The behavior of conditional Wiener integrals on product Wiener space

Abstract

Key words Conditional Wiener integral, multiple conditional analytic Feynman integral, multiple conditional analytic Fourier-Feynman transform, multiple conditional analytic convolution product, first variation, n-dimensional Wiener directional derivative MSC (2010) 28C20, 60J65In this paper we first investigate the behavior of conditional Wiener integrals on product Wiener spaces and then proceed to establish a Fubini theorem for conditional Wiener integrals. We then define a very general multiple conditional analytic Fourier-Feynman transform and a multiple conditional analytic convolution product for functionals on Wiener space and then establish a relationship between them. Next we obtain a basic formula for the conditional Wiener integral involving the first variation and the n-dimensional Wiener directional derivative. Finally we establish several evaluation formulas for conditional Wiener and Feynman integrals involving the n-dimensional Wiener directional derivative of functionals on Wiener space.

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

“…In various papers the authors have studied the analytic Fourier–Feynman transform on Wiener space. Furthermore various theories involving the Fubini theorem, the Bearman theorem and the convolution product with respect to the analytic Fourier–Feynman transform were studied , , , –, –. Recently, the authors introduced a new concept of the generalized analytic Feynman integral and the analytic Fourier–Feynman transform.…”