  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 is 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.
