The Laplace method for Gaussian measures and integrals in Banach spaces

We prove results on tight asymptotics of probabilities and integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_A (uD)andJ_u (D) = \int\limits_D {f(x)\exp \{ - u^2 F(x)\} dP_A (ux)} $\end{document}, where PA 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.