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arXiv · 1712.04325

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

Constantin Glenz·Nicola Kistler·Marius A. Schmidt
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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.

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Constantin Glenz, Nicola Kistler, Marius A. Schmidt. 2017-12-12. On McKean's martingale in the Bovier-Hartung extremal process. https://arxiv.org/abs/1712.04325

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