The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction
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.
Keywords
Cite
@article{arxiv.0808.4024,
title = {The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction},
author = {Janos Englander},
journal= {arXiv preprint arXiv:0808.4024},
year = {2008}
}