Almost sure path localisation for the derivative martingale of branching Brownian motion

Abstract

The evolution of the front of branching Brownian motion is determined by the limit of the derivative martingale. In this work, we characterise which particles contribute to this limit. Precisely, we establish a sharp almost sure path localisation result which shows that the limit is determined by those particles whose trajectory stays within a thin tube at distance $s^{1/2}$ from the extremal particle.