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

Abstract

Let Lambda be a finite measure on the unit interval. A Lambda-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions (Lambda-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where Lambda is the Dirac measure in 0. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions (Xi-coalescent) with mutation, the Xi-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 Xi-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 populations is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial Xi-Fleming-Viot processes naturally arise as limiting models.