Dong Hyun Cho, Department of Mathematics, Kyonggi University, Kyonggido Suwon 443-760, Korea, e-mail: j94385@kyonggi.ac.kr
Abstract: Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi$ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots< t_n <t_{n+1}= T$ of $[0, T]$, let $X_n C[0,T]\to\mathbb R^{n+1}$ and $X_{n+1} C [0, T]\to\mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots, x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots, x(t_{n+1}))$, respectively. In this paper, using a simple formula for the conditional $w_\varphi$-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi$-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi$-integral of the functions of the form $F_m(x) = \int_0^T (x(t))^m \dd t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi$-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi}$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi$-integrals of the functions in $\mathcal S_{w_\varphi}$.
Keywords: analogue of Wiener measure, Cameron-Martin translation theorem, conditional analytic Feynman $w_\varphi$-integral, conditional Wiener integral, Kac-Feynman formula, simple formula for conditional $w_\varphi$-integral
Classification (MSC 2000): 28C20
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