Existence, uniqueness and the strong Markov property of solutions to Kimura diffusions with singular drift

TL;DR

This paper investigates the existence, uniqueness, and strong Markov property of solutions to degenerate Kimura stochastic differential equations with singular drift, motivated by population genetics applications.

Contribution

It establishes conditions for existence, uniqueness, and the Markov property of weak solutions to a class of degenerate SDEs with singular boundary behavior.

Findings

Proved existence of weak solutions under regularity assumptions.

Established uniqueness in law for solutions.

Demonstrated the strong Markov property for solutions.

Abstract

Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set $[0,\infty)^n\times\mathbb{R}^m$, and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero proportional to the distance to the boundary of the domain, and the drift coefficients are allowed to have power-type singularities in a neighborhood of the boundary of the domain. Under suitable regularity assumptions on the coefficients, we establish existence of weak solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.