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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

In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in $\mathbb{Z}$ with long jumps. The transition probability of the jump from $x$ to $y$ is proportional to $|x-y|^{-\gamma-1}$. Here we restrict to the choice $\gamma \geq 2$ so that the system has a diffusive behavior. Moreover, when particles move between $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, the jump rates are slowed down by a factor $\alpha n^{-\beta}$, where $\alpha>0$, $\beta\geq 0$ and $n$ is the scaling parameter. Depending on the values of $\beta$ and $\gamma$, we obtain several stochastic partial differential equations, corresponding to a heat equation without boundary conditions, or with Robin boundary conditions or Neumann boundary conditions.

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 $\alpha\neq 1$.

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 $\beta\geq 0$), 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 $0\leq \beta<1$, but for $\beta\geq 1$ boundary conditions appear, namely, we get Robin (linear) boundary conditions if $\beta=1$ and Neumann boundary conditions if $\beta>1$.