Mean Sojourn Time Problems and Eigenvalue Asymptotics of the Laplacian

Abstract

In this thesis, the theory of pseudo-differential operators, an element of microlocal analysis is used in order to construct explicit Greens function expansions in a variety of standard geometric settings in order to derive mean sojourn time asymptotics for a test Brownian motion. In addition, we apply these Greens function expansions to derive a spectral asymptotic result on the variation of Neumann eigenvalues when a geometrically small Dirichlet boundary perturbation is introduced. What appears ubiquitously throughout our results are special geometric quantities such as the mean and principal curvatures, as well as the geodesic distance and volumes of the geometries in question.