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 spacial domain is divided into sequence of two subdomains \begin{document}$ A_n \cup B_n $\end{document} and have three kernels, one controls jumps from id="M2">\begin{document}$ to id="M3">\begin{document}$ $\end{document}, second id="M4">\begin{document}$ id="M5">\begin{document}$ third governs interactions between id="M6">\begin{document}$ id="M7">\begin{document}$ $\end{document}. Assuming id="M8">\begin{document}$ \chi_{A_n} (x) \to X(x) weakly-* in id="M9">\begin{document}$ L^\infty (and then id="M10">\begin{document}$ \chi_{B_n} (1-X)(x) id="M11">\begin{document}$ $\end{document}) as id="M12">\begin{document}$ n \infty show there an homogenized limit system which kernels function id="M13">\begin{document}$ X appear. deal with both Neumann Dirichlet boundary conditions. Moreover, also provide probabilistic interpretation our results.