We prove the hydrodynamical limit for weakly asymmetric simple exclusion processes. A large deviation property with respect to this limit is established for the symmetric case. We treat also the situation where a slow reaction (creation and annihilation of particles) is present. IntroductionWhen we study the hydrodynamical limit (reduced description) of an infinite particle system we can obtain the same limiting equation for different systems. For instance, both symmetric simple exclusion and a system of independent symmetric random walks give rise to the heat equation. However, in the study of the large deviations, the rate functions are different for these two systems.In this spirit the large deviations from the hydrodynamical limit for independent Brownian motions were studied in [7], using an approach that can be also used for strongly interacting systems, like simple exclusion, as we show in this paper.The case of independent particles is simple because only the empirical density field appears in the martingales considered. In the case of strongly interacting systems the density field is no longer autonomous and we need to control the deviations of the empirical correlation fields. In Theorem 2.1 we prove that the probability that any correlation field deviates from an appropriate function of the density field is superexponentially small, whereas the large deviations of the density fields are only exponentially small.The other ingredient needed for the lower bound is a large number of hydrodynamical limit theorems for several small perturbations of the original

Anomalous large thermal conductivity has been observed numerically and experimentally in one- and two-dimensional systems. There is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimensions 1 and 2 if momentum is conserved, while it remains finite in dimension d > or = 3. We consider a system of harmonic oscillators perturbed by a nonlinear stochastic dynamics conserving momentum and energy. We compute explicitly the time correlation function of the energy current C(J)(t), and we find that it behaves, for large time, like t(-d/2) in the unpinned cases, and like t(-d/2-1) when an on-site harmonic potential is present. This result clarifies the role of conservation of momentum in the anomalous thermal conductivity in low dimensions.

We study the large scale space-time fluctuations of an interface which is modeled by a massless scalar field with reversible Langevin dynamics. For a strictly convex interaction potential we prove that on a large space-time scale these fluctuations are governed by an infinite-dimensional Ornstein-Uhlenbeck process. Its effective diffusion type covariance matrix is characterized through a variational formula.