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

Abstract

In this article, we establish the Cameron-Martin translation theorems for the analytic Fourier-Feynman transform of functionals on the productT ] is induced by the generalized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0, T ]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class F a,b A1,A2 of functionals on C 2 a,b [0, T ], which is a generalization of the Kallianpur and Bromley-Fresnel class F A1,A2 . We then proceed to establish the translation theorems for the functionals in F a,b A1,A2 .