Multiple generalized analytic Fourier--Feynman transform via rotation of Gaussian paths on function space

Abstract

The main purpose of this article is to develop the generalized analytic Fourier-Feynman transform theory. We introduce a generalized analytic Fourier-Feynman transform and a multiple generalized analytic Fourier-Feynman transform with respect to Gaussian processes on the function space C a,b [0, T ] induced by a generalized Brownian motion process. We then establish a relationship between these two generalized analytic transforms.

Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$

“…On the other hand, the translation theorem for the function space integral and the generalized analytic Fourier-Feynman transform (GFFT) have been developed for the functionals on the very general function space C a,b [0, T ] in [9], [10], and [13]. The function space C a,b [0, T ], induced by the generalized Brownian motion process (GBMP), was introduced by Yeh [24], [25] and used extensively in [8], [12]- [16], and [23] (for the precise definition of GBMP, see [24] and [25]).…”

“…We then proceed to establish a general translation theorem on the product function space C 2 a,b [0, T ]. The Wiener process used in [2]- [7], [17], [18], [20], and [21] is stationary in time and free of drift, while the stochastic process used in this article, as well as in [8]- [16], [23], and [24], is nonstationary in time, subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field (see [22]).…”

Generalized transforms and generalized convolution products associated with Gaussian paths on function space

“…On the other hand, in [5,6,8], the authors introduced the FFTs on the very general function space C a,b [0, T ] (rather than the Wiener space C 0 [0, T ]), and studied their properties and related topics. The function space C a,b [0, T ], induced by a generalized Brownian motion process (GBMP), was introduced by Yeh [22,23] and was used extensively in [4,5,6,7,8,9].…”

“…The Wiener process used in [1,2,3,12,13,14,16,20] is stationary in time and is free of drift, while the Gaussian process used in [10,11,15,19] is non-stationary in time and is free of drift. However the stochastic processes used in this paper, as well as in [4,5,6,7,8,9], are non-stationary in time and are subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [17]. But We then complete this function space to obtain the complete probability measure space…”

“…Gaussian processes. In order to present our results involving the analytic GFFT and the GCP, we follow the exposition of [5].…”