  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.
