Conditions for uniform in time convergence: applications to averaging, numerical discretisations and mean-field systems

We establish general conditions under which there exists uniform in time convergence between a stochastic process and its approximated system. These standardised conditions consist of a local in time estimate between the original and the approximated process as well as of a contraction property for one of the processes and a uniform control for the other one. Specifically, the results we present provide global in time error bounds for multiscale methods and numerical discretisations as well as uniform in time propagation of chaos bounds for mean-field particle systems. We provide a general method of proof which can be applied to many types of approximation. In all three scenarios, examples where the joint conditions are verified and uniform in time convergence is achieved are given.