Stationarity and uniform in time convergence for the graphon particle system

Understand

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.