The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 22, Number 4 (2012), 1693-1711.
Poissonian statistics in the extremal process of branching Brownian motion
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.
Ann. Appl. Probab., Volume 22, Number 4 (2012), 1693-1711.
First available in Project Euclid: 10 August 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012), no. 4, 1693--1711. doi:10.1214/11-AAP809. https://projecteuclid.org/euclid.aoap/1344614208