Degenerate Diffusions on Manifolds with Corners

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Models for many problems in Biology and Economics involve the evolution of a collection of variables that are constrained to lie in a domain of Euclidean space bounded by a collection of hypersurfaces. For example the value of a security, or its variance might be constrained to be non-negative, whereas the frequency of a gene must lie between 0 and 1. The actual path that the value of a particular security, or the prevalence of an allele in a single population follow is very complicated and, indeed, fundamentally unpredictable. Collectively these paths constitute what is called a stochastic process. While the time course of a single path is unpredictable, the statistical properties of the whole family of possible paths can often be shown to satisfy partial differential equations, which allow for their detailed analysis. The main goal of Dr. Epstein's research is to understand the solutions of the types of equations that arise in these contexts. This is challenging because the equations display degeneracies connected to the fact that the paths are constrained to lie in certain regions of space, like triangles and tetrahedra, which themselves have non-smooth boundaries. Dr. Epstein is trying to develop a detailed enough understanding of these equations to develop incisive numerical tools for use by biologists and population geneticists. A principal focus of Dr. Epstein's research is to understand the detailed analytic properties of the solutions to Kimura diffusion equations, which arise as limiting cases of Wright-Fisher Markov models. Working jointly with collaborators for the past seven years, he has established analytic foundations for this natural class of equations, on a natural class of domains. In their recent work it became clear that the class of operators needed to be expanded to include equations with somewhat singular coefficients, which are needed for the analysis of problems that arise in more realistic applications to Probability, Mathematical Finance and Population Biology. Much of the work proposed herein deals with developing analytic tools to address these sorts of real world problems. This will include the analysis of such quantities as the heat kernel, stationary measures, probabilities of fixation, and times to fixation. Beyond the abstract analytic work, Dr. Epstein will also develop numerical algorithms to accurately solve Kimura diffusion equations and the associated elliptic problems that arise in the study of statistical property of such processes in cases of genuine applied interest. Dr. Epstein will also pursue the analytic aspects of stable, accurate numerical methods for solving time-dependent problems in electromagnetics. This involves finding novel representations of solutions to the wave equation and full Maxwell equations, which in turn lead to numerical methods with better accuracy and stability properties than pre-existing approaches.