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

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 , ∞ ) n × R m [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 at a rate 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 solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.