2013/04/04 by Julien Berestycki, Berestycki, Julien, Leif Doering +5
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #60G18 #60J80 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1304.1342
openalex publication_date 2013/04/04 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
If mathbf Y is a standard Fleming-Viot process with constant mutation rate\n(in the infinitely many sites model) then it is well known that for each t>0\nthe measure mathbf Yt is purely atomic with infinitely many atoms. However,\nSchmuland proved that there is a critical value for the mutation rate under\nwhich almost surely there are exceptional times at which mathbf Y is a\nfinite sum of weighted Dirac masses. In the present work we discuss the\nexistence of such exceptional times for the generalized Fleming-Viot processes.\nIn the case of Beta-Fleming-Viot processes with index \α\∈ ,]1,2[ we\nshow that - irrespectively of the mutation rate and \α - the number of\natoms is almost surely always infinite. The proof combines a Pitman-Yor type\nrepresentation with a disintegration formula, Lamperti's transformation for\nself-similar processes and covering results for Poisson point processes.\n