The Laplace method for Gaussian measures and integrals in Banach spaces

We prove results on tight asymptotics of probabilities and integrals of the form P-A(uD) and J(u)(D) = integral(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 is an element of 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 ->infinity is a large parameter.