A note on Fokker-Planck equations and graphons

Fabio Coppini (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 have been introduced in the 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 systems focusing on the relation between initial conditions and interaction network: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available. This in turn implies that very different graphons can lead to exactly the same particle behavior, shedding some light on the network influence on the dynamics. Examples of such representation are provided. In particular, step kernels represent a class of graphons to which our result applies: this in turn opens the way to approximate the graphon particle system with a finite system of Fokker-Planck equations. As a byproduct, we show that when the initial condition is uniform, every graphon with constant degree leads to a behavior indistinguishable from the well-known mean-field limit.

Mots clés: Fokker-Planck equation; graphons; mean-field systems; Interacting particles; McKean-Vlasov; graphon particle system; step kernels

Domaines: Probabilités [math.PR]

Fabio Coppini. A note on Fokker-Planck equations and graphons. 2021. ⟨hal-03138054v1⟩