Date of Award

8-1-2019

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Hughes, Harry

Abstract

This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.

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