Relative entropy and modulated free energy without confinement via self-similar transformation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914688183304192 |
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| author | Rosenzweig, Matthew Serfaty, Sylvia |
| author_facet | Rosenzweig, Matthew Serfaty, Sylvia |
| contents | This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13977 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative entropy and modulated free energy without confinement via self-similar transformation Rosenzweig, Matthew Serfaty, Sylvia Analysis of PDEs Mathematical Physics Probability 35Q70, 35Q82, 82C22, 82C40, 82C31, 35Q92, 94A17, 39B62 This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper. |
| title | Relative entropy and modulated free energy without confinement via self-similar transformation |
| topic | Analysis of PDEs Mathematical Physics Probability 35Q70, 35Q82, 82C22, 82C40, 82C31, 35Q92, 94A17, 39B62 |
| url | https://arxiv.org/abs/2402.13977 |