The Laplace method for Gaussian measures and integrals in Banach spaces

Abstract

We prove results on tight asymptotics of probabilities and integrals of the form $P_A (uD)andJ_u (D) = \int\limits_D {f(x)\exp \{ - u^2 F(x)\} dP_A (ux)} $ , where P A is a Gaussian measure in an infinite-dimensional Banach space B, D = {x ? B: Q(x) ? 0} is a Borel set in B, Q and F are continuous functions which are smooth in neighborhoods of minimum points of the rate function, f is a continuous real-valued function, and u?? is a large parameter.