A simpler path to Ergodic Theorems for the Frontier of Branching Brownian Motion

TL;DR

This paper presents a simplified and more direct proof of the ergodic theorem for the frontier of branching Brownian motion, extending its applicability and addressing previous gaps.

Contribution

It offers a shorter, more straightforward proof of the ergodic theorem for BBM and broadens its scope to include various functionals of the recentred maximum.

Findings

Provided a shorter proof of the ergodic theorem for BBM.

Extended the ergodic theorem to a broad class of functionals.

Addressed a gap in the path localization argument of prior work.

Abstract

We revisit the ergodic theorem for the frontier of branching Brownian motion (BBM). Motivated by the proof of Arguin, Bovier, and Kistler \cite{arguin2012ergodic}, we provide a shorter and more direct argument. It relies on two observations: pairs of extremal particles observed at well-separated times must have branched early, and pairs of early-branching extremal particles have negatively correlated positions. This yields the ergodic theorem for BBM and extends it to a broad class of functionals of the recentred maximum. We also address a gap in the path localization argument of \cite{arguin2012ergodic}.