On the tightness of the maximum of branching Brownian motion in random environment
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914845552541696 |
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| author | Černý, Jiří Drewitz, Alexander Oswald, Pascal |
| author_facet | Černý, Jiří Drewitz, Alexander Oswald, Pascal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12390 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the tightness of the maximum of branching Brownian motion in random environment Černý, Jiří Drewitz, Alexander Oswald, Pascal Probability 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. |
| title | On the tightness of the maximum of branching Brownian motion in random environment |
| topic | Probability |
| url | https://arxiv.org/abs/2212.12390 |