The Annals of Probability
- Ann. Probab.
- Volume 45, Number 5 (2017), 3336-3384.
The Feynman–Kac formula and Harnack inequality for degenerate diffusions
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.
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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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. https://projecteuclid.org/euclid.aop/1506132040