Sub-exponential tails in biased run and tumble equations with unbounded velocities
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918018438660096 |
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| author | Bouin, Émeric Evans, Josephine Ziviani, Luca |
| author_facet | Bouin, Émeric Evans, Josephine Ziviani, Luca |
| contents | Run and tumble equations are widely used models for bacterial chemotaxis. In this paper, we are interested in the long time behaviour of run and tumble equations with unbounded velocities. We show existence, uniqueness and quantitative convergence towards a steady state. In contrast to the bounded velocity case, the equilibrium has sub-exponential tails and we have sub-exponential rate of convergence to equilibrium. This produces additional technical challenges. We are able to successfully adapt both Harris' type and $\sfL^2-$ hypocoercivity \textit{a la} Dolbeault-Mouhot-Schmeiser techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sub-exponential tails in biased run and tumble equations with unbounded velocities Bouin, Émeric Evans, Josephine Ziviani, Luca Analysis of PDEs Run and tumble equations are widely used models for bacterial chemotaxis. In this paper, we are interested in the long time behaviour of run and tumble equations with unbounded velocities. We show existence, uniqueness and quantitative convergence towards a steady state. In contrast to the bounded velocity case, the equilibrium has sub-exponential tails and we have sub-exponential rate of convergence to equilibrium. This produces additional technical challenges. We are able to successfully adapt both Harris' type and $\sfL^2-$ hypocoercivity \textit{a la} Dolbeault-Mouhot-Schmeiser techniques. |
| title | Sub-exponential tails in biased run and tumble equations with unbounded velocities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.08061 |