High-order convergence rates of periodic homogenization for symmetric Lévy type operators

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical $\alpha$-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales – a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.