Homogenization of a class of integro-differential equations with L{\'e}vy operators

TL;DR

This paper investigates the homogenization of a class of integro-differential equations involving Lévy operators, using asymptotic expansion and ergodic theory to establish rigorous results.

Contribution

It introduces a rigorous homogenization framework for integro-differential equations with Lévy operators leveraging symmetry and asymptotic methods.

Findings

Established connection between homogenization and ergodic cell problem

Developed a rigorous proof using perturbed test functions

Extended homogenization techniques to Lévy operator equations

Abstract

The periodic homogenization problem of integro-differential equations of the alpha stable L{\'e}vy operators is studied in this paper. Thanking to the symmetry of the L{\'e}vy density, we can use the method of the formal asymptotic expansion, to connect the problem to the ergodic cell problem. A rigorous proof is given by the perturbed test function's method.