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**A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks.**
*(English)*
Zbl 1162.60342

Summary: 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 P. Donnelly and T. G. Kurtz [Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)]. 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.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60G09 | Exchangeability for stochastic processes |

92D10 | Genetics and epigenetics |

60C05 | Combinatorial probability |

92D15 | Problems related to evolution |