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Multiple scale analysis of clusters in spatial branching models. (English) Zbl 0909.60078

Author’s abstract: 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.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G57 Random measures
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