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**Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration.**
*(English)*
Zbl 1411.60122

Summary: 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 T. Shiga [J. Math. Kyoto Univ. 30, No. 2, 245–279 (1990; Zbl 0751.60044)] and M. Birkner et al. [Electron. J. Probab. 10, Paper No. 9, 303–325 (2005; Zbl 1066.60072)] 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.

### MSC:

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60G09 | Exchangeability for stochastic processes |

60G52 | Stable stochastic processes |

60G57 | Random measures |

92D25 | Population dynamics (general) |