A stochastic Burgers equation from a class of microscopic interactions

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on Z, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order O (n^-) for 1/2<1, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein–Uhlenbeck process. However, at the critical weak asymmetry when =1/2, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp “Boltzmann–Gibbs” estimate which improves on earlier bounds.