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

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