Degenerate Diffusion Operators Arising in Population Biology

By Charles L. Epstein, Rafe Mazzeo. 2011

110 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/51d39a82cd501c1a427bb56628f66b695c2fe032

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In Mathematical Physics

In Modeling and Simulation

Citations in the corpus

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions (2019, background): “[3] for a very much related model) play a crucial role.”
Finite state mean field games with Wright–Fisher common noise (2019, methodology): “[27] C. L. Epstein and R. Mazzeo.”