Periodic Homogenization of Nonlocal Operators with a Convolution-Type Kernel

The paper deals with a homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behavior of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in L2 space and the space of continuous functions and show that for the related family of Markov processes the invariance principle holds.