The Laplace method for Gaussian measures and integrals in Banach spaces

We prove results on tight asymptotics of probabilities and integrals of the form
$$ \mathbf P_A(uD)\quad\text{and}\quad J_u(D)=\int_D f(x)\exp\{-u^2F(x)\}\,d \mathbf P_A(ux), $$
where $\mathbf P_A$ is a Gaussian measure in an infinite-dimensional Banach space $B$, $D=\{x\in B\colon Q(x)\ge0\}$ 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 $u\to\infty$ is a large parameter.