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

TL;DR

This paper proves that in a one-dimensional branching Brownian motion within a random environment, the distribution of the maximum particle remains tight over time, contrasting with the unbounded transition fronts in the corresponding F-KPP equation.

Contribution

It demonstrates the tightness of the maximum particle distribution in BBMRE for almost every environment, revealing complex behavior introduced by randomness.

Findings

Maximum particle distribution remains tight over time in BBMRE.

Contrasts with unbounded transition fronts in homogeneous settings.

Random environment induces intricate behavior in branching processes.

Abstract

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.