Quantitative convergence in relative entropy for a moderately interacting particle system on $\mathbb{R}^d$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913354148216832 |
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| author | Chen, Li Holzinger, Alexandra Huo, Xiaokai |
| author_facet | Chen, Li Holzinger, Alexandra Huo, Xiaokai |
| contents | This article shows how to combine the relative entropy method by D. Bresch, P.-E. Jabin, and Z. Wang in arXiv:1706.09564, arXiv:1906.04093 and the regularized $L^2(\mathbb{R}^d)$-estimate by Oelschläger (Probability theory and related fields, 1987) to prove a strong propagation of chaos result for the viscous porous medium equation from a moderately interacting particle system in $L^\infty(0,T; L^1(\mathbb{R}^d))$-norm. In the moderate interacting setting, the interacting potential is a smoothed Dirac Delta distribution, however, current results regarding the relative entropy methods for singular potentials do not apply. The result holds on $\mathbb{R}^d$ for any dimension $d\geq 1$ and provides a quantitative result where the rate of convergence depends on the moderate scaling parameter and the dimension $d\geq 1$. Additionally, the presented method can be adapted for moderately interacting systems for which a certain convergence probability holds -- thus a propagation of chaos result in relative entropy can be obtained for kernels approximating Coulomb potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative convergence in relative entropy for a moderately interacting particle system on $\mathbb{R}^d$ Chen, Li Holzinger, Alexandra Huo, Xiaokai Analysis of PDEs Probability 35K55, 35K65, 65C35, 60H30 This article shows how to combine the relative entropy method by D. Bresch, P.-E. Jabin, and Z. Wang in arXiv:1706.09564, arXiv:1906.04093 and the regularized $L^2(\mathbb{R}^d)$-estimate by Oelschläger (Probability theory and related fields, 1987) to prove a strong propagation of chaos result for the viscous porous medium equation from a moderately interacting particle system in $L^\infty(0,T; L^1(\mathbb{R}^d))$-norm. In the moderate interacting setting, the interacting potential is a smoothed Dirac Delta distribution, however, current results regarding the relative entropy methods for singular potentials do not apply. The result holds on $\mathbb{R}^d$ for any dimension $d\geq 1$ and provides a quantitative result where the rate of convergence depends on the moderate scaling parameter and the dimension $d\geq 1$. Additionally, the presented method can be adapted for moderately interacting systems for which a certain convergence probability holds -- thus a propagation of chaos result in relative entropy can be obtained for kernels approximating Coulomb potentials. |
| title | Quantitative convergence in relative entropy for a moderately interacting particle system on $\mathbb{R}^d$ |
| topic | Analysis of PDEs Probability 35K55, 35K65, 65C35, 60H30 |
| url | https://arxiv.org/abs/2311.01980 |