CLT for Ornstein-Uhlenbeck branching particle system

TL;DR

This paper establishes a central limit theorem for a branching Ornstein-Uhlenbeck particle system, revealing three distinct regimes with different limit behaviors and normalizations, including Gaussian and non-Gaussian limits.

Contribution

It provides the first CLT for this system, characterizing the limit distributions and normalizations across different branching rate regimes.

Findings

Gaussian limit in small and critical cases

Non-Gaussian limit in large branching rate case

Spatial fluctuations are asymptotically independent of total particle fluctuations

Abstract

In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary, supercritical branching with a constant rate $\lambda>0$. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalisation fall into three qualitatively different classes. In, what we call, the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalisation is the square root of the size of the system. In the critical case the limit is still Gaussian, however the normalisation requires an additional term. Finally, when branching has large rate the situation is completely different. The limit is no longer Gaussian, the normalisation is substantially larger than the classical one and the convergence holds in probability. We prove also that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process).