The extremal process of branching Brownian motion with absorption

TL;DR

This paper analyzes the extremal behavior of branching Brownian motion with absorption, showing convergence to a decorated Poisson process and describing the distribution of the rightmost particle.

Contribution

It establishes the convergence of the extremal process and the distribution of the maximum for branching Brownian motion with absorption, extending previous results to this absorbed case.

Findings

Extremal process converges to a decorated Poisson point process.

Distribution of the rightmost particle converges to a shifted Gumbel law.

Provides a detailed probabilistic description of extremal particles with absorption.

Abstract

In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions with drift and are killed upon reaching the origin. We prove that the extremal process of this branching Brownian motion with absorption converges to a random shifted decorated Poisson point process. Furthermore, we show that the law of the right-most particle converges to the law of a random shifted Gumbel random variable.