An ergodic theorem for the frontier of branching Brownian motion

TL;DR

This paper proves that the empirical distribution of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution with a random shift, confirming a longstanding conjecture.

Contribution

It establishes an ergodic theorem for the maximum of branching Brownian motion, using path localization and decorrelation techniques to prove convergence.

Findings

Empirical distribution converges to Gumbel distribution almost surely.

Method relies on decorrelation of maximal displacements.

Path localization is key to the proof.

Abstract

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribution with a random shift. The method of proof is based on the decorrelation of the maximal displacements for appropriate time scales. A crucial input is the localization of the paths of particles close to the maximum that was previously established by the authors [Comm. Pure Appl. Math. 64 (2011)].