论文标题
弗拉索夫方程式有关定向超图措施的方程式
Vlasov Equations on Directed Hypergraph Measures
论文作者
论文摘要
在本文中，我们提出了一个框架，以研究定向超图上相互作用粒子系统的平均场极限（MFL）。我们提供了一个非平凡的措施理论观点，并按照定向超图测量（DHGM）对有向超图的扩展，这是紧凑型公制空间上的测量值函数。这些DHGM可以被视为超图极限，其中包括一系列稀疏，密集或中间密度的序列的限制。我们的主要结果表明，DHGM上的VLASOV方程均匀，可以通过高阶相互作用的大型网络的经验分布来近似其解决方案。结果应用于物理学，流行病网络和生态网络的库拉莫托网络，所有这些网络都包括高阶相互作用。为了证明DHGMS上VLASOV方程的近似值和适当性的主要结果，我们坚定地概括了[Kuehn，Xu的方法。 vlasov方程式有关，JDE，339（2022），261--349]到更高维度。
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.
