Large deviations of empirical measures of diffusions in weighted topologies

International audienceWe consider large deviations of the empirical measure of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) in a Wasserstein topology. In particular, we derive a precise class of unbounded functions for which the LDP holds. This provides an analogue to the standard Cramer condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten-Schrodinger operator. Secondly, we study more precisely the properties of the Donsker-Varadhan rate functional associated to the LDP. We revisit and generalize some standard duality results as well as a more original decomposition of the rate functional with respect to the symmetric and antisymmetric parts of the dynamics. Finally, we apply our results to overdamped and underdamped Langevin dynamics, showing the applicability of our framework in both unbounded and degenerate situations