Limit theorems for some branching measure-valued processes

TL;DR

This paper studies the long-term behavior of a continuous-time particle system with spatial motion and nonlocal branching, providing asymptotic results and a large population approximation through a growth-fragmentation equation.

Contribution

It introduces a size-biased auxiliary process to analyze the asymptotics of measure-valued branching processes, extending understanding of their long-term behavior.

Findings

Asymptotic behavior characterized using size-biased process

Large population limit described by a growth-fragmentation equation

Examples illustrating the theoretical results

Abstract

We consider a particle system in continuous time, discrete population, with spatial motion and nonlocal branching. The offspring's weights and their number may depend on the mother's weight. Our setting captures, for instance, the processes indexed by a Galton-Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine this asymptotic behaviour. We also obtain a large population approximation as weak solution of a growth-fragmentation equation. Several examples illustrate our results.