Extended Convergence of the Extremal Process of Branching Brownian Motion

TL;DR

This paper extends the understanding of the extremal process of branching Brownian motion by incorporating particle location, resulting in a new cluster point process limit characterized by a Poisson process with a random intensity measure.

Contribution

It introduces an extended convergence result for the extremal process of branching Brownian motion, including an additional spatial dimension and explicit construction of the limiting measure.

Findings

Limit is a cluster point process on  times;  with a Poisson structure.

The limiting measure is explicitly constructed from the derivative martingale.

The work parallels results for the Gaussian free field.

Abstract

We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_+\times \mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_+\times \mathbb{R}$ with a random intensity measure $Z(dz) \times Ce^{-\sqrt 2x}dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor.