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The Feynman--Kac Formula and Harnack Inequality for Degenerate Diffusions

Connections for Women: Dispersive and Stochastic PDE August 19, 2015 - August 21, 2015

August 19, 2015 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Camelia Pop (University of Pennsylvania)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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. 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. This is joint work with Charles Epstein

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