Translation Theorems for the Fourier-Feynman Transform on the Product Function Space C2 a,b, [0,T]

Authors

Seung Jun Chang, Dankook UniversityFollow
Jae Gil Choi, Dankook UniversityFollow
David Skoug, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

11-6-2018

Citation

2019 by the Tusi Mathematical Research Group

Comments

Banach J. Math. Anal. 13 (2019), no. 1, 192{216 https://doi.org/10.1215/17358787-2018-0022 ISSN: 1735-8787 (electronic) http://projecteuclid.org/bjma

Abstract

In this article, we establish the Cameron{Martin translation theo- rems for the analytic Fourier{Feynman transform of functionals on the product function space C2 a;b[0; T]. The function space Ca;b[0; T] is induced by the gener- alized 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 Fa;b A1;A2 of functionals on C2 a;b[0; T], which is a generaliza- tion of the Kallianpur and Bromley{Fresnel class FA1;A2 . We then proceed to establish the translation theorems for the functionals in Fa;b A1;A2 .