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 distribution of the maximal particle of the BBMRE re-centred around its median is tight. This result is in stark contrast to the fact that the transition fronts in the solution to the randomised F-KPP equation are not bounded uniformly in time. In particular, this highlights that -- when compared to the setting of homogeneous branching -- the introduction of a random environment leads to a much more intricate situation.