Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state.

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions $d \ge 3$, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a non linear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation