A MICROSCOPIC MODEL FOR A ONE PARAMETER CLASS OF FRACTIONAL LAPLACIANS WITH DIRICHLET BOUNDARY CONDITIONS
Résumé
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 α at the left of the system and β at the right of the system. The strength of the reservoirs is ruled by κN −θ > 0. Here N is the size of the system, κ > 0 and θ ∈. Our results are valid for θ ≤ 0. For θ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter κ and with Dirichlet boundary conditions. Their solutions also depend on κ. For θ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case θ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case θ = 0 when we send the parameter κ to zero. Indeed, we conjecture that the limiting profile when κ → 0 is the one that we should obtain when taking small values of θ > 0.
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