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In this paper we establish periodic homogenization forHamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators ofintegro-differential type. We consider the case when the fractional diffusionhas the same order as the drift term, and is weakly elliptic. The outcome ofthe paper is twofold. One one hand, we provide Lipschitz regularity results forweakly elliptic non-local HJB, extending the results previously obtained in[8]. On the other hand, we establish a convergence result, based on halfrelaxed limits and a comparison principle for the effective problem. The latterstrongly relies on the regularity and the ellipticity properties of theeffective Hamiltonian, for which a fine Lipschitz estimate of the correctorplays a crucial role.

Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion