Stationarity and uniform in time convergence for the graphon particle system

We consider the long time behavior of heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, we show the exponential ergodicity for both systems, establish the uniform in time law of large numbers as the number of particles increases, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided.