A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE

Abstract

Let <TEX>$C[0,t]$</TEX> denote the function space of all real-valued continuous paths on <TEX>$[0,t]$</TEX>. Define <TEX>$X_n:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$</TEX> by <TEX>$Xn(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$</TEX>, where <TEX>$0=t_0$</TEX> < <TEX>$t_1$</TEX> < <TEX>${\cdots}$</TEX> < <TEX>$t_n$</TEX> < <TEX>$t$</TEX> is a partition of <TEX>$[0,t]$</TEX>. In the present paper, using a simple formula for the conditional expectation given the conditioning function <TEX>$X_n$</TEX>, we evaluate the <TEX>$L_p(1{\leq}p{\leq}{\infty})$</TEX>-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form <TEX>$$f((v_1,x),{\cdots},(v_r,x))\;for\;x{\in}C[0,t]$$</TEX>, where {<TEX>$v_1,{\cdots},v_r$</TEX>} is an orthonormal subset of <TEX>$L_2[0,t]$</TEX> and <TEX>$f{\in}L_p(\mathbb{R}^r)$</TEX>. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.