A Note on Fokker–Planck Equations and Graphons
A Note on Fokker–Planck Equations and Graphons
复制标题
关于福克-普朗克方程和图元的注释
DOI：
10.1007/s10955-022-02905-7
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发表时间：
2021
影响因子：
1.6
通讯作者：
Fabio Coppini
中科院分区：
文献类型：
--
作者：
Fabio Coppini
Fokker-Planck方程代表了一个合适的描述的有限时间行为的一大类粒子系统的规模趋于无穷大。最近，图极限理论被引入到经典的平均场框架中来解释粒子间的非均匀相互作用。在许多情况下，这样的网络异质性被保留在极限中，该极限从单个福克-普朗克方程（也称为McKean-Vlasov）变成通过图子耦合的非线性偏微分方程（PDE）的无限系统。虽然从应用的角度来看，有吸引力的一些严格的结果存在的graphon粒子系统。本文讨论了这样一个极限系统，着重讨论了相互作用网络与初始条件之间的关系：如果系统的初始数据和图子度满足适当的条件，则可以得到一个简单得多的解的表示.从这个结果中可以得出有趣的结论：基本上不同的图子可以产生完全相同的粒子行为，例如，如果初始条件与图子无关，只要图子具有常数度，粒子行为就无法与众所周知的平均场极限区分开来;阶梯核，图极限理论的构建块，可以用来近似具有有限福克-普朗克方程系统和显式收敛率的图子粒子系统。
Fokker–Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits has been introduced in the classical mean-field framework to account for heterogeneous interactions among particles. In many instances, such network heterogeneity is preserved in the limit which turns from being a single Fokker–Planck equation (also known as McKean–Vlasov) to an infinite system of non-linear partial differential equations (PDE) coupled by means of a graphon. While appealing from an applied viewpoint, few rigorous results exist on the graphon particle system. This note addresses such limit system focusing on the relation between interaction network and initial conditions: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available.Interesting conclusions can be drawn from this result: substantially different graphons can give rise to exactly the same particle behavior, e.g., if the initial condition is independent of the graphon, the particle behavior is indistinguishable from the well-known mean-field limit as long as the graphon has constant degree; step-kernels, a building block of graph limit theory, can be used to approximate the graphon particle system with a finite system of Fokker–Planck equations and an explicit rate of convergence.
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