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May, 1983 High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions
Luis G. Gorostiza
Ann. Probab. 11(2): 374-392 (May, 1983). DOI: 10.1214/aop/1176993603


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.


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Luis G. Gorostiza. "High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions." Ann. Probab. 11 (2) 374 - 392, May, 1983.


Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0519.60085
MathSciNet: MR690135
Digital Object Identifier: 10.1214/aop/1176993603

Primary: 60F17
Secondary: 60G15 , 60G20 , 60G60 , 60J65 , 60J80

Keywords: Branching Brownian motion , demographic variation , generalized Gaussian process , generalized Langevin equation , limit theorems , Random field , renormalization

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 2 • May, 1983
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