Scalings limits for the exclusion process with a slow site

We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, $\mathit{\text{the}}\mathit{\text{slow}}\mathit{\text{site}}$, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)$. Two cases are treated, namely $g(n)=1+o(1)$, and $g(n)=\alpha n^{-\beta }$ with $\beta >1$, $\alpha >0$. In the former, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation (with periodic boundary conditions when in finite volume). In the latter, they are driven by the heat equation with Neumann boundary conditions. We therefore establish the existence of a dynamical phase transition. The critical behavior remains open.