Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property

Labbé, Cyril

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 732-769.

Nous construisons la généalogie d’un processus de branchement à espace d’états et temps continus associé à un mécanisme de branchement $𝛹$ - ou $𝛹\text{-CSBP}$ - à l’aide d’un flot stochastique de partitions. Cette construction est valable quel que soit le mécanisme de branchement et permet de définir un objet remarquablement efficace pour étudier les comportements asymptotiques et les convergences. En particulier, nous étudions la propriété d’Eve - l’existence d’un ancêtre dont descend asymptotiquement toute la population - et donnons une condition nécessaire et suffisante sur le $𝛹\text{-CSBP}$ pour que cette propriété soit vérifiée. Finalement, nous montrons que le flot de partitions unifie la représentation lookdown et le flot de subordinateurs lorsque la propriété d’Eve est vérifiée.

We encode the genealogy of a continuous-state branching process associated with a branching mechanism $𝛹$ - or $𝛹\text{-CSBP}$ in short - using a stochastic flow of partitions. This encoding holds for all branching mechanisms and appears as a very tractable object to deal with asymptotic behaviours and convergences. In particular we study the so-called Eve property - the existence of an ancestor from which the entire population descends asymptotically - and give a necessary and sufficient condition on the $𝛹\text{-CSBP}$ for this property to hold. Finally, we show that the flow of partitions unifies the lookdown representation and the flow of subordinators when the Eve property holds.

Classification : 60J80, 60G09, 60J25
Mots clés : continuous-state branching process, measure-valued process, genealogy, partition, stochastic flow, lookdown process, subordinator, EVE

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