Additive martingales of the branching Brownian motion
Abstract
In this thesis, we study asymptotic properties of the standard branching Brownian motion, with a specific emphasis on the additive martingales at high temperature. We start by presenting classic and fundamental tools for our investigation. Subsequently, we establish various convergence results that enhance our understanding of the model. In particular, these results include the determination of particles contributing to the additive martingales, the description of the fluctuations of these martingales around their limits, and an approximation of the so-called overlap distribution. Regarding the latter, we believe this is the first time that such an approximation is given. Remarkably, we identify a specific regime in which stable distributions emerge.
Keywords
Cite
@article{arxiv.2407.20227,
title = {Additive martingales of the branching Brownian motion},
author = {Louis Chataignier},
journal= {arXiv preprint arXiv:2407.20227},
year = {2024}
}
Comments
Master thesis completed from April to August 2023 under the supervision of Michel Pain and Pascal Maillard, some results are new and will be included in a future publication, 47 pages, 1 figure