  We provide a detailed description of the structure of the transition
probabilities and of the hitting distributions of 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.
