The branching Brownian motion seen from its tip

TL;DR

This paper provides a detailed description of the limiting extremal point process of branching Brownian motion from its tip, confirming its structure as a decorated Poisson point process with an explicit construction.

Contribution

It offers a complete, explicit construction of the decoration point process describing the limit of the extremal process in branching Brownian motion.

Findings

The extremal process converges to a decorated Poisson point process.

The decoration process is explicitly constructed and characterized.

The work confirms and extends previous proofs of the limit structure.

Abstract

It has been conjectured since the work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, 2010, Arguin et al., 2010, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (2011).