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Coalescent distingués échangeables et processus de Fleming-Viot généralisés avec immigration.

Abstract : The purpose of the dissertation is to study stochastic coalescent processes modelling the genealogy of an exchangeable population with immigration. We represent the popu- lation by the set of integers N = {1, 2...}. Suppose that we sample n individuals in the population of today. We group together individuals with the same parent at preceding generations. Due to the immigration, some individuals, from a certain generation, may have no ancestor in the population. By convention, we will gather these individuals in a block to which we add 0. We talk about the distinguished block. Processes called distinguished exchangeable coalescents are valued in the space of the partitions of Z+ = {0, 1, ..}. At each time t, we consider a distinguished exchangeable par- tition. That is a partition whose law is invariant under the action of permutations leaving 0 at 0. The presence of the distinguished block adds new coagulation events, which do not exist in the classic coalescent processes. We determine a sufficient condition (neces- sary under some hypotheses) for a distinguished coalescent to come down from infinity, meaning that immediately after 0, there is only a finite number of blocks. On the other hand, there is a duality between these coalescent processes and some pro- cesses valued in the probability-measures space, called generalized Fleming-Viot processes with immigration. We establish links between them and the continuous branching pro- cesses with immigration. In the case of a process with α-stable reproduction and (α −1)- stable immigration, we show that the corresponding measure-valued process, properly renormalized, is a time-changed Fleming-Viot process with immigration.
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Submitted on : Wednesday, September 12, 2012 - 5:27:05 PM
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Clément Foucart. Coalescent distingués échangeables et processus de Fleming-Viot généralisés avec immigration.. Probabilités [math.PR]. Faculté des sciences de l'Université de Paris, 2012. Français. ⟨tel-00731260⟩

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