Uniform in time weak propagation of chaos on the torus

Understand

We address the long time behaviour of weakly interacting diffusive particle systems on the d-dimensional torus.

• Our main result is to show that, under certain regularity conditions, the weak error between the empirical distribution of the particle system and the theoretical law of the limiting process (governed by a McKean-Vlasov stochastic differential equation) is of the order O(1/N), uniform in time on [0, infinity), where N is the number of particles in the interacting diffusion.
• This comprises general interaction terms with a small enough mean-field dependence together with interactions terms driven by an H-stable potential.
• Our approach relies on a systematic analysis of the long-time behaviour of the derivatives of the semigroup generated by the McKean-Vlasov SDE, which may be explicitly computed through the linearised Fokker-Planck equation.