Conditional Integral Transforms on a Function Space

• Journal title : Kyungpook mathematical journal
• Volume 52, Issue 4,  2012, pp.413-431
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2012.52.4.413
Title & Authors
Conditional Integral Transforms on a Function Space
Cho, Dong Hyun;

Abstract
Let $\small{C^r[0,t]}$ be the function space of the vector-valued continuous paths $\small{x:[0,t]{\rightarrow}\mathbb{R}^r}$ and define $\small{X_t:C^r[0,t]{\rightarrow}\mathbb{R}^{(n+1)r}}$ and $\small{Y_t:C^r[0,t]{\rightarrow}\mathbb{R}^{nr}}$ by $\small{X_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}),\;x(t_n))}$ and $\small{Y_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}))}$, respectively, where $\small{0=t_0}$ < $\small{t_1}$ < $\small{{\cdots}}$ < $\small{t_n=t}$. In the present paper, using two simple formulas for the conditional expectations over $\small{C^r[0,t]}$ with the conditioning functions $\small{X_t}$ and $\small{Y_t}$, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form $\small{{\exp}\{{\int_o}^t{\theta}(s,\;x(s))\;d{\eta}(s)\}{\psi}(x(t)),\;x{\in}C^r[0,t]}$ where $\small{{\eta}}$ is a complex Borel measure on [0, t] and both $\small{{\theta}(s,{\cdot})}$ and $\small{{\psi}}$ are the Fourier-Stieltjes transforms of the complex Borel measures on $\small{\mathbb{R}^r}$.
Keywords
Analogue of Wiener measure;Conditional Feynman integral;Conditional Fourier-Feynman transform;Conditional Wiener integral;Simple formula for conditional Wiener integral;
Language
English
Cited by
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