The Annals of Probability

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

Luis G. Gorostiza

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.

Article information

Source
Ann. Probab., Volume 11, Number 2 (1983), 374-392.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993603

Digital Object Identifier
doi:10.1214/aop/1176993603

Mathematical Reviews number (MathSciNet)
MR690135

Zentralblatt MATH identifier
0519.60085

JSTOR