Sharp laplace asymptotics in infinite dimensions

Abstract

In this thesis we study the sharp asymptotic behavior of perturbations of Gaussian measures on infinite dimensional spaces. We first introduce Gaussian measures on fractional Sobolev spaces over the d-dimensional torus. Depending on the space dimension d, these spaces are spaces of functions or distributions. We then construct some perturbations of Gaussian measures for d=1,2. In the case of d=1, the perturbation is constructed using the Sobolev embedding. For d=2, problems arise since the measure we are interested in can only be defined on spaces of distributions. We remedy this by what is called renormalization procedure in literature. Key steps are the hypercontractivity estimate for the Ornstein-Uhlenbeck semigroup and Nelson's estimate. We finally determine the sharp asymptotic behavior of the perturbations constructed.