Convergence to equilibrium for granular media equations and their Euler schemes

Abstract

We introduce a new interacting particle system to investigate the behavior of the nonlinear, nonlocal diffusive equation already studied by Benachour et al. [3, 4]. We first prove an uniform (with respect to time) propagation of chaos. Then, we show that the solution of the nonlinear PDE converges exponentially fast to equilibrium recovering a result established by an other way by Carrillo, McCann and Vilanni [7]. At last we provide explicit and Gaussian confidence intervals for the convergence of an implicit Euler scheme to the stationary distribution of the nonlinear equation.