Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

Abstract

We consider the weakly asymmetric simple exclusion process on the discrete\nspace $\\{1,...,n-1\\}$, in contact with stochastic reservoirs, both with density\n$\\rho\\in{(0,1)}$ at the extremity points, and starting from the invariant\nstate, namely the Bernoulli product measure of parameter $\\rho$. Under time\ndiffusive scaling $tn^2$ and for $\\rho=\\frac12$, when the asymmetry parameter\nis taken of order $1/ \\sqrt n$, we prove that the density fluctuations at\nstationarity are macroscopically governed by the energy solution of the\nstochastic Burgers equation with Dirichlet boundary conditions, which is shown\nto be unique and different from the Cole-Hopf solution.\n