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

Camelia A. Pop

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arXiv · 1406.0742

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

Camelia A. Pop

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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.

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Camelia A. Pop. 2014-06-03. $C^0$-estimates and smoothness of solutions to the parabolic equation defined by Kimura operators. https://arxiv.org/abs/1406.0742

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