The Annals of Probability

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

Luis G. Gorostiza

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G20: Generalized stochastic processes 60G60: Random fields 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G15: Gaussian processes 60J65: Brownian motion [See also 58J65]

Keywords
Branching Brownian motion demographic variation random field generalized Gaussian process generalized Langevin equation limit theorems renormalization

Citation

Gorostiza, Luis G. High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions. Ann. Probab. 11 (1983), no. 2, 374--392. doi:10.1214/aop/1176993603. https://projecteuclid.org/euclid.aop/1176993603


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Corrections

  • See Correction: Luis G. Gorostiza. Correction: Correction to "High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions". Ann. Probab., Volume 12, Number 3 (1984), 926--927.