The boundary driven zero-range process

We study the asymptotic behaviour of the symmetric zero-range process in the finite lattice $\{1,\ldots, N-1\}$ with slow boundary, in which particles are created at site $1$ or annihilated at site $N\!-\!1$ with a rate proportional to $N^{-\theta}$, for $\theta\geq 1$. We present the invariant measure for this model and obtain the hydrostatic limit. In order to understand the asymptotic behaviour of the spatial-temporal evolution of this model under the diffusive scaling, we start to analyze the hydrodynamic limit, exploiting attractiveness as an essential ingredient. We obtain that the hydrodynamic equation has boundary conditions that depend on the value of $\theta$.