An evaluation formula for a generalized conditional expectation over paths in abstract Wiener space and its applications

Abstract

Let $C^{\mathbb B}[a,b]$ denote an analogue of Weiner space over paths in abstract Wiener space $\mathbb B$, the space of $\mathbb B$-valued continuous functions on $[a,b]$. Let $W(x)=\{x(t_n)\}_{n=0}^\infty\in \mathbb B^{\aleph_0}$ for $x\in C^{\mathbb B}[a,b]$, where $\{t_n\}_{n=0}^\infty$ is a strictly increasing sequence in $[a,b]$ with $t_0=a$ and $\lim_{n\to\infty}t_n=b$. In this paper, we introduce a positive finite measure with a scale on $C^{\mathbb B}[a,b]$ which is a generalized analogue of Wiener measure. Then we extend the time integral (Riemann integral) to the Riemann-Stieltjes integral which is more generalized time integral. We also derive a simple evaluation formula for calculating Radon-Nikodym derivatives similar to the conditional Wiener integrals of functions on $C^{\mathbb B}[a,b]$ given $W$ which has an initial weight. As applications of the formula, we evaluate the derivatives of various functions containing the generalized time integral which is interested in quantum mechanics, especially in Feynman integration theory.