Poissonian statistics in the extremal process of branching Brownian motion

TL;DR

This paper rigorously establishes that the extremal particles of branching Brownian motion form a Poisson point process with exponential density in the large time limit, advancing understanding of the process's limiting extremal structure.

Contribution

It proves that the extremal particles, maximal within their clusters, converge to a Poisson point process, providing rigorous support for the cluster-based extremal process picture.

Findings

Extremal particles form a Poisson point process in the limit.

The convergence is for particles maximal within their clusters.

Supports the cluster process model of the extremal process.

Abstract

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large $t$ to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [Ann. Probab. 15 (1987) 1052-1061] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [Math. Nachr. 149 (1990) 41-59] on branching Brownian motion with atypical displacement.