Non equilibrium stationary state for the SEP with births and deaths

We consider the symmetric simple exclusion process in the interval ${\text{La}}_N:=[-N,N]\cap \mathbb{Z}$ with births and deaths taking place respectively on suitable boundary intervals $I_{+}$ and $I_{-}$, as introduced in De Masi et al. (J. Stat. Phys. 2011). We study the stationary measure density profile in the limit $N\to \mathrm{\infty }$