# Homogenization of Non-symmetric Jump Processes

## TL;DR

This paper investigates the homogenization of non-symmetric jump processes with complex jump intensities, showing convergence to Lévy processes and extending to stable-like processes, with numerical demonstrations for exit time approximations.

## Contribution

It introduces a homogenization framework for non-symmetric jump processes with mixed periodic and aperiodic components, generalizing previous results and including numerical applications.

## Key findings

- Convergence of non-symmetric jump processes to Lévy processes as scale vanishes
- Extension to stable-like processes with variable order
- Numerical methods for approximating first exit times

## Abstract

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a L\'evy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.