Branching diffusion with interactions

TL;DR

This paper studies a $d$-dimensional branching diffusion with interactions and Ornstein-Uhlenbeck drift, revealing how the system's long-term behavior depends on the interplay of attraction, repulsion, and drift, including convergence and escape phenomena.

Contribution

It provides a detailed analysis of the large-time behavior of interacting branching diffusions with Ornstein-Uhlenbeck drift, including new results on the center of mass and conjectures on long-term dynamics.

Findings

Center of mass converges to origin for inward drift

Center of mass escapes exponentially for outward drift

Provides SLLN for non-interacting inward O-U process

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 time tends to infinity, the center of mass of $Z$ (i) converges to the origin, when $b>0$; (ii) 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 show a number of results/conjectures regarding the long term behavior of the system; some of these are scaling limits, while some others concern local extinction.