Hydrodynamics of a $d$-dimensional long jumps symmetric exclusion with a slow barrier

We obtain the hydrodynamic limit of symmetric long-jumps exclusion in ${\mathbb{Z}}^d$ (for $d\ge 1$), where the jump rate is inversely proportional to a power of the jump’s length with exponent $\gamma +1$, where $\gamma \ge 2$. Moreover, movements between ${\mathbb{Z}}^{d-1}\times {\mathbb{Z}}_{-}^{\ast }$ and ${\mathbb{Z}}^{d-1}\times \mathbb{N}$ are slowed down by a factor $\alpha n^{-\beta }$ (with $\alpha >0$ and $\beta \ge 0$). In the hydrodynamic limit we obtain the heat equation in ${\mathbb{R}}^d$ without boundary conditions or with Neumann boundary conditions, depending on the values of $\beta$ and $\gamma$. The (rather restrictive) condition in \cite{casodif} (for $d=1$) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.