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.
Highlights
The connections between the Fleming-Viot processes and the continuous-state branching processes have been intensively studied
This result has been generalized in 2005 by Birkner et al in [7] in the setting of Λ-generalized Fleming-Viot processes and continuous-state branching processes (CBs for short). In that paper they proved that the ratio process associated with an α-stable branching process is a time-changed Beta(2 − α, α)-Fleming-Viot process for α ∈ (0, 2)
The notation M stands for a couple of finite measures (Λ0, Λ1) encoding respectively the rates of immigration and of reproduction
Summary
The connections between the Fleming-Viot processes and the continuous-state branching processes have been intensively studied. Shiga established in 1990 that a Fleming-Viot process may be recovered from the ratio process associated with a Feller diffusion up to a random time change, see [23] This result has been generalized in 2005 by Birkner et al in [7] in the setting of Λ-generalized Fleming-Viot processes and continuous-state branching processes (CBs for short). The continuous-state branching processes with immigration (CBIs for short) are a class of time-homogeneous Markov processes with values in R+ They have been introduced by Kawazu and Watanabe in 1971, see [16], as limits of rescaled Galton-Watson processes with immigration.
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