Some asymptotic formulas for Wiener integrals

Abstract

Introduction. In this paper asymptotic formulas for Wiener integrals similar to the Laplace asymptotic formulas for integrals in one dimension will be proved [5], and some applications to analysis of these formulas will be given.Let C[0, T] be the space of continuous functions defined on the interval [0, 7"] which vanish at zero, and let ||z|| be the sup norm of a function z.A functional H(z) will be said to have a proper maximum at a point x if H{z) < H(x) for z^x.Let C*[0, T]<=C[0, T] be the subset of functions on [0, T] which are absolutely continuous and whose derivatives are in 2[0, T],Suppose F(z) is a functional on C[0, T] such that (z) -\ \\ {z'(t))2 dr has a proper maximum on C*[0, T\.Call this maximizing function x(t).With further restrictions on (z)(2) but not supposing F(z) to have any Frechet derivatives, and for a large class of functionals G(z), it will be shown(2) that