CLT for Ornstein-Uhlenbeck branching particle system

Abstract

In this paper we consider a branching particle system consisting of particles moving according to an Ornstein-Uhlenbeck process in $\mathbb{R}d$ and undergoing 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 normalization 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 normalization is the square root of the size of the system. In the critical case the limit is still Gaussian, but the normalization requires an additional term. Finally, when branching has a large rate the situation is completely different. The limit is no longer Gaussian, the normalization is substantially larger than the classical one and the convergence holds in probability. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process).