An ergodic theorem for the extremal process of branching Brownian motion

TL;DR

This paper extends previous results on the maximum of branching Brownian motion, showing that the time-averaged distribution of extremal particles converges to a Poisson cluster process, revealing detailed structure of extremal configurations.

Contribution

It proves the convergence of the entire extremal particle system's distribution to a Poisson cluster process, extending earlier maximum distribution results.

Findings

Time-average distribution of extremal particles converges to a Poisson cluster process.

Extension of maximum distribution convergence to the entire extremal system.

Provides a detailed probabilistic description of extremal particle configurations.

Abstract

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.