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.
Flots stochastiques et représentation lookdown