converges to the SHE, and in particular we show that already at the microscopic level the Cole-Hopf transformation changes the boundary conditions from Neumann to Robin’s type.

Model and definitions {#s2}
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The microscopic dynamics: WASEP in contact with stochastic reservoirs
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For $n$ a positive integer, we define $\Lambda_n = \{1,\ldots, n-1\}$ and $\Omega_n=\{0,1\}^{\Lambda_n}$ as the state space of a Markov process $\{\eta_t^n\,; \, t {\geqslant}0\}$, whose dynamics is entirely encoded into its infinitesimal generator, denoted below by ${{\mathcal L}}_n$. More precisely, our process belongs to the family of well-known *weakly asymmetric simple exclusion processes*. Here we consider that the strength of asymmetry is ruled by a parameter $\gamma {\geqslant}\frac12$, and we put the system of particles in contact with two stochastic reservoirs, whose rates of injection/removal of particles from the bulk depend on a parameter $\rho\in{(0,1)}$, which is fixed. To keep notation simple we omit the dependence on $\rho$ in all the quantities that we define ahead and in order to facilitate future computations, we write the generator as $$\mathcal L_n  =  {{\mathcal L}}_n^{\textrm{bulk}} + {{\mathcal L}}_n^{\mathrm{bnd}}\,,$$ where ${{\mathcal L}}_n^{\textrm{bulk}}$ and ${{\mathcal L}}_n^{\mathrm{bnd}}$ given below encode respectively the dynamics on the bulk, and on the left/right boundaries. For 