Feynman integral and a change of scale formula about the first variation and a Fourier Stieltjes transform

Abstract

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x) = exp{integral(T)(0) theta(t,x(t))dt} successfully exist under the certain condition, where theta(t,u) = integral(R) exp{iuv}d sigma(t)(v) is a Fourier-Stieltjes transform of a complex Borel measure sigma(t)is an element of M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.