Nonlinear Graphon mean-field systems

Abstract

We address a system of weakly interacting particles where the heterogeneous connections among the particles are described by a graph sequence and the number of particles grows to infinity. Our results extend the existing law of large numbers and propagation of chaos results to the case where the interaction between one particle and its neighbours is expressed as a nonlinear function of the local empirical measure. In the limit of the number of particles which tends to infinity, if the graph sequence converges to a graphon, then we show that the limit system is described by an infinite collection of processes and can be seen as a process in a suitable $L^2$ space constructed via a Fubini extension. The proof is built on decoupling techniques and careful estimates of the Wasserstein distance.