The extremal process of super-Brownian motion

TL;DR

This paper studies the asymptotic behavior of the maximum support of super-Brownian motion, establishing limit theorems, large deviation results, and describing the extremal process as a decorated Poisson random measure.

Contribution

It introduces new limit theorems and describes the extremal process of super-Brownian motion, extending results known for branching Brownian motion.

Findings

Convergence in law of the centered support supremum and process.

Large deviation estimates for the support maximum.

The extremal process converges to a decorated Poisson random measure.

Abstract

In this paper, we establish limit theorems for the supremum of the support, denoted by $M_t$, of a supercritical super-Brownian motion $\{X_t, t\ge0\}$ on $\mathbb{R}$. We prove that there exists an $m(t)$ such that $(X_t-m(t), M_t-m(t))$ converges in law, and give some large deviation results for $M_t$ as $t\to\infty$. We also prove that the limit of the extremal process $\mathcal{E}_t:=X_t-m(t)$ is a Poisson random measure with exponential intensity in which each atom is decorated by an independent copy of an auxiliary measure. These results are analogues of the results for branching Brownian motions obtained in Arguin et al. (Probab. Theory Relat. Fields 157 (2013), 535-574), A\"id\'ekon et al. (Probab. Theory Relat. Fields 157 (2013), 405-451) and Roberts (Ann. Probab. 41 (2013), 3518-3541).