Abstract

In this paper, we define the conditional Fourier-Feynman transform and the conditional convolution product over Wiener paths in abstract Wiener space. Using a simple formula, we obtain conditional Feynman integrals of Fourier-Feynman transform and convolution product of cylinder type functions. For these functions, we evaluate the conditional Fourier-Feynman transforms and the conditional convolution products, and show that the conditional Fourier-Feynman transform of the conditional convolution product is a product of the conditional Fourier-Feynman transforms.

A Time-Independent Conditional Fourier-Feynman Transform and Convolution Product on an Analogue of Wiener Space

“…In that paper, they also examined the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them. Further works were studied by Chang, Cho, Kim, Song and Yoo [3,8]. In fact, Cho and his coauthors [3] introduced the L 1 -analytic conditional Fourier-Feynman transform and the conditional convolution product over Wiener paths in abstract Wiener space and then, established their relationships between them of certain cylinder type functions.…”

“…Further works were studied by Chang, Cho, Kim, Song and Yoo [3,8]. In fact, Cho and his coauthors [3] introduced the L 1 -analytic conditional Fourier-Feynman transform and the conditional convolution product over Wiener paths in abstract Wiener space and then, established their relationships between them of certain cylinder type functions. Cho [8] extended the relationships between the conditional convolution product and the L p (1 ≤ p ≤ 2)-analytic conditional Fourier-Feynman transform of the same kind of cylinder functions.…”

“…In this paper, we further develop the relationships in [3,5,6,8,14] on the more generalized space (C[0, t], w ϕ ), the analogue of the Wiener space associated with the probability measure ϕ on the Borel class B(R) of R [12,16,17]. For the conditioning function X n : C[0, t] → R n+1 given by X n (x) = (x(t 0 ), x(t 1 ), · · · , x(t n )) which is independent of the present time t, we proceed to study the relationships between the conditional convolution product and the analytic conditional FourierFeynman transform of the cylinder functions defined on C[0, t].…”

Conditional Fourier-Feynman Transforms and Convolutions of Unbounded Functions on a Generalized Wiener Space

“…In those papers, they examined the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them. Further works were produced by Chang, Kim, Skoug, Song, Yoo and the author of [3,13,19]. In fact, they [3] introduced the L 1 -analytic conditional FourierFeynman transform and the conditional convolution product over Wiener paths in abstract Wiener space and established the relationships between the transform and convolutions of certain functions similar to cylinder functions.…”

A Translation Theorem for the Generalized Fourier-Feynman Transform Associated With Gaussian Process on Function Space

“…For an elementary introduction to the analytic Fourier-Feynman transform, see [29] and the references cited therein. Various kinds of the study for the analytic FourierFeynman transform and related topics were developed on abstract Wiener space [1,2,11,12,13,25], space of abstract Wiener space valued continuous functions on compact interval in R [8,9,10,17,18,19], and the analogue of Wiener space [20,28].…”