Propagation of Chaos for Singular Interactions via Regular Drivers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913950147280896 |
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| author | Qi, Qian |
| author_facet | Qi, Qian |
| contents | We introduce a framework to prove propagation of chaos for interacting particle systems with singular, density-dependent interactions, a classical challenge in mean-field theory. Our approach is to define the dynamics implicitly via a regular driver function. This regular driver is engineered to generate a singular effective interaction, yet its underlying regularity provides the necessary analytical control. Our main result establishes propagation of chaos under a key dissipation condition. The proof hinges on deriving a priori bounds, uniform in a regularization parameter, for the densities of the associated non-linear Fokker-Planck equations. Specifically, we establish uniform bounds in $L^\infty([0,T]; H^1(\R) \cap L^\infty(\R))$. These are established via an energy method for the excess mass to secure the $L^\infty$ bound, and a novel energy estimate for the $H^1$ norm that critically leverages the stability condition. These bounds provide the compactness necessary to pass to the singular limit, showing convergence to a McKean-Vlasov SDE with a singular, density-dependent diffusion coefficient. This work opens a new path to analyzing singular systems and provides a constructive theory for a class of interactions that present significant challenges to classical techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Propagation of Chaos for Singular Interactions via Regular Drivers Qi, Qian Analysis of PDEs We introduce a framework to prove propagation of chaos for interacting particle systems with singular, density-dependent interactions, a classical challenge in mean-field theory. Our approach is to define the dynamics implicitly via a regular driver function. This regular driver is engineered to generate a singular effective interaction, yet its underlying regularity provides the necessary analytical control. Our main result establishes propagation of chaos under a key dissipation condition. The proof hinges on deriving a priori bounds, uniform in a regularization parameter, for the densities of the associated non-linear Fokker-Planck equations. Specifically, we establish uniform bounds in $L^\infty([0,T]; H^1(\R) \cap L^\infty(\R))$. These are established via an energy method for the excess mass to secure the $L^\infty$ bound, and a novel energy estimate for the $H^1$ norm that critically leverages the stability condition. These bounds provide the compactness necessary to pass to the singular limit, showing convergence to a McKean-Vlasov SDE with a singular, density-dependent diffusion coefficient. This work opens a new path to analyzing singular systems and provides a constructive theory for a class of interactions that present significant challenges to classical techniques. |
| title | Propagation of Chaos for Singular Interactions via Regular Drivers |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.14981 |