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

2022/12/23 by Jǐŕı Černý, Černý, Jiří, Alexander Drewitz +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2212.12390

openalex publication_date 2022/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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