Large deviations for interacting particle systems: joint mean-field and small-noise limit

We consider a system of stochastic interacting particles in ${\mathbb{R}}^d$ and we describe large deviations asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviations principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics.