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Journal Articles Annals of Applied Probability Year : 2011

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

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Abstract

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.
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Dates and versions

hal-00431118 , version 1 (10-11-2009)
hal-00431118 , version 2 (02-08-2010)

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Vincent Bansaye, Jean-François Delmas, Laurence Marsalle, Viet Chi Tran. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. Annals of Applied Probability, 2011, 21 (6), pp.2263-2314. ⟨10.1214/10-AAP757⟩. ⟨hal-00431118v2⟩
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