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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
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
Summary:
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. \endgraf 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 $F_m(x) = \int _0^T (x(t))^m d t$ 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 $\mathcal S_{w_\varphi }$.
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