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

Abstract

Abstract. In this paper we define a generalized analytic Fourier-Feynman transform associated with Gaussian process on the function space C a,b [0, T ]. We establish the existence of the generalized analytic FourierFeynman transform for certain bounded functionals on C a,b [0, T ]. We then proceed to establish a translation theorem for the generalized transform associated with Gaussian process.

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]).…”

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

“…On the other hand, in [5][6][7]9,14,15], the authors defined the generalized analytic Feynman integral and the generalized analytic Fourier-Feynman transform on the function space C a,b [0, T ], and studied their properties and related topics. The function space C a,b [0, T ], induced by a GBMP, was introduced by Yeh in [30], and was used extensively in [4,8,[10][11][12][13]20].…”

“…In this note, we set c = a(0) = b(0) = 0. Then the function space C a,b [0, T ] induced by the GBMP Y determined by the a(·) and b(·) can be considered as the space of continuous sample paths of Y , see [4][5][6][7][8][9][10][11][12][13][14][15][16]20], and one can see that for each t ∈ [0, T ],…”

“…However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”