Electronic Communications in Probability

High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process

Constantin Glenz, Nicola Kistler, and Marius A. Schmidt

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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.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 86, 12 pp.

Received: 19 December 2017
Accepted: 31 October 2018
First available in Project Euclid: 23 November 2018

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Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

(in)homogeneous Branching Brownian motions McKean martingale Bovier-Hartung extremal process

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Glenz, Constantin; Kistler, Nicola; Schmidt, Marius A. High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process. Electron. Commun. Probab. 23 (2018), paper no. 86, 12 pp. doi:10.1214/18-ECP187. https://projecteuclid.org/euclid.ecp/1542942175

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