Homogenization of Levy-type operators with oscillating coefficients

TL;DR

This paper studies the homogenization process of Levy-type operators with oscillating coefficients, covering periodic and random micro-structures, and extends results to nonlinear cases, revealing the limiting Levy-operator.

Contribution

It provides a comprehensive analysis of Levy-type operator homogenization for both periodic and random micro-structures, including symmetric and non-symmetric kernels, and introduces a nonlinear homogenization framework.

Findings

Homogenization leads to a Levy-operator limit.

Periodic case includes symmetric and non-symmetric kernels.

Random case focuses on symmetric kernels.

Abstract

The paper deals with homogenization of Levy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Levy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.