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

TL;DR

This paper proves a convergence result for nonlinear stochastic oscillators with mean-field drift, showing how they approximate diffusions with mean-field effects using the Smoluchowski-Kramers limit.

Contribution

It establishes a uniform L^2 convergence for stochastic Newton equations with mean-field drift, extending the understanding of the Smoluchowski-Kramers limit in this context.

Findings

Uniform L^2 convergence of solutions

Approximation of mean-field diffusions by nonlinear oscillators

Application of Gronwall's inequality for convergence proof

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