Hydrodynamics of a d-dimensional long jumps diffusive symmetric exclusion with a slow barrier

Abstract

We obtain the hydrodynamic limit of symmetric long-jumps exclusion in Zd (for d ≥ 1), where the jump rate is inversely proportional to a power of the jump’s length with exponent y+1, where y ≥ 2. Moreover, movements between Zd-1 x Z*_ and Zd-1 xN are slowed down by a factor an -B (with a > 0 and B ≥ 0). In the hydrodynamic limit we obtain the heat equation in Rd without boundary conditions or with Neumann boundary conditions, depending on the values of B and y. The (rather restrictive) condition in [7] (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.