Critical homogenization of Levy process driven SDEs in random medium

TL;DR

This paper establishes a homogenization result for Levy process driven SDEs in random media, showing convergence to a limit with both Brownian and jump components, advancing understanding of stochastic homogenization in complex environments.

Contribution

It introduces a critical homogenization theorem for SDEs driven by Levy processes in random media, addressing the case where the limit includes both diffusion and jump parts.

Findings

Proves annealed convergence of the SDEs to a combined Brownian and jump process

Connects stochastic homogenization with integral PDE homogenization

Provides a rigorous framework for critical Levy-driven SDEs in random environments

Abstract

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations