CONDITIONAL INTEGRAL TRANSFORMS OF FUNCTIONALS ON A FUNCTION SPACE OF TWO VARIABLES

Let C(Q) denote Yeh-Wiener space, the space of all real-valued con-tinuous functions x(s, t) on Q equivalent to [0, S] x [0, T] with x(s, 0) = x(0, t) = 0 for every (s, t) E Q. For each partition tau = tau m,n = {(si, tj)|i = 1, ... , m, j = 1, ... , n} of Q with 0 = s0 < s1 < ... < sm = S and 0 = t0 < t1 < ... < tn = T, define a random vector X tau : C(Q) -> Rmn by X tau(x) = (x(s1, t1), ... , x(sm, tn)). In this paper we study the conditional integral transform and the conditional convolution product for a class of cylinder type functionals defined on K(Q) with a given conditioning function X tau above, where K(Q)is the space of all complex valued continuous functions of two variables on Q which satify x(s, 0) = x(0, t) = 0 for every (s, t) E Q. In particular we derive a useful equation which allows to calculate the conditional integral transform of the conditional convolution product without ever actually calculating convolution product or conditional convolution product.