Generalized stepping stone model with $Ξ$-resampling mechanism

Abstract

In this paper we formulate a generalized stepping stone model with Ξresampling mechanism to describe the evolution of relative frequencies for different types of alleles in a population with migration between two colonies. For a Ξ-coalescent and a jump type mutation generator A, such a probability-measure-valued Markov process is dual to the (Ξ, A)-coalescent process with geographical labels and migration. The existence of the generalized stepping stone model is directly established from a moment duality by verifying a multidimensional Hausdorff moment problem, and its probability law is also uniquely determined by the moment duality. Further, we characterize the stationary distribution for this model and show that the model is not reversible when the mutation operator is of uniform type.