Flots stochastiques et représentation lookdown

Abstract : This thesis focuses on mathematical properties of two population models, namely the generalised Fleming-Viot process and the branching process. In both cases, the population is composed of infinitely many individuals characterised by a genetic type. As time passes, the asymptotic frequencies of the types within the population evolve stochastically through reproduction events where a uniformly chosen individual gives birth to a progeny with the same genetic type. Mathematically these two models are defined by measure-valued processes. In order to give a meaning to the genealogy of the underlying population, several approaches have been proposed these last fifteen years. One of the main contributions of this thesis is to unify two constructions: the lookdown representation introduced by Peter Donnelly and Thomas Kurtz in 1999 and the stochastic flow of bridges (or subordinators) introduced by Jean Bertoin and Jean-François Le Gall in 2000. This unification relies on the definition of new objects (the Eves, the stochastic flow of partitions) and necessitates a fine study of the asymptotic behaviours of the two aforementioned population models. In particular we define the Eve property as follows: if there is a genetic type whose asymptotic frequency tends to $1$ as $t$ becomes large then the population asymptotically descends from a single ancestor called the Eve of the population. In the case of the branching process, we obtain a necessary and sufficient condition on the branching mechanism ensuring the Eve property. We also provide a complete classification of all the possible asymptotic behaviours according to the branching mechanism. In the case of the generalised Fleming-Viot process, we obtain a partial classification of the possible asymptotic behaviours. Finally when the Eve property is fulfilled we present a pathwise construction of the lookdown representation from a stochastic flow of bridges (or subordinators). We also present a complete study of the explosive branching process conditioned to the non-explosion and provide an infinite collection of quasi-stationary distributions for this conditioned process. Finally we study the process of lengths of the evolving Kingman coalescent and propose an alternative construction to that of Pfaffelhuber, Wakolbinger and Weisshaupt.
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Cyril Labbé. Flots stochastiques et représentation lookdown. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 2013. English. ⟨tel-00874551⟩

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