A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $\alpha$ at the left of the system and $\beta$ at the right of the system. The strength of the reservoirs is ruled by $\kappa$N --$\theta$ > 0. Here N is the size of the system, $\kappa$ > 0 and $\theta$ $\in$. Our results are valid for $\theta$ $\le$ 0. For $\theta$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $\kappa$ and with Dirichlet boundary conditions. Their solutions also depend on $\kappa$. For $\theta$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $\theta$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $\theta$ = 0 when we send the parameter $\kappa$ to zero. Indeed, we conjecture that the limiting profile when $\kappa$ $\rightarrow$ 0 is the one that we should obtain when taking small values of $\theta$ > 0.