Stochastic Burgers and KPZ equations from particle systems

We consider the weakly asymmetric exclusion process on the one dimensional lattice. It has been proven that, in the diiusive scaling limit, the density eld evolves according to the Burgers equation 8, 19, 14] and the uctuation eld converges to a generalized Ornstein-Uhlenbeck process 8, 10]. We analyze instead the density uctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. We also study an interface growth model, for which the microscopic dynamics is a Solid-On-Solid type deposition process. We prove that the uctua-tion eld, if suitably rescaled, converges to the Kardar-Parisi-Zhang equation. This provides a microscopic justiication of the so called kinetical roughening, i.e. the non Gaussian uctuations in some nonequilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also exploit the (known) connection between the two microscopic models and develop a mathematical theory for the macroscopic equations.