Electronic Journal of Probability

The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction

Janos Englander

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Consider the center of mass of a supercritical branching-Brownian motion. In this article we first show that it is a Brownian motion being slowed down such that it tends to a limiting position almost surely, and that this is also true for a model where the branching-Brownian motion is modified by attraction/repulsion between particles.  We then put this observation together with the description of the interacting system as viewed from its center of mass, and get the following asymptotic behavior: the system asymptotically becomes a branching Ornstein-Uhlenbeck process (inward for attraction and outward for repulsion), but (i) the origin is shifted to a random point which has normal distribution, and (ii) the Ornstein-Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less than their number by precisely one.  The main result of the article then is a scaling limit theorem for the local mass, in the attractive case. A conjecture is stated for the behavior of the local mass in the repulsive case.  We also consider a supercritical super-Brownian motion. Again, it turns out that, conditioned on survival, its center of mass is a continuous process having an a.s. limit.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 63, 1938-1970.

Accepted: 18 November 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching Brownian motion super-Brownian motion center of mass self-interaction Curie-Weiss model McKean-Vlasov limit branching Ornstein-Uhlenbeck process spatial branching processes H-transform

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Englander, Janos. The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction. Electron. J. Probab. 15 (2010), paper no. 63, 1938--1970. doi:10.1214/EJP.v15-822. https://projecteuclid.org/euclid.ejp/1464819848

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