Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations

Romain Abraham (1) , Jean-François Delmas (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

Mots clés: continuous state branching process; immigration; continuous random tree; Williams' decomposition; probability of extinction; neutral mutation

Domaines: Probabilités [math.PR]

Romain Abraham, Jean-François Delmas. Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations. 2007. ⟨hal-00141154v1⟩