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

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