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September 2017 The Feynman–Kac formula and Harnack inequality for degenerate diffusions
Charles L. Epstein, Camelia A. Pop
Ann. Probab. 45(5): 3336-3384 (September 2017). DOI: 10.1214/16-AOP1138


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.


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Charles L. Epstein. Camelia A. Pop. "The Feynman–Kac formula and Harnack inequality for degenerate diffusions." Ann. Probab. 45 (5) 3336 - 3384, September 2017.


Received: 1 May 2015; Revised: 1 July 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812207
MathSciNet: MR3706745
Digital Object Identifier: 10.1214/16-AOP1138

Primary: 35J90
Secondary: 60J60

Keywords: Anisotropic Hölder spaces , Degenerate diffusions , Degenerate elliptic equations , Feynman–Kac formulas , generalized Kimura diffusions , Girsanov formula , Markov processes , weighted Sobolev spaces

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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