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

2013/03/01 by Al-Talibi, Haidar, Hilbert, Astrid, Kolokoltsov, Vassili
#60H05 #60K99 #65C30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1303.0110

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 L² sense is achieved by applying L²-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

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