Some asymptotic formulas for Wiener integrals

October 2. Some asymptotic formulas for Wiener integrals.The main results of this paper are : Theorem A(3).Suppose F(z) and G(z) are two real valued functionals defined on C[0, T], and suppose that the functional F(z) -\ ¡T0 [z'(t)]2 dr has a proper maximum at x over C*[0, T].If F(z) and G(z) satisfy conditions 1-5 below, then E?{G()exp(\-2F())} _ ÍÜS E0>{exp(\-2F())} ~U{X)-Conditions 1-5 are: 1. G(z) and F(z) are Wiener measurable.2. (z)\ ^ K1 exp (K2\\\2) almost everywhere with respect to Wiener measure where K1 and K2 are any positive real numbers.3. F(z) í¡ Lx + L21| z ||2 almost everywhere with respect to Wiener measure where Lj is any positive real number but L2 < T/4. 4. G(z) is continuous at x in the sup norm. 5. F(z) is continuous for \\\ < |(é-£1+l)/(2La-l/2r)|1/a> and upper semicontinuous almost everywhere in the sup norm.For convenience the following notation is introduced.If _8KF(x + v)_ d(x(s1) + tjCji)) • • • 8(x(sK) + 7¡(sK))is the Kth Frechet derivative of the functional F(z) at the point x + r¡ (r¡ e C[0, T]), then the expressionwill be denoted by fK(v)zK-Theorem B. Suppose F(z) and G(z) are defined on C[0, T] and satisfy the hypothesis of Theorem A, and suppose also, that 1. F(z) has two continuous Frechet derivatives in a uniform neighborhood of the maximizing function x.2. Ez{exp ((1 + e)f2(q)z2)} is uniformly bounded for r¡inao uniform neighborhood of 0, for some e > 0.(3) See definitions in the Introduction.