Long time limits and concentration bounds for graphon mean field systems

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random 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 convexity/dissipativity assumptions, we show the exponential ergodicity for both systems, establish a uniform-in-time law of large numbers for the empirical measure of particle states, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided. We also provide uniform-in-time exponential concentration bounds for the rate of the LLN convergence under additional integrability conditions. Based on joint works with Erhan Bayraktar and Suman Chakraborty.