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

Jiří Černý; Alexander Drewitz; Pascal Oswald

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arXiv · 2212.12390

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

Jiří Černý·Alexander Drewitz·Pascal Oswald

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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.

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Jiří Černý, Alexander Drewitz, Pascal Oswald. 2022-12-23. On the tightness of the maximum of branching Brownian motion in random environment. https://arxiv.org/abs/2212.12390

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