Convergence of nonlocal diffusion models on lattices

In this paper we look at models of nonlocal (or anomalous) diffusion which are defined on subsets of the lattice ϵZnϵZn, for some ϵ>0ϵ>0, and ask if they can be approximated by continuum models. The answer is given by an operator semigroup convergence theorem. As an application, we establish hypotheses under which a discrete model of nonlocal diffusion satisfying an absorbing boundary condition has a continuum limit which is conservative.