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 mean field type with weights characterized by an underlying graphon. limit given a graphon system consisting independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, including certain convexity condition, we show exponential ergodicity for both systems, establish uniform-in-time law numbers marginal as number particles increases, introduce Euler approximation. precise rate convergence approximation provided.