Vlasov Equations on Directed Hypergraph Measures
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912232356446208 |
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| author | Kuehn, Christian Xu, Chuang |
| author_facet | Kuehn, Christian Xu, Chuang |
| contents | In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measure-theoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349] to higher-dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_03806 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Vlasov Equations on Directed Hypergraph Measures Kuehn, Christian Xu, Chuang Analysis of PDEs Classical Analysis and ODEs Dynamical Systems Functional Analysis Probability 35R02, 35Q84, 05C65 In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measure-theoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349] to higher-dimensions. |
| title | Vlasov Equations on Directed Hypergraph Measures |
| topic | Analysis of PDEs Classical Analysis and ODEs Dynamical Systems Functional Analysis Probability 35R02, 35Q84, 05C65 |
| url | https://arxiv.org/abs/2207.03806 |