Alpha-stable branching and beta-coalescents. (English) Zbl 1066.60072

Birkner, Matthias; Blath, Jochen; Capaldo, Marcella; Etheridge, Alison M.; Möhle, Martin; Schweinsberg, Jason; Wakolbinger, Anton

Electron. J. Probab. 10, Paper No. 9, 303-325 (2005).

Summary: 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 \(\alpha\)-stable branching mechanisms. The random ancestral partition is then a time-changed \(\Lambda\)-coalescent, where \(\Lambda\) is the Beta-distribution with parameters \(2-\alpha\) and \(\alpha\), and the time change is given by \(Z^{1-\alpha}\), where \(Z\) is the total population size. For \(\alpha = 2\) (Feller’s branching diffusion) and \(\Lambda = \delta_0\) (Kingman’s coalescent), this is in the spirit of (a non-spatial version of) Perkins’ disintegration theorem. For \(\alpha =1\) and \(\Lambda\) the uniform distribution on \([0,1]\), this is the duality discovered by J. Bertoin and J.-F. Le Gall [Probab. Theory Relat. Fields 117, No. 2, 249–266 (2000; Zbl 0963.60086)] 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 P. Donnelly and T. G. Kurtz [Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)]; the other is based on direct calculations with generators.

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J25	Continuous-time Markov processes on general state spaces
60G09	Exchangeability for stochastic processes
60G52	Stable stochastic processes
92D25	Population dynamics (general)

Keywords:

coalescent; genealogy; lookdown construction

Citations:

Zbl 0963.60086; Zbl 0956.60081