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

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

2009-11-10

arxiv.org/abs/0911.1973v3

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.

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