## Electronic Journal of Probability

### Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration

#### Abstract

Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the $\alpha$-stable continuous state branching process with the $Beta(2-\alpha,\alpha)$-generalized Fleming-Viot process. In a recent work, a new class of probability-measure valued processes, called $M$-generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called $M$ coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the $\alpha$-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a $Beta(2-\alpha, \alpha-1)$-coalescent.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 23, 21 pp.

Dates
Accepted: 12 February 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064248

Digital Object Identifier
doi:10.1214/EJP.v18-2024

Mathematical Reviews number (MathSciNet)
MR3035751

Zentralblatt MATH identifier
06247192

Subjects
Primary: 60J25 60G09 92D25

Rights

#### Citation

Foucart, Clément; Hénard, Olivier. Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration. Electron. J. Probab. 18 (2013), paper no. 23, 21 pp. doi:10.1214/EJP.v18-2024. https://projecteuclid.org/euclid.ejp/1465064248

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