WIAS Preprint No. 1352, (2008)

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



Authors

  • Birkner, Matthias
  • Blath, Jochen
  • Möhle, Martin
  • Steinrücken, Matthias
  • Tams, Johanna

2010 Mathematics Subject Classification

  • 60K35 60G09 92D10 60C05 92D15

Keywords

  • Coalescent, duality, Fleming-Viot process, measure-valued process, modified lookdown construction

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.

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