Conditional Foruier-Feynman transform and convolution product for a vector valued conditioning function

Abstract

Let $C_0[0,T]$ denote the Wiener space, the space of continuous functions $x(t)$ on $[0,T]$ such that $x(0)=0$. Define a random vector $Z_{\overrightarrow{e},k}:C_0[0,T]\to {\mathbb{R}}^k$ by
$$Z_{\overrightarrow{e},k}(x)=({\int }_0^T{e}_1(t)dx(t),\dots ,{\int }_0^T{e}_k(t)dx(t))$$
where $e_j\in L_2[0,T]$ with $e_j\ne 0$ a.e., $j=1,\dots ,k$.
In this paper we study the conditional Fourier-Feynman transform and a conditional convolution product for a cylinder type functionals defined on $C_0[0,T]$ with a general vector valued conditioning functions $Z_{\overrightarrow{e},k}$ above which need not depend upon the values of $x$ at only finitely many points in $(0,T]$ rather than a conditioning function $X(x)=(x(t_1),\dots ,x(t_n))$ where $0<t_1<\dots <t_n=T$. In particular we show that the conditional Fourier-Feynman transform of the conditional convolution product is the product of conditional Fourier-Feynman transforms.