Harnack inequalities for degenerate diffusions
We will present probabilistic and analytic properties of a
class of degenerate diffusion operators arising in population genetics,
the so-called generalized Kimura diffusion operators. Such processes
arise as models for the evolution of gene frequencies. We will start by
highlighting the main questions of interest and the mathematical
difficulties in addressing them. 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.