Hydrodynamic behavior of long-range symmetric exclusion with a slow barrier: diffusive regime

Abstract

In this article we analyse the hydrodynamical behavior of the symmetric exclusion process with long jumps and in the presence of a slow barrier. The jump rates for fast bonds are given by a transition probability $p(\cdot )$ which is symmetric and has finite variance, while for slow bonds the jump rates are given $p(\cdot )\alpha n^{-\beta }$ (with $\alpha >0$ and $\beta \ge 0$), and correspond to jumps from ${\mathbb{Z}}_{-}^{\ast }$ to $\mathbb{N}$. We prove that: if there is a fast bond from ${\mathbb{Z}}_{-}^{\ast }$ and $\mathbb{N}$, then the hydrodynamic limit is given by the heat equation with no boundary conditions; otherwise, it is given by the previous equation if $0\le \beta <1$, but for $\beta \ge 1$ boundary conditions appear, namely, we get Robin (linear) boundary conditions if $\beta =1$ and Neumann boundary conditions if $\beta >1$.