Applied Mathematics Seminar | Camelia Pop, Harnack Inequalities for degenerate diffusions

Monday, January 26, 2015 1:30 pm - 1:30 pm EST (GMT -05:00)

MC 5479 {Old numbering MC 5136B}


Dr. Camelia Pop
Department of Mathematics | University of Pennsylvania


Harnack Inequalities for degenerate diffusions


We will present 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 on the a priori regularity of the weak solutions. This is joint work with Charles Epstein.