@article{EJP822,
author = {Janos Englander},
title = {The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {15},
year = {2010},
keywords = {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},
abstract = {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.},
pages = {no. 63, 1938-1970},
issn = {1083-6489},
doi = {10.1214/EJP.v15-822},
url = {http://ejp.ejpecp.org/article/view/822}}