On the Laplace method for Gaussian measures in a Banach space

In this paper, we prove results on sharp asymptotics for the probabilities $P_A(uD)$, as $u\to\infty$, where $P_A$ is the Gaussian measure in an infinite-dimensional Banach space $B$ with zero mean and nondegenerate covariance operator $A$, $D=\{x\in B:Q(x)\geqslant 0\}$ is a Borel set in $B$, and $Q$ is a smooth function. We analyze the case where the action functional attains its minimum on some set $D$ on a one-dimensional manifold. We make use of the Laplace method in Banach spaces for Gaussian measures. Based on the general result obtained, for $0<p\leqslant6$ we find a sharp asymptotics for large deviations of distributions of $L^p$-functionals for the centered Brownian bridge which arises as the limit while studying the Watson statistics. Explicit constants are given for the cases $p=1$ and $p=2$.