## Abstract

If Y is a standard Fleming–Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each t > 0 the measure Y_{t} is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which the stationary version of Y is a finite sum of weighted Dirac masses. In the present work we discuss the existence of such exceptional times for the generalized Fleming–Viot processes. In the case of Beta-Fleming–Viot processes with index α ∈ ]1, 2[ we show that—irrespectively of the mutation rate and α—the number of atoms is almost surely always infinite. The proof combines a Pitman– Yor type representation with a disintegration formula, Lamperti’s transformation for self-similar processes and covering results for Poisson point processes.

Original language | English |
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Pages (from-to) | 84-120 |

Number of pages | 37 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2014 |

## Keywords

- Exceptional times
- Excursion theory
- Fleming
- Jump-type SDE
- Mutations
- Self-similarity
- Viot processes

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics