  The large deviation properties of equilibrium (reversible) lattice gases are
mathematically reasonably well understood. Much less is known in
non--equilibrium, namely for non reversible systems. In this paper we consider
a simple example of a non--equilibrium situation, the symmetric simple
exclusion process in which we let the system exchange particles with the
boundaries at two different rates. We prove a dynamical large deviation
principle for the empirical density which describes the probability of
fluctuations from the solutions of the hydrodynamic equation. The so called
quasi potential, which measures the cost of a fluctuation from the stationary
state, is then defined by a variational problem for the dynamical large
deviation rate function. By characterizing the optimal path, we prove that the
quasi potential can also be obtained from a static variational problem
introduced by Derrida, Lebowitz, and Speer.
