A Cameron--Storvick theorem on $C_{a,b}^2[0,T]$ with applications

Commun. Korean Math. Soc. 2021 Vol. 36, No. 4, 685-704 https://doi.org/10.4134/CKMS.c200297 Published online March 10, 2021 Printed October 31, 2021

Jae Gil Choi, David Skoug Dankook University; University of Nebraska-Lincoln

Abstract : The purpose of this paper is to establish a very general Cameron--Storvick theorem involving the generalized analytic Feynman integral of functionals on the product function space $C_{a,b}^2[0,T]$. The function space $C_{a,b}[0,T]$ can be induced by the generalized Brownian motion process associated with continuous functions $a$ and $b$. To do this we first introduce the 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 a Cameron--Storvick theorem on the product function space $C_{a,b}^2[0,T]$. Finally we use our Cameron--Storvick theorem to obtain several meaningful results and examples.

Keywords : Generalized analytic Feynman integral, product function space generalized Brownian motion process, Kallianpur and Bromley Fresnel class, Cameron--Storvick theorem