  We consider a diffusion process on $\mathbb R^n$ and prove a large deviation
principle for the empirical process in the joint limit in which the time window
diverges and the noise vanishes. The corresponding rate function is given by
the expectation of the Freidlin-Wentzell functional per unit of time. As an
application of this result, we obtain a variational representation of the rate
function for the Gallavotti-Cohen observable in the small noise and large time
limits.

Probability;Statistical Mechanics