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2010 The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction
Janos Englander
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Electron. J. Probab. 15: 1938-1970 (2010). DOI: 10.1214/EJP.v15-822


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.


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Janos Englander. "The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction." Electron. J. Probab. 15 1938 - 1970, 2010.


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

zbMATH: 1226.60118
MathSciNet: MR2738344
Digital Object Identifier: 10.1214/EJP.v15-822

Primary: 60J60
Secondary: 60J80

Keywords: Branching Brownian motion , branching Ornstein-Uhlenbeck process , Center of mass , Curie-Weiss model , H-transform , McKean-Vlasov limit , self-interaction , Spatial branching processes , Super-Brownian motion

Vol.15 • 2010
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