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

We consider one-dimensional branching Brownian motion in spatially random branching environment (BBMRE) and show that for almost every realisation of the environment, the distributions of the maximal particle of the BBMRE re-centred 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.