  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, including a certain convexity condition, we show the
exponential ergodicity for both systems, establish the uniform-in-time law of
large numbers for marginal distributions 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.
