The all-time maximum for branching Brownian motion with absorption conditioned on long-time survival

TL;DR

This paper investigates the maximum displacement in a branching Brownian motion with absorption, conditioned on long-term survival, revealing that the furthest particle from the origin occurs around time t^{5/6}.

Contribution

It introduces a detailed analysis of the process conditioned on survival, showing the maximum displacement scales as t^{5/6} and employing a spine decomposition method.

Findings

Maximum displacement occurs around time t^{5/6}.

The process conditioned on survival exhibits specific spatial behavior.

The analysis uses spine decomposition to understand long-time survival.

Abstract

We consider branching Brownian motion in which initially there is one particle at $x$, particles produce a random number of offspring with mean $m+1$ at the time of branching events, and each particle branches at rate $\beta = 1/2m$. Particles independently move according to Brownian motion with drift $-1$ and are killed at the origin. It is well-known that this process eventually dies out with positive probability. We condition this process to survive for an unusually large time $t$ and study the behavior of the process at small times $s \ll t$ using a spine decomposition. We show, in particular, that the time when a particle gets furthest from the origin is of the order $t^{5/6}$.