Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917265141661696 |
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| author | Baker, Graeme Hambly, Ben Jettkant, Philipp |
| author_facet | Baker, Graeme Hambly, Ben Jettkant, Philipp |
| contents | We study a system of reflected Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. If this local time drift is too strong, such systems exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterisation of this finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric setting, which admits a McKean--Vlasov representation, and establish propagation of chaos. In the absence of breakdowns, the McKean--Vlasov equation exhibits multiple stationary and unique self-similar solutions and we prove convergence to these profiles. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarisation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08837 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times Baker, Graeme Hambly, Ben Jettkant, Philipp Probability 60H10, 60F17, 60J55, 05C50 We study a system of reflected Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. If this local time drift is too strong, such systems exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterisation of this finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric setting, which admits a McKean--Vlasov representation, and establish propagation of chaos. In the absence of breakdowns, the McKean--Vlasov equation exhibits multiple stationary and unique self-similar solutions and we prove convergence to these profiles. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarisation. |
| title | Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times |
| topic | Probability 60H10, 60F17, 60J55, 05C50 |
| url | https://arxiv.org/abs/2503.08837 |