Degenerate Diffusion Operators Arising in Population Biology

TL;DR

This paper studies degenerate elliptic PDEs from population genetics and finance, establishing existence, uniqueness, and regularity of solutions, and analyzing the associated stochastic processes on manifolds with corners.

Contribution

It introduces anisotropic Hölder spaces and proves fundamental properties of these degenerate operators, including semigroup generation and martingale problem solutions.

Findings

Existence and uniqueness of solutions to heat and resolvent equations.

Regularity results for solutions in anisotropic Hölder spaces.

Characterization of the nullspace of the forward Kolmogorov operator.

Abstract

We analyze a class of partial differential equations that arise as "backwards Kolmogorov operators" in infinite population limits of the Wright-Fisher models in population genetics and in mathematical finance. These are degenerate elliptic operators defined on manifolds with corners. The classical example is the Kimura diffusion operator, which acts on functions defined on the simplex in R^n. We introduce anisotropic Holder spaces, and prove existence, uniqueness and regularity results for the heat and resolvent equations defined by this class of operators. This suffices to prove that the C^0-graph closure generates a strongly continuous semigroup, and that the associated Martingale problem has a unique solution. We give a detailed description of the nullspace of the forward Kolmogorov operator.