CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE.

Let Cr [0, t] be the function space of the vector-valued continuous paths x : [0, t]→Rr and define Xt : Cr [0; t]→R(n+1)r by Xt (x) = (x(0), x(t1), … , x(tn)), where 0 < t1 < ⋯ < tn = t. In this paper, using a simple formula for the conditional expectations of the functions on Cr [0, t] given Xt , we evaluate the conditional analytic Feynman integral Ean fq [Ft ∣Xt ] of Ft given by (These character(s) cannot be represented in ASCII text), where θ(s, ∙) are the Fourier-Stieltjes transforms of the complex Borel measures on Rr, and provide an inversion formula for Ean fq [Ft ∣Xt ]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Ean fq [Ft ∣Xt ] and a probability distribution on Rr when Xt (x) = (x(0), x(t)). [ABSTRACT FROM AUTHOR]