Persistent anisotropy in multi-dimensional branching Brownian motion

We study angular variation in the frontier of branching Brownian motion (BBM) in multiple dimensions. The time average of this frontier is tied to the long-time limit of the critical derivative martingale, which is a random integrable function on the sphere. We show that this function is nowhere locally bounded. As a consequence, BBM exhibits strong persistent anisotropy: there are dense arbitrarily large gaps between the BBM frontier in different directions.