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

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 $p(\cdot)αn^{-β}$ (with $α>0$ and $β\geq 0$), and correspond to jumps from $\mathbb{Z}_{-}^{*}$ to $\mathbb N$. We prove that: if there is a fast bond from $\mathbb{Z}_{-}^{*}$ 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 $0\leq β<1$, but for $β\geq 1$ boundary conditions appear, namely, we get Robin (linear) boundary conditions if $β=1$ and Neumann boundary conditions if $β>1$.

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