CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS OF BOUNDED FUNCTIONS ON AN ANALOGUE OF WIENER SPACE

Abstract

Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define Xn : C[0, t] → Rn+1 and Xn+1 : C[0, t] → Rn+2 by Xn(x) = (x(t0), x(t1), · · · , x(tn)) and Xn+1(x) = (x(t0), x(t1), · · · , x(tn), x(tn+1)), where 0 = t0 < t1 < · · · < tn < tn+1 = t. In the present paper, using simple formulas for the conditional expectations with the conditioning functions Xn and Xn+1, we evaluate the Lp(1 ≤ p ≤ ∞)-analytic conditional FourierFeynman transforms and the conditional convolution products of the functions which have the form

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