Communications of the
Korean Mathematical Society CKMS

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

MSC numbers: Primary 46G12; Secondary 28C20, 60J65