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

TL;DR

This paper delivers a shorter, more direct proof of the ergodic theorem for the BBM frontier by exploiting two structural facts: extremal pairs observed at distant times must have branched early, and such early-branching extremal pairs exhibit negative correlation. The authors develop a two-time-scale approach, introduce $\alpha$-localization, and prove key bounds on late-branching probabilities, enabling almost-sure convergence of the empirical distribution of the front to $\exp(-C Z_\infty e^{-\sqrt{2} x})$ in the sense of $F_T(x)$. They also provide a rigorous path-localization framework and a decorrelation mechanism via a van den Berg–Kesten–Reimer-type inequality, with extensions to Laplace functionals and other observables. The work resolves a gap in prior proofs and strengthens the understanding of the extremal genealogy in BBM, with broader implications for ergodic properties of front propagation in branching systems.

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 distinct times far apart must have branched early, and pairs of early-branching extremal particles have negatively correlated positions. This yields the ergodic theorem for BBM. We also address a gap in the path localization argument of \cite{arguin2012ergodic}.

Theorems & Definitions (28)

• Theorem 1.1: Ergodic Theorem for $M_t$, Theorem 1 in arguin2012ergodic
• Theorem 1.2: Early branching across timescales
• Definition 2.1: $\alpha$-$t$-localized particle
• Lemma 2.2
• Lemma 2.3
• proof : Proof of Theorem \ref{['thm:earlybr']}
• Theorem 2.4
• Proposition 2.5
• Lemma 2.6: BKR corollary for cross--index unions
• proof : Proof of Proposition \ref{['thm:decorrelation']}