Document Type

Article

Publication Date

2016

ISSN

1083-6489

Volume

21

Issue

67

First Page

1

Last Page

25

DOI

http://dx.doi.org/10.1214/16-EJP4782

Abstract

A d-dimensional branching diffusion, Z, is investigated, where the linear attraction or repulsion between particles is competing with an Ornstein-Uhlenbeck drift, with parameter b (we take b > 0 for inward O-U and b < 0 for outward O-U). This work has been motivated by [4], where a similar model was studied, but without the drift component. We show that the large time behavior of the system depends on the interaction and the drift in a nontrivial way. Our method provides, inter alia, the SLLN for the non-interactive branching (inward) O-U process. First, regardless of attraction (γ > 0) or repulsion (γ < 0), a.s., as t → ∞, the center of mass of Zt converges to the origin when b > 0, while escapes to infinity exponentially fast (rate |b|) when b < 0. We then analyze Z as viewed from the center of mass, and finally, for the system as a whole, we provide a number of results and conjectures regarding the long term behavior of the system; some of these are scaling limits, while some others concern local extinction.

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