On exceptional times for generalized fleming–viot processes with mutations

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.