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 product function space $C_{a,b}^2[0,T]$. The function space $C_{a,b}[0,T]$ 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 ${\mathcal{F}}_{A_1,A_2}^{a,b}$ of functionals on $C_{a,b}^2[0,T]$, which is a generalization of the Kallianpur and Bromley–Fresnel class ${\mathcal{F}}_{A_1,A_2}$. We then proceed to establish the translation theorems for the functionals in ${\mathcal{F}}_{A_1,A_2}^{a,b}$.