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