A transition function expansion for a diffusion model with selection

Using duality, an expansion is found for the transition function of the reversible K-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time t of a specified K-type birth-and-death process starting at .infinity.. The latter process is constructed by means of a coupling argument and characterized as the Ray process corresponding to the Ray.Knight compactification of the K-dimensional nonnegative-integer lattice.