论文标题
库拉莫托型模型的图形均值限制
Graphop Mean-Field Limits for Kuramoto-Type Models
论文作者
论文摘要
最初是在统计物理学中的相互作用粒子系统，网络/图上的微分方程中出现的，已渗透到许多数学领域以及许多应用中。该领域中的一个中心问题是找到动力学的合适近似值，因为节点/顶点的数量倾向于无穷大，即在较大的图形限制中。在这种情况下的基石是vlasov-fokker-planck方程（VFPE），描述了平均场水平上的粒子密度。对于全耦合系统，证明VFPE的严格近似是许多类别的粒子系统，这是相当古典的。对于汇聚到Graphon限制的密集图，人们还知道，对于某些类别的模型（例如，对于图形上的库拉莫托模型），平均场近似具有。然而，中间密度和稀疏图的空间显然非常相关。在这里，我们证明，库拉莫托模型可以在平均场限制中近似于比图形限制的一般图形限制。特别是，我们的贡献如下。 （i）我们展示了如何通过考虑图形机，将操作员理论更加抽象地介绍到VFP中。图形机最近被提出为图形限制理论的统一方法，在这里我们表明它们可用于图形上的微分方程。 （ii）对于图表上的库拉莫托模型，我们严格地证明有一个vfpe方程在均值场上近似。 （iii）此平均场VFPE涉及图形机，我们证明了弱解的存在，独特性和连续的图形依赖性。 （iv）在技术层面上，我们的结果依赖于设计新的合适图形收敛度指标，并在紧凑型Abelian组上采用傅立叶分析来使用可总结内核来近似图形。
Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many applications. One central problem in the field is to find suitable approximations of the dynamics as the number of nodes/vertices tends to infinity, i.e., in the large graph limit. A cornerstone in this context are Vlasov-Fokker-Planck equations (VFPEs) describing a particle density on a mean-field level. For all-to-all coupled systems, it is quite classical to prove the rigorous approximation by VFPEs for many classes of particle systems. For dense graphs converging to graphon limits, one also knows that mean-field approximation holds for certain classes of models, e.g., for the Kuramoto model on graphs. Yet, the space of intermediate density and sparse graphs is clearly extremely relevant. Here we prove that the Kuramoto model can be be approximated in the mean-field limit by far more general graph limits than graphons. In particular, our contributions are as follows. (I) We show, how to introduce operator theory more abstractly into VFPEs by considering graphops. Graphops have recently been proposed as a unifying approach to graph limit theory, and here we show that they can be used for differential equations on graphs. (II) For the Kuramoto model on graphs we rigorously prove that there is a VFPE equation approximating it in the mean-field sense. (III) This mean-field VFPE involves a graphop, and we prove the existence, uniqueness, and continuous graphop-dependence of weak solutions. (IV) On a technical level, our results rely on designing a new suitable metric of graphop convergence and on employing Fourier analysis on compact abelian groups to approximate graphops using summability kernels.
