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

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

ALEA, Lat. Am. J. Probab. Math. Stat. 6, 25-61 (2009).

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

Keywords:

coalescent; duality; Fleming-Viot process; measure-valued process; modified lookdown construction

Citations:

Zbl 0956.60081