2014/06/18 by Charles L. Epstein, Epstein, Charles L., Camelia A. Pop +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #math.AP #math.PR
paper · pdf · doi:10.48550/arxiv.1406.4759
57 pages
arxiv created 2014/06/18 · openalex publication_date 2014/06/18 · arxiv updated 2014/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.