boundary conditions, with the same parameters $A=1$, $B=E$ and $C=\frac12$.

A closely related convergence result was recently shown by Corwin and Shen [@Corwin2016], who consider the height function associated to a variant of the WASEP in contact with reservoirs, apply Gärtner’s microscopic Cole-Hopf transform, and show that in the scaling limit the process converges to a solution of the SHE with Robin boundary conditions. However, the model we study here has parameters that are not admissible in [@Corwin2016], because in their formulation the parameters $A$ and $B$ have to be positive (see e.g. Proposition 2.7 and p.14 in [@Corwin2016]), while we would get $A = B = -E^2/8$ (see Proposition \[prop:link\] below and also Remark \[rmk:CS\]). Apart from that, our methods are very different from the ones used in [@Corwin2016] and we study the microscopic density fluctuation field without relying on the Cole-Hopf transform.

### Uniqueness of energy solutions and boundary behavior at the macroscopic scale

The convergence proofs described above show relative compactness of the microscopic density fluctuations field (resp. the microscopic height fluctuation field) under rescaling, and that any limiting point is an energy solution to the stochastic Burgers equation with Dirichlet boundary conditions (resp. the KPZ equation 