The Atlas Model and SDEs with Boundary Interaction
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917153046790144 |
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| author | Jettkant, Philipp |
| author_facet | Jettkant, Philipp |
| contents | We study the mean-field limit of the Atlas model and its connection to SDEs with dependence on the distribution of hitting and local times. The Atlas model describes a system of Brownian particles on the real line, where only the lowest ranked particle receives a positive drift, proportional to the number of particles. We show that in the mean-field limit the particle system converges to a novel SDE with reflection at a moving boundary, whose motion is such that the average local time spent at the boundary grows at a constant rate. In general, the boundary is represented by a measure, so the reflection must be interpreted in a relaxed sense. However, for sufficiently regular initial particle profiles, we prove that the boundary is a continuous function. Our analysis relies on a reformulation of the problem via McKean--Vlasov SDEs with interaction through hitting and local times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Atlas Model and SDEs with Boundary Interaction Jettkant, Philipp Probability We study the mean-field limit of the Atlas model and its connection to SDEs with dependence on the distribution of hitting and local times. The Atlas model describes a system of Brownian particles on the real line, where only the lowest ranked particle receives a positive drift, proportional to the number of particles. We show that in the mean-field limit the particle system converges to a novel SDE with reflection at a moving boundary, whose motion is such that the average local time spent at the boundary grows at a constant rate. In general, the boundary is represented by a measure, so the reflection must be interpreted in a relaxed sense. However, for sufficiently regular initial particle profiles, we prove that the boundary is a continuous function. Our analysis relies on a reformulation of the problem via McKean--Vlasov SDEs with interaction through hitting and local times. |
| title | The Atlas Model and SDEs with Boundary Interaction |
| topic | Probability |
| url | https://arxiv.org/abs/2511.17486 |