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2018 High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process
Constantin Glenz, Nicola Kistler, Marius A. Schmidt
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP187

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.

Citation

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Constantin Glenz. Nicola Kistler. Marius A. Schmidt. "High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP187

Information

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

zbMATH: 07023472
MathSciNet: MR3882227
Digital Object Identifier: 10.1214/18-ECP187

Subjects:
Primary: 60G70, 60J80, 82B44

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