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

TL;DR

This paper proves that the center of mass of supercritical branching Brownian motions converges almost surely and explores the asymptotic behavior of the system viewed from this center, including models with particle attraction or repulsion.

Contribution

It establishes the almost sure convergence of the center of mass for supercritical branching processes and characterizes the asymptotic behavior of the system with self-interaction.

Findings

Center of mass converges almost surely for supercritical processes.

System asymptotically behaves like a branching Ornstein-Uhlenbeck process.

The origin of the process shifts to a normally distributed random point.

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.