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