Diffusive fluctuations of long-range symmetric exclusion with a slow barrier

In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in Z with long jumps. The transition probability of the jump from x to y is proportional to |x−y|−γ−1. Here we restrict to the choice γ≥2 so that the system has a diffusive behaviour. Moreover, when particles move between negative integers sites and sites in N, the jump rates are slowed down by a factor αn−β, where α>0, β≥0 and n is the scaling parameter. Depending on the values of β and γ, we obtain several stochastic partial differential equations, corresponding to a heat equation without boundary conditions, or with Robin boundary conditions or Neumann boundary conditions.