The Feynman–Kac formula and Harnack inequality for degenerate diffusions
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Charles L. Epstein [1] ; Camelia A. Pop [2]
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[1] University of Pennsylvania
University of Pennsylvania
City of Philadelphia, Estados Unidos
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[2] University of Minnesota
University of Minnesota
City of Minneapolis, Estados Unidos
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 3336-3384
- Idioma: inglés
- DOI: 10.1214/16-AOP1138
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Resumen
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.
