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

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