Alpha-stable branching and Beta coalescents

Abstract: We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from α-stable branching mechanisms. The random ancestral partition is then a time-changed Λ-coalescent, where Λ is the Beta-distribution with parameters 2 − α and α, and the time change is given by Z1−α, where Z is the total population size. For α = 2 (Feller’s branching diffusion) and Λ = δ0 (Kingman’s coalescent), this is in the spirit of (a non-spatial version of) Perkins ’ Disintegration Theorem. For α = 1 and Λ the uniform distribution on [0, 1], this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu’s continuous state branching process and the Bolthausen-Sznitman coalescent. We present two approaches: one, exploiting the ‘modified lookdown construction’, draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators