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

TL;DR

This paper establishes limit theorems for particle systems evolving on continuous-time Galton--Watson trees, including laws of large numbers and a central limit theorem, with applications to various stochastic processes.

Contribution

It introduces a probabilistic framework for analyzing empirical measures and limit theorems in Markov processes on branching trees, extending existing results to continuous-time models.

Findings

Law of large numbers for empirical measures

Central limit theorem in a special case

Applications to splitting diffusions and branching Lévy processes

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. 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.