The weakly asymmetric simple exclusion process

The one dimensional n.n. simple exclusion process with generator \epsilon^{-2}L_0+\epsilon^{-1}1L_a, \epsilon > 0, is considered, L_0 and L_a respectively the generators of the symmetric and totally asymmetric simple exclusion processes. Propagation of chaos and convergence to the Burgers equation with viscosity are proven in the limit when \epsilon goes to zero. The density fluctuation field is shown to converge to a generalized Ornstein Uhlenbeck process with mean zero. The time asymptotic covariance kernel is explicitly computed for traveling wave profiles and the result indicates that the shock profile is stable while its space location fluctuates around its average position like a brownian motion. Its diffusion coefficient is explicitly computed.