We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.
Bibliographical noteFunding Information:
C. L. Epstein’s research is partially supported by NSF Grant DMS-1507396. C.A. Pop’s research is partially supported by NSF Grant DMS-1714490.
- Caloric measure
- Compact manifold with corners
- Degenerate elliptic operators
- Dirichlet heat kernel
- Fundamental solution
- Hitting distributions
- Markov processes
- Transition probabilities