Fine asymptotics for the consistent maximal displacement of branching Brownian motion

TL;DR

This paper investigates the behavior of particles near the frontier in branching Brownian motion, improving bounds on their paths and revealing qualitative differences between existing analytical approaches.

Contribution

It provides refined asymptotic bounds for the maximal displacement and compares two different analytical methods, highlighting their qualitative differences.

Findings

Improved bounds on the maximal displacement paths.

Identification of qualitative differences between two analytical approaches.

Enhanced understanding of particle behavior near the frontier.

Abstract

It is well-known that the maximal particle in a branching Brownian motion sits near $\sqrt2 t - \frac{3}{2\sqrt2}\log t$ at time $t$. One may then ask about the paths of particles near the frontier: how close can they stay to this critical curve? Two different approaches to this question have been developed. We improve upon the best-known bounds in each case, revealing new qualitative features including marked differences between the two approaches.