A measure-valued diffusion process describing the stepping stone model with infinitely many alleles

Abstract

In this paper we formulate the stepping stone model in population genetics as a measure-valued diffusion process. In order to formulate this model we introduce an appropriate martingale problem and show that it is well-posed. In the selectively neutral case an ergodic property of the process corresponding to the solution of the martingale problem is proved, under a suitable assumption on the mechanism of mutation. If in addition the mutation mechanism is of jump type, then simple calculations involving the generator associated with our martingale problem give us equations for the probabilities of identity at equilibrium.