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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.10515 |
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| _version_ | 1866915389289529344 |
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| author | Addario-Berry, Louigi Arias, Arturo Arellano Lin, Jessica |
| author_facet | Addario-Berry, Louigi Arias, Arturo Arellano Lin, Jessica |
| contents | We consider the long-time behaviour of binary branching Brownian motion (BBM) where the branching rate depends on a periodic spatial heterogeneity. We prove that almost surely as $t\to\infty$, the heterogeneous BBM at time $t$, normalized by $t$, approaches a deterministic convex shape with respect to Hausdorff distance. Our approach relies on establishing tail bounds on the probability of existence of BBM particles lying in half-spaces, which in particular yields the asymptotic speed of propagation of projections of the BBM in every direction. Our arguments are primarily probabilistic in nature, but additionally exploit the existence of a "front speed" (or minimal speed of a pulsating traveling front solution) for the Fisher-KPP reaction-diffusion equation naturally associated to the BBM. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A shape theorem for BBM in a periodic environment Addario-Berry, Louigi Arias, Arturo Arellano Lin, Jessica Probability 60J80, 60F15, 35K57, 35B40 We consider the long-time behaviour of binary branching Brownian motion (BBM) where the branching rate depends on a periodic spatial heterogeneity. We prove that almost surely as $t\to\infty$, the heterogeneous BBM at time $t$, normalized by $t$, approaches a deterministic convex shape with respect to Hausdorff distance. Our approach relies on establishing tail bounds on the probability of existence of BBM particles lying in half-spaces, which in particular yields the asymptotic speed of propagation of projections of the BBM in every direction. Our arguments are primarily probabilistic in nature, but additionally exploit the existence of a "front speed" (or minimal speed of a pulsating traveling front solution) for the Fisher-KPP reaction-diffusion equation naturally associated to the BBM. |
| title | A shape theorem for BBM in a periodic environment |
| topic | Probability 60J80, 60F15, 35K57, 35B40 |
| url | https://arxiv.org/abs/2507.10515 |