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

By Vincent Bansaye, Jean-François Delmas, Laurence Marsalle, Viet Chi Tran. 2009

86 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4ff6c7e68f17978f03cc013a6bf28a0f9c857dd9

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In Condensed Matter Physics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Detection of cellular aging in a Galton-Watson process (2008, result): “We give in a paper with Bansaye and Tran [7] a partial extension of the present results to a continuous time setting.”
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior (2015, methodology): “Nevertheless this difficulty can be overcome by proving (see [6]) that the common ancestor of two individuals lives almost-surely at the beginning of the continuous time Galton-Watson tree.”
The many-to-few lemma and multiple spines (2011, result): “ in [7]. For the parabolic Anderson model with Weibull upper tails, Albeverio et al. [3] gave a similar result by considering existence and uniqueness of solutions to a Cauchy problem. Bansaye et al. [4] develop a quite general many-to-two lemma for Markov branching processes, allowing particles to ”
Limit theorems for some branching measure-valued processes (2011, methodology): “Since this process is homogeneous, we can see it as a process indexed by a tree [4].”