Phase transition of the consistent maximal displacement of branching Brownian motion

TL;DR

This paper investigates the behavior of particles in branching Brownian motion near a shifted linear boundary, revealing a phase transition at a critical time scale related to the shift magnitude.

Contribution

It identifies a phase transition in the maximal displacement of particles staying close to a line with slope \\sqrt{2} + \\varepsilon, depending on the time scale.

Findings

Discovered a phase transition at time scale \\varepsilon^{-3/2}

Characterized the behavior of particles near the shifted boundary

Determined initial positions for survival in drifted branching Brownian motion

Abstract

Consider branching Brownian motion in which we begin with one particle at the origin, particles independently move according to Brownian motion, and particles split into two at rate one. It is well-known that the right-most particle at time $t$ will be near $\sqrt{2} t$. Roberts considered the so-called consistent maximal displacement and showed that with high probability, there will be a particle at time $t$ whose ancestors stayed within a distance $ct^{1/3}$ of the curve $s \mapsto \sqrt{2} s$ for all $s \in [0, t]$, where $c = (3 \pi^2)^{1/3}/\sqrt{2}$. We consider the question of how close the trajectory of a particle can stay to the curve $s \mapsto (\sqrt{2} + \varepsilon) s$ for all $s \in [0, t]$, where $\varepsilon> 0$ is small. We find that there is a phase transition, with the behavior changing when $t$ is of the order $\varepsilon^{-3/2}$. This result allows us to determine, for branching Brownian motion in which particles have a drift to the left of $\sqrt{2} + \varepsilon$ and are killed at the origin, the position at which a particle needs to begin at time zero for there to be a high probability that the process avoids extinction until time $t$.