Heat Equation and Brownian Motion on Riemannian Manifolds

Abstract

Riemannian geometry is used in many theoretical models such as general relativity. Theoretical physicists and mathematicians have researched this extensively to describe the behaviour of particles and interactions in spacetime. These interactions include solutions to the heat equation and Brownian motion. In this thesis, we give an elaborate theoretical description of differential geometry starting from basic topology and expand it to develop a thorough conceptualization of Riemannian geometry including geodesics and Riemannian distance. We formalize the heat equation on compact Riemannian manifolds and show the existence and uniqueness of the solution that commences as a point source called the fundamental solution and derive an explicit expression using the Euclidean case. In addition, we show that the transition density function of Brownian motion that determines the likelihood of terminating within a certain region on compact Riemannian manifolds is precisely given by the fundamental solution of the heat equation. Finally, we show some applications of these two results and give an overview of present and future research.