Some effect of drift of the generalized Brownian motion process: Existence of the operator-valued generalized Feynman integral

Abstract

In this paper an analytic operator-valued generalized Feynman integral was studied on a very general Wiener space Ca,b[0, T]. The general Wiener space Ca,b[0, T] is a function space which is induced by the generalized Brownian motion process associated with continuous functions a and b. The structure of the analytic operator-valued generalized Feynman integral is suggested and the existence of the analytic operator-valued generalized Feynman integral is investigated as an operator from L1(R, ??,a) to L?(R) where ??,a is a ?-finite measure on R given by d??,a = exp{?Var(a)u2}du, where ? > 0 and Var(a) denotes the total variation of the mean function a of the generalized Brownian motion process. It turns out in this paper that the analytic operator-valued generalized Feynman integrals of functionals defined by the stochastic Fourier-Stieltjes transform of complex measures on the infinite dimensional Hilbert space C?a,b[0, T] are elements of the linear space ? ?>0 L(L1(R, ??,a), L?(R)).