## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 15 (2010), paper no. 63, 1938-1970.

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

#### 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.

#### Article information

**Source**

Electron. J. Probab., Volume 15 (2010), paper no. 63, 1938-1970.

**Dates**

Accepted: 18 November 2010

First available in Project Euclid: 1 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1464819848

**Digital Object Identifier**

doi:10.1214/EJP.v15-822

**Mathematical Reviews number (MathSciNet)**

MR2738344

**Zentralblatt MATH identifier**

1226.60118

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65]

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**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

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Englander, Janos. The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction. Electron. J. Probab. 15 (2010), paper no. 63, 1938--1970. doi:10.1214/EJP.v15-822. https://projecteuclid.org/euclid.ejp/1464819848