Critical branching Brownian motion with absorption: particle configurations

TL;DR

This paper analyzes the behavior of critical branching Brownian motion with absorption, focusing on particle configurations, number, and positions before extinction as initial position tends to infinity.

Contribution

It provides new asymptotic estimates for particle counts and configurations in critical branching Brownian motion with absorption.

Findings

Estimated particle numbers at given times

Determined the position of the right-most particle

Described typical particle configurations

Abstract

We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.