High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions

Abstract

The fluctuations of an infinite system of unscaled branching Brownian motions in $R^d$ are shown to converge weakly under a spatial central limit normalization when the initial density of particles tends to infinity. The limit is a generalized Gaussian process $M$ which can be written as $M = M^I + M^{II}$, where $M^I$ is the fluctuation limit of a Poisson system of Brownian motions obtained by Martin-Lof, and $M^{II}$ arises from the spatial central limit normalization of the "demographic variation process" of the system. In the critical case $M^I$ and $M^{II}$ are independent and $M^{II}$ coincides with the generalized Ornstein-Uhlenbeck process found by Dawson and by Holley and Stroock as the renormalization limit of an infinite system of critical branching Brownian motions when $d \geq 3$. Generalized Langevin equations for $M, M^I$ and $M^{II}$ are given.