Conditional Fourier--Feynman transform and conditional convolution product given infinite-dimensional vector-valued conditioning function

Abstract

Let $C_0[0,T]$ denote Wiener space. Define an infinite-dimensional random vector $X_{\mathcal E,\infty}: C_0[0,T]\to \mathbb R^{\infty}$ by
$X_{\mathcal E,\infty}(x)=(\langle{e_1,x}\rangle, \langle{e_2,x}\rangle,\ldots)$
where $\mathcal E =\{e_n\}_{n=1}^{\infty}$ is an orthonormal sequence in $L_2[0,T]$ and $\langle{e,x}\rangle$ denotes the Paley--Wiener--Zygmund (PWZ) stochastic integral. In this paper, we study a conditional Fourier--Feynman transform (CFFT) and a conditional convolution product (CCP) for scale-invariant measurable (SIM) functionals on $C_0[0,T]$ with the very general conditioning function $X_{\mathcal E,\infty}$. In particular, we show that the CFFT of the CCP is a product of CFFTs.