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 $d \geq 1$), where the jump rate is inversely proportional to a power of the jump's length with exponent $\gamma+1$, where $\gamma \geq 2$. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb N$ are slowed down by a factor $\alpha n^{-\beta}$ (with $\alpha>0$ and $\beta\geq 0$). 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.
Comment: 42 pages, 1 figure