95-451 L. Bertini, G. Giacomin

Stochastic Burgers and KPZ equations from particle systems (124K, LaTeX) Oct 12, 95

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Abstract. We consider the weakly asymmetric exclusion process on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations 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 fluctuation field, if suitably rescaled, converges to the Kardar-Parisi-Zhang equation. This provides a microscopic justification of the so called {\em kinetical roughening}, i.e.the non Gaussian fluctuations 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.

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