Non-persistence of two-level branching particle systems in low dimensions

Summary

Abstract

We show that in dimensions d ≤ 4, no non-trivial finite intensity equilibria of a critical two-level binary branching Brownian particle system exist. Our method relies on the analysis of backward trees: in dimensions d ≤ 4, they are shown to exhibit an infinite clumping of mass, contradicting the existence of such equilibria.

Introduction

Hierarchically structured multilevel branching particle systems were first introduced by Dawson, Hochberg and Wu and have since been the object of much investigation. Such a two-level system consists of individuals or “firstlevel particles” undergoing some spatial motion and branching, which are, on the second level, grouped into clusters. Each of these clusters constitutes a “superparticle” which is simultaneously affected by some other, independent branching mechanism. The idea is to describe not only reproduction or death of individuals, but also replication or catastrophic elimination of whole families or clans. See for several examples arising in various fields of applications.

In this paper, we consider a system of Brownian particles in IR^d with particularly simple reproduction mechanisms – namely, critical binary branching at each level – and we show that in dimensions d ≤ 4, these systems do not persist, in the sense that no nontrivial equilibria with finite first moments exist. (In light of a recent result of Bramson, Cox and Greven on nonexistence of equilibria of branching Brownian particle systems in dimensions one and two, the restriction “with finite first moments” might be superfluous here; the determination of this, however, is beyond the scope of the techniques used in this paper.)