Homogenization for nonlocal problems with smooth kernels

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains An ∪ Bn and we have three different smooth kernels, one that controls the jumps from An to An, a second one that controls the jumps from Bn to Bn and the third one that governs the interactions between An and Bn. Assuming that χAn(x) → X(x) weakly-* in L∞ (and then χBn(x) → (1 − X)(x) weakly-* in L∞) as n → ∞ we show that there is an homogenized limit system in which the three kernels and the limit function X appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.