We analyse the hydrodynamical behavior of the long jumps symmetric exclusion process in the presence of a slow barrier. The jump rates are given by a symmetric transition probability $p(\cdot)$ with infinite variance. When jumps occur from $\mathbb{ Z}_{ -}^{ *}$ to $\mathbb N$ the rates are slowed down by a factor $\alpha n^{ -\beta}$ (with $\alpha>0$ and $\beta\geq 0$). We obtain several partial differential equations given in terms of the regional fractional Laplacian on $\mathbb R^*$ and with different boundary conditions. Surprisingly, in opposition to the diffusive regime, we get different regimes depending on whether $\alpha=1$ (all bonds with the same rate) or $\alpha\neq 1$.
Categories: math-ph math.AP math.MP math.PR