The Annals of Probability

The Feynman–Kac formula and Harnack inequality for degenerate diffusions

Charles L. Epstein and Camelia A. Pop

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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.

Article information

Ann. Probab., Volume 45, Number 5 (2017), 3336-3384.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J90
Secondary: 60J60: Diffusion processes [See also 58J65]

Degenerate elliptic equations degenerate diffusions generalized Kimura diffusions Markov processes Feynman–Kac formulas Girsanov formula weighted Sobolev spaces anisotropic Hölder spaces


Epstein, Charles L.; Pop, Camelia A. The Feynman–Kac formula and Harnack inequality for degenerate diffusions. Ann. Probab. 45 (2017), no. 5, 3336--3384. doi:10.1214/16-AOP1138.

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