On Exceptional Times for generalized Fleming-Viot Processes with Mutations

TL;DR

This paper investigates the occurrence of exceptional times in generalized Fleming-Viot processes, showing that for Beta-Fleming-Viot processes with certain parameters, the measure always remains infinitely atomic regardless of mutation rates.

Contribution

It extends the understanding of atomic structures in Fleming-Viot processes by analyzing exceptional times in generalized models, including Beta-Fleming-Viot processes with index .

Findings

For Beta-Fleming-Viot processes with , the number of atoms is almost surely always infinite.

The study demonstrates the existence of exceptional times in generalized Fleming-Viot processes.

The proof employs advanced probabilistic tools like Pitman-Yor representation and Lamperti's transformation.

Abstract

If $\mathbf Y$ 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 $\mathbf Y_t$ 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 $\mathbf Y$ 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 $\alpha\in\,]1,2[$ we show that - irrespectively of the mutation rate and $\alpha$ - 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.