Mathematics – Probability
Scientific paper
2009-11-10
Annals of Applied Probability 2011, Vol. 21, No. 6, 2263-2314
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/10-AAP757
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. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events 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 L\'{e}vy processes.
Bansaye Vincent
Delmas Jean-François
Marsalle Laurence
Tran Viet Chi
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