A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks

Abstract

Let $Λ$ be a finite measure on the unit interval. A $Λ$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($Λ$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $Λ$ is the Dirac measure in 0. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($Ξ$-coalescent) with mutation, the $Ξ$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $Ξ$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a population is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $Ξ$-Fleming-Viot processes naturally arise as limiting models.