Homogenizations of integro-differential equations with L{\'e}vy operators with asymmetric and degenerate densities

TL;DR

This paper develops a homogenization framework for asymmetric Lévy operators with degenerate densities, extending classical methods applicable to symmetric cases by rescaling and extracting singular parts of measures.

Contribution

It introduces conditions under which homogenization is possible for asymmetric Lévy operators, broadening the scope of existing homogenization techniques.

Findings

Established conditions (A) and (B) for homogenization of asymmetric Lévy operators.

Extended homogenization methods to include asymmetric and degenerate densities.

Provided a systematic approach to handle singular parts of Lévy measures in homogenization.

Abstract

We consider periodic homogenization problems for the L{\'e}vy operators with asymmetric L{\'e}vy densities. The formal asymptotic expansion used for the $\a$-stable (symmetric) L{\'e}vy operators ($\a\in (0,2)$) is not applicable directly to such asymmetric cases. We rescale the asymmetric densities, extract the most singular part of the measures, which average out the microscopic dependences in the homogenization procedures. We give two conditions (A) and (B), which characterize such a class of asymmetric densities, that the above "rescaled" homogenization is available.