The Feynman-Kac formula and Harnack inequality for degenerate diffusions

Charles L. Epstein, Camelia A. Pop

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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 languageEnglish (US)
Pages (from-to)3336-3384
Number of pages49
JournalAnnals of Probability
Volume45
Issue number5
DOIs
StatePublished - 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

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