On the tightness of the maximum of branching Brownian motion in random environment

We consider one-dimensional branching Brownian motion in a spatially random branching environment (BBMRE) and show that for almost every realisation of the environment, the distributions of the maximal particle of the BBMRE recentred around its median are tight as time evolves. This result is in stark contrast to the fact that the transition fronts in the solution to the randomised Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation are, in general, not bounded uniformly in time. In particular, this highlights that—when compared to the settings of homogeneous branching Brownian motion and the F-KPP equation in a homogeneous environment—the introduction of a random environment leads to a much more intricate behaviour.