Mean-field limit versus small-noise limit for some interacting particle systems

TL;DR

This paper investigates whether the mean-field limit and small-noise limit can be interchanged in interacting particle systems, proving convergence of rate functions in large particle systems within a nonlinear diffusion framework.

Contribution

It establishes the convergence of the rate function of the first particle in a mean-field system to that of the hydrodynamic limit as the number of particles grows.

Findings

Rate function of the first particle converges as particles increase.

Interchanging mean-field and small-noise limits is justified.

Provides theoretical foundation for analyzing particle system limits.

Abstract

In the nonlinear diffusion framework, stochastic processes of McKean-Vlasov type play an important role. In some cases they correspond to processes attracted by their own probability distribution: the so-called self-stabilizing processes. Such diffusions can be obtained by taking the hydrodymamic limit in a huge system of linear diffusions in interaction. In both cases, for the linear and the nonlinear processes, small-noise asymptotics have been emphasized by specific large deviation phenomenons. The natural question, therefore, is: is it possible to interchange the mean-field limit with the small-noise limit? The aim here is to consider this question by proving that the rate function of the first particle in a mean-field system converges to the rate function of the hydrodynamic limit as the number of particles becomes large.