STOCHASTIC BURGERS AND KPZ EQUATIONS FROM PARTICLE-SYSTEMS

We consider two strictly related models: a solid on solid interface gr owth model and the weakly asymmetric exclusion process, both on the on e dimensional lattice. It has been proven that, in the diffusive scali ng limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation [8, 13, 18] and the fluctua tion field converges to a generalized Omstein-Uhlenbeck process [8, 10 ], We analyze instead the density fluctuations beyond the hydrodynamic al scale and prove that their limiting distribution solves the (non li near) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar-Paris i-Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e, the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformati on and its microscopic version, We also develop a mathematical theory for the macroscopic equations.