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

Abstract

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 ${\mathbb{Z}}_{-}^{\ast }$ to $\mathbb{N}$ the rates are slowed down by a factor $\alpha n^{-\beta }$ (with $\alpha >0$ and $\beta \ge 0$). We obtain several partial differential equations given in terms of the regional fractional Laplacian on ${\mathbb{R}}^{\ast }$ 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 $\alpha \ne 1$.