Title :

Limit theorems for Markov processes indexed by continuous time Galton-Watson trees

Author(s) :

Bansaye, Vincent [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Delmas, Jean-François [Auteur]
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Marsalle, Laurence [Auteur correspondant]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Tran, Chi [Auteur correspondant]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

The Annals of Applied Probability

Pages :

2263-2314

Publisher :

Institute of Mathematical Statistics (IMS)

Publication date :

2011-12

ISSN :

1050-5164

English keyword(s) :

Branching Markov process
Branching diffusion
Limit theorems
Size biased reproduction distribution

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]

English abstract : [en]

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, ...
Show more >We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes and ancestral lineages.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Modélisation stochastique et inférence statistique pour la propagation des maladies infectieuses transmissibles: du microscopique au macroscopique

Comment :

41 pages, 2 figures

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL