MC 5479Ìý{Old numberingÌýMCÌý5136B}
Speaker
¶Ù°ù.Ìý°ä²¹³¾±ð±ô¾±²¹Ìý±Ê´Ç±è
Department
of
Mathematics
|ÌýUniversity
of
Pennsylvania
Title
HarnackÌýInequalities for degenerate diffusions
Abstract
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-invariantÌýHarnackÌýinequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results onÌýFeynman-KacÌýformulas concerning the assumptions on the degeneracy of the diffusion matrix, theÌýboundednessÌýof the drift coefficients, and on the a priori regularity of the weak solutions. This is joint work with CharlesÌýEpstein.