## 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 Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568-608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are 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 the a priori regularity of the weak solutions.

Original language | English (US) |
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Pages (from-to) | 3336-3384 |

Number of pages | 49 |

Journal | Annals of Probability |

Volume | 45 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2017 |

### Bibliographical note

Funding Information:Received May 2015; revised July 2016. 1Supported in part by NSF Grants DMS12-05851, DMS-1507396, and ARO Grant W911NF-12-1-0552. MSC2010 subject classifications. Primary 35J90; secondary 60J60. Key words and phrases. Degenerate elliptic equations, degenerate diffusions, generalized Kimura diffusions, Markov processes, Feynman–Kac formulas, Girsanov formula, weighted Sobolev spaces, anisotropic Hölder spaces.

Publisher Copyright:

© Institute of Mathematical Statistics, 2017.

## Keywords

- Anisotropic Hölder spaces
- Degenerate diffusions
- Degenerate elliptic equations
- Feynman-Kac formulas
- Generalized Kimura diffusions
- Girsanov formula
- Markov processes
- Weighted Sobolev spaces