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

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