Conditional integral transforms of functionals on a function space of two variables

Abstract

Let $C(Q)$ denote Yeh-Wiener space, the space of all real-valued continuous functions $x(s,t)$ on $Q\equiv [0,S]\times[0,T]$ with $x(s,0)=x(0,t)=0$ for every $(s, t) \in Q$. For each partition $\tau=\tau_{m,n} =\{ (s_i , t_j ) | i=1, \ldots , m, j=1, \ldots , n \}$ of $Q$ with $0=s_0 < s_1 < \ldots <s_m =S$ and $0= t_0 <t_1 < \ldots <t_n =T$, define a random vector $X_{\tau} : C(Q) \rightarrow {\mathbb R}^{mn}$ by $$ X_{\tau} (x) =(x(s_1 , t_1), \ldots,x(s_m , t_n ) ).$$

In this paper we study the conditional integral transform and the conditional convolution product for a class of cylinder type functionals defined on $K(Q) $ with a given conditioning function $X_{\tau}$ above, where $K(Q) $is the space of all complex valued continuous functions of two variables on $Q$ which satify $x(s, 0)=x(0, t) =0$ for every $(s, t) \in Q$. In particular we derive a useful equation which allows to calculate the conditional integral transform of the conditional convolution product without ever actually calculating convolution product or conditional convolution product.