Abstract

Starting from models that describe microscopic phenomena, homogenization aims at building less complex models that capture the macroscopic aspects, averaging any microscopic oscillations. In this talk, we deal with periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. The main aim is establishing a convergence result for the solutions of a class of nonlocal HJB equations depending on a small parameter to an effective equation. The nonlocal operators that we consider are linked to the infinitesimal generator of Lévy processes. In particular, we deal with the case where the nonlocal diffusion has the same order as the Hamiltonian (order 1). A typical example is the square root of the fractional laplacian. Our outcomes are two-fold. One one hand, we provide Lipschitz regularity results for weakly elliptic nonlocal HJB. On the other hand, we establish a convergence result, based on a comparison principle for the effective problem. The latter strongly relies on the regularity and the ellipticity properties of the effective Hamiltonian, for which our fine Lipschitz estimate of the solutions plays a crucial role.