Applications of probability to partial differential equations and infinite dimensional analysis

Abstract

This thesis consists of two parts. The first part applies a probabilistic approach to the study of the Wright-Fisher equation, an equation which is used to model demographic evolution in the presence of diffusion. The fundamental solution to the Wright-Fisher equation is carefully analyzed by relating it to the fundamental solution to a model equation which has the same degeneracy at one boundary. Estimates are given for short time behavior of the fundamental solution as well as its derivatives near the boundary. The second part studies the probabilistic extensions of the classical Cauchy functional equation for additive functions both in finite and infinite dimensions. The connection between additivity and linearity is explored under different circumstances, and the techniques developed in the process lead to results about the structure of abstract Wiener spaces. Both parts are joint work with Daniel W. Stroock.