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.
Citation
Achim Klenke. "Multiple scale analysis of clusters in spatial branching models." Ann. Probab. 25 (4) 1670 - 1711, October 1997. https://doi.org/10.1214/aop/1023481107
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