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

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 population is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $\Xi$-Fleming-Viot processes naturally arise as limiting models.Comment: 35 pages, 2 figure

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

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