Degenerate Diffusions on Manifolds with Corners

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Dr. Epstein's work concerns the analysis of Markov Processes that arise as infinite population limits of Wright-Fisher-like Markov chains. Such models also arise in Epidemiology and Mathematical Finance. These models are diffusion-like processes, but with variables that are constrained to lie in certain regions of space, e.g. the non-negative orthant for a Finance problem, or a simplex for a Genetics problem. Hence the paths of these Markov processes must also remain within the feasible part of space. This forces the generator to be an elliptic operator whose leading part degenerates along the boundary. A distinguishing feature of these processes is that, in the absence of an outside force, like mutation, a typical path reaches the boundary of the feasible region in finite time, but cannot cross it. This implies that the degeneracies are rather different from any that have heretofore been successfully analyzed. While a great deal is known about some classes of degenerate operators, very little is known either about the class of operators that arise here, or about degenerate PDEs on domains with boundaries as singular as that of a simplex or other polyhedra. A principal focus of Dr. Epstein's research is to understand the detailed analytic properties of the solutions to equations of this type. His recent work, with Rafe Mazzeo, establishes the existence and uniqueness of regular solutions for this natural class of equations, on a natural class of domains, and lays the foundation for a detailed study of the qualitative properties of these models. In applications one needs to solve equations that these prior results show cannot have regular solutions. Thus Dr. Epstein will now turn his attention to the analysis of the singular solutions that arise in applications to Probability, Mathematical Finance and Population Biology, etc. Amongst other things, this will entail an elaboration of the functional analytic framework used to analyze regular solutions, to address the singularities that arise in these applications. Combining these analytic techniques with methods used in Probability Theory, he hopes to precisely describe the structure of the heat kernel itself near to time zero, and along the boundaries of the feasible region, where it can exhibit various types of singularities.Mathematical models for many problems in Population Biology, Epidemiology and Finance involve the time evolution of a collection of discrete variables under both random and deterministic forces. These variables are frequently constrained: for example a population with a given genotype, or the number of people infected with a pathogen must be a non-negative integer, and the value of a stock is usually assumed to be non-negative. The dynamical behavior of such models is often described by a Markov chain. These are discrete models with "no history," meaning that the statistical properties of the current generation determine those of the next. Discrete models of this type are difficult to directly analyze, and so they are often replaced by continuum limits that are described by partial differential equations, which is also the language of classical and quantum physics. While much work has been done to study such models when there is a single variable, the purpose of this project is to develop analytic and computational tools to study the qualitative behavior of solutions to these equations when there are many, possibly interacting, variables. In applications to Population Genetics, such tools can be used to understand when random aspects of reproduction dominate the evolution of a population, and when deterministic forces like natural selection dominate. The investigator hopes to unravel the effects of fitness interactions among a small group mutations on the evolution of simple organisms. Similar methods could be applied to study the early stages of an epidemic, when it might be possible to make a small intervention that could control the growth of the infected population.