  We consider large deviations of empirical measures of diffusion processes. In
a first part, we present conditions to obtain a large deviations principle
(LDP) for a precise class of unbounded functions. This provides an analogue to
the standard Cram\'er condition in the context of diffusion processes, which
turns out to be related to a spectral gap condition for a Witten-Schr\"odinger
operator. Secondly, we study more precisely the properties of the
Donsker-Varadhan rate functional associated with 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 for degenerate diffusions in unbounded configuration spaces.
