Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift

Abstract

We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators, where the nonlinear drift includes a mean field term of McKean type and the driving noise is Gaussian. Uniform convergence in L^2 sense is achieved by applying L^2-type estimates and the Gronwall Theorem. The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by stochastic nonlinear oscillators with differentiable trajectories