$C^0$-estimates and smoothness of solutions to the parabolic equation defined by Kimura operators

TL;DR

This paper establishes $C^0$-estimates for solutions to Kimura diffusion equations, proving their smoothness up to the boundary despite degeneracies, even with only continuous initial data.

Contribution

It provides new $C^0$-estimates and demonstrates boundary smoothness of solutions to Kimura operators, advancing understanding of degenerate elliptic PDEs in population genetics.

Findings

Solutions are smooth up to the boundary where the operator degenerates.

$C^0$-estimates enable boundary regularity results.

Solutions remain well-behaved even with continuous initial data.

Abstract

Kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics. Their infinitesimal generator is an elliptic differential operator whose second-order coefficients matrix degenerates on the boundary of the domain. In this article, we consider the inhomogeneous initial-value problem defined by generators of Kimura diffusions, and we establish $C^0$-estimates, which allows us to prove that solutions to the inhomogeneous initial-value problem are smooth up to the boundary of the domain where the operator degenerates, even when the initial data is only assumed to be continuous.