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.
A fractional Ficks law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived.
We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.
