Multiple scale analysis of clusters in spatial branching models

Abstract

In this paper we will investigate the long time behavior of critical branching Brownian motion and (finite variance) super-Brownian motion (the so-called Dawson-Watanabe process) on $\mathbb{R}$^d$. These processes are known to be persistent if $d \geq 3$; that is, there exist nontrivial equilibrium measures. If $d \leq 2$, they cluster; that is, the processes converge to the 0 configuration while the surviving mass piles up in so-called clusters.

We study the spatial profile of the clusters in the “critical” dimension $d = 2$ via multiple space scale analysis. We will also investigate the long-time behavior of these models restricted to finite boxes in $d \geq 2$. On the way, we develop coupling and comparison methods for spatial branching models.