The genealogy of extremal particles of Branching Brownian Motion

TL;DR

This paper investigates the detailed statistics of the extremal particles in Branching Brownian Motion, revealing their ancestral origins and providing insights into the limiting extremal process.

Contribution

It characterizes the paths and ancestral origins of the extremal particles, advancing understanding of their full distribution and the structure at the edge of the process.

Findings

Extremal particles mostly originate from ancestors near initial or final times.

Paths of extremal particles are characterized up to a certain precision.

Provides a heuristic picture of the extremal process at the edge.

Abstract

Branching Brownian Motion describes a system of particles which diffuse in space and split into offsprings according to a certain random mechanism. In virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher-KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first-, second-, third- etc. largest). In particular, we prove that in the large $t-$limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order one from time 0, or within a distance of order one from time $t$. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of Branching Brownian Motion "at the edge" emerges, which sheds light on the still unknown limiting extremal process.