Waiting times for particles in a branching Brownian motion to reach the rightmost position

TL;DR

This paper investigates the asymptotic behavior of the earliest time at which all particles in a branching Brownian motion have descendants reaching the rightmost position, extending understanding of particle reachability over time.

Contribution

It provides asymptotic estimates for the first time when all particles at a given time have descendants at the frontier, a novel analysis in branching Brownian motion.

Findings

Asymptotic estimates for waiting times as s approaches infinity

Extension of Lalley and Sellke's result to collective particle reachability

Insights into the temporal dynamics of particle descendants reaching the frontier

Abstract

It has been proved by Lalley and Sellke [13] that every particle born in a branching Brownian motion has a descendant reaching the rightmost position at some future time. The main goal of the present paper is to estimate asymptotically as s goes to infinity, the first time that every particle alive at the time s has a descendant reaching the rightmost position.