Additive martingales of the branching Brownian motion

TL;DR

This thesis investigates the asymptotic behavior of additive martingales in branching Brownian motion, revealing convergence properties, particle contributions, fluctuations, and a novel approximation of the overlap distribution, including stable distribution regimes.

Contribution

It provides new convergence results, characterizes particle contributions, describes fluctuations, and offers the first approximation of the overlap distribution in branching Brownian motion.

Findings

Identification of particles contributing to martingales

Description of fluctuations around martingale limits

First approximation of the overlap distribution

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.