  We consider a class of nearest-neighbor weakly asymmetric mass conservative
particle systems evolving on $\mathbb{Z}$, which includes zero-range and types
of exclusion processes, starting from a perturbation of a stationary state.
When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, 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 $\gamma=1/2$, 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.
