Small noise asymptotic of the Gallavotti-Cohen functional for diffusion processes

We analyze the Gallavotti-Cohen functional, defined as the empirical power dissipated by the non conservative part of the drift, fora diffusion process in R-n. In particular we prove a large deviation principle in the limit in which the noise vanishes and the time interval diverges. The corresponding rate functional, which satisfies the fluctuation theorem, is expressed in terms of a variational problem on the classical Freidlin-Wentzell functional. As shown in an example, the rate functional can be not strictly convex.