CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE

Title & Authors
CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE
Cho, Dong Hyun;

Abstract
Let C[0, $\small{t}$] denote the function space of real-valued continuous paths on [0, $\small{t}$]. Define $\small{X_n\;:\;C[0,t}$$\small{]}$$\small{{\rightarrow}\mathbb{R}^{n+1}}$ and $\small{X_{n+1}\;:\;C[0,t}$$\small{]}$$\small{{\rightarrow}\mathbb{R}^{n+2}}$ by $\small{X_n(x)=(x(t_0),x(t_1),{\ldots},x(t_n))}$ and $\small{X_{n+1}(x)=(x(t_0),x(t_1),{\ldots},x(t_n),x(t_{n+1}))}$, respectively, where $0=t_0 <; t_1 <{\ldots} < t_n < t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $\small{X_n}$ and $\small{X_{n+1}}$, we evaluate the $\small{L_p(1{\leq}p{\leq}{\infty})}$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form $\small{fr((v_1,x),{\ldots},(v_r,x)){\int}_{L_2}_{[0,t]}\exp\{i(v,x)\}d{\sigma}(v)}$ for $\small{x{\in}C[0,t]}$, where $\small{\{v_1,{\ldots},v_r\}}$ is an orthonormal subset of $L_2[0,t]$, $\small{f_r{\in}L_p(\mathbb{R}^r)}$, and $\small{{\sigma}}$ is the complex Borel measure of bounded variation on $\small{L_2[0,t]}$. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.
Keywords
analogue of Wiener space;analytic conditional Feynman integral;analytic conditional Fourier-Feynman transform;analytic conditional Wiener integral;conditional convolution product;Wiener space;
Language
English
Cited by
1.
A TRANSLATION THEOREM FOR THE GENERALIZED FOURIER-FEYNMAN TRANSFORM ASSOCIATED WITH GAUSSIAN PROCESS ON FUNCTION SPACE, Journal of the Korean Mathematical Society, 2016, 53, 5, 991
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