Critical branching Brownian motion with absorption: survival probability

TL;DR

This paper studies the survival probability of critical branching Brownian motion with absorption at zero, providing improved bounds that support theoretical predictions and extend previous results.

Contribution

It offers new upper and lower bounds on survival probability, advancing understanding of critical branching Brownian motion with absorption.

Findings

Bounds on survival probability are tightened compared to previous work.

Results partially confirm nonrigorous predictions of Derrida and Simon.

The process almost surely dies out, with quantified survival chances over time.

Abstract

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-\sqrt{2}$. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).