Transition probabilities for degenerate diffusions arising in population genetics

TL;DR

This paper analyzes the transition probabilities and boundary hitting distributions of a generalized Wright-Fisher process, a degenerate Markov process relevant in population genetics, using advanced PDE regularity techniques.

Contribution

It provides a detailed description of transition probabilities and hitting distributions for a class of degenerate diffusions on manifolds with corners, extending classical models.

Findings

Characterization of transition probabilities for degenerate processes

Analysis of boundary hitting distributions in complex geometries

Regularity results for solutions to degenerate Kolmogorov equations

Abstract

We provide a detailed description of the structure of the transition probabilities and of the hitting distributions of boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright-Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.