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arXiv · 0808.4024

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

Janos Englander
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Abstract

In this paper we prove that the center of mass of a supercritical branching-Brownian motion, or that of a supercritical super-Brownian motion tends to a limiting position almost surely, which, in a sense complements a result of Tribe on the final behavior of a critical super-Brownian motion. This is shown to be true also for a model where 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 the origin is shifted to a random point which has normal distribution, and the Ornstein Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less by their number by precisely one.

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Janos Englander. 2008-08-29. The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction. https://arxiv.org/abs/0808.4024

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