On exceptional times for generalized Fleming–Viot processes with mutations

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 the stationary version of \(\mathbf Y\) 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.