Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

Abstract : Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Imagine an infinite population with immigration labelled at each generation by N:={1,2,...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focussing on simple distinguished coalescents, i.e such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0,1] denoted by M=(\Lambda_{0},\Lambda_{1}). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the \Lambda-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures \Lambda_{0} and \Lambda_{1} specify respectively the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time, all individuals are immigrant children.
Type de document :
Pré-publication, Document de travail
30 pages. 2010
Liste complète des métadonnées

Littérature citée [19 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-00487275
Contributeur : Clément Foucart <>
Soumis le : mardi 1 mars 2011 - 20:55:49
Dernière modification le : jeudi 27 avril 2017 - 09:46:23
Document(s) archivé(s) le : lundi 30 mai 2011 - 03:04:32

Fichiers

CoalDist.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-00487275, version 3
  • ARXIV : 1006.0581

Collections

UPMC | INSMI | USPC | PMA

Citation

Clément Foucart. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. 30 pages. 2010. 〈hal-00487275v3〉

Partager

Métriques

Consultations de
la notice

117

Téléchargements du document

132