We prove the hydrostatics of boundary driven gradient exclusion processes, Fick's law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.

We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.where ∆ represents the Laplacian and where B and D are non-negative polynomials.We investigate the static large deviations of the empirical measure under the stationary state. In contrast with the previous dynamics [10,21], in which the hydrodynamic equation 2010 Mathematics Subject Classification. Primary 82C22, secondary 60F10, 82C35.

Fix a functionwhere d ≥ 1, and each function W k : R → R is strictly increasing, right continuous with left limits. We prove the equilibrium fluctuations for exclusion processes with conductances, induced by W , in random environments, when the system starts from an equilibrium measure. The asymptotic behavior of the empirical distribution is governed by the unique solution of a stochastic differential equation taking values in a certain nuclear Fréchet space.

We prove the dynamical large deviations for a particle system in which particles may have different velocities. We assume that we have two infinite reservoirs of particles at the boundary: this is the so-called boundary driven process. The dynamics we considered consists of a weakly asymmetric simple exclusion process with collision among particles having different velocities.2000 Mathematics Subject Classification. Primary 82C22; Secondary 60F10, 82C35.