On McKean's martingale in the Bovier-Hartung extremal process

TL;DR

This paper offers a new interpretation of McKean's martingale in the context of branching Brownian motion, linking it to a law of large numbers for high-points and highlighting its broader relevance.

Contribution

It provides an alternative interpretation of McKean's martingale as a law of large numbers for high-points in BBM, extending its conceptual understanding.

Findings

McKean's martingale relates to high-point statistics in BBM.

The interpretation suggests McKean-like martingales are universal in BBM models.

Abstract

It has been proved by Bovier & Hartung [Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean's martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean's martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean's martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, 'McKean-like martingales' are naturally expected to arise in all models belonging to the BBM-universality class.