## Electronic Communications in Probability

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

#### 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.

#### Article information

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

Dates
Accepted: 31 October 2018
First available in Project Euclid: 23 November 2018

https://projecteuclid.org/euclid.ecp/1542942175

Digital Object Identifier
doi:10.1214/18-ECP187

#### Citation

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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