Branching diffusion with particle interactions

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 ($\gamma >0$) or repulsion ($\gamma <0$), a.s., as $t\to \infty $, the center of mass of $Z_t$ 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.