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arXiv · 1406.4759

Harnack Inequalities for Degenerate Diffusions

Charles L. Epstein·Camelia A. Pop
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Abstract

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators. Our main results is a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman-Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients, and on the a priori regularity of the weak solutions.

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Charles L. Epstein, Camelia A. Pop. 2014-06-18. Harnack Inequalities for Degenerate Diffusions. https://arxiv.org/abs/1406.4759

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