Exclusion process with slow boundary

We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-\theta}$, where $\theta > 0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $\theta$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $\theta$. If $\theta \in(0,1)$, we get Dirichlet boundary conditions, (which is the same behavior if $\theta=0$, see \cite{f}); if $\theta=1$, we get Robin boundary conditions; and, if $\theta\in(1,\infty)$, we get Neumann boundary conditions.