Some effect of drift of the generalized Brownian motion process II: Pseudo homomorphism structures of the transforms on function space

Abstract

In this paper, we analyze pseudo homomorphism structures of the generalized Fourier--Feynman transform (GFFT) on the function space $C_{a,b}[0,T]$
which is induced by a generalized Brownian motion process (GBMP). The Fourier--Feynman transform (FFT) of functionals on classical Wiener space
satisfies a homomorphism property with a convolution product (CP). Consequently speaking, the GFFTs have no heuristic homomorphism structure
with their CP, because the stochastic process defining the GFFT and the CP in this paper is not centered. In order to develop the structure, we modified
the definition of the GFFT. We then proceed to investigate pseudo homomorphism structures between the transform and the convolution.