Periodic Homogenization of a L\'evy-Type Process with Small Jumps

TL;DR

This paper studies the homogenization of a Lévy-type process with small jumps, showing that under periodic conditions, the process converges to a Brownian motion with a specific covariance matrix.

Contribution

It extends classical homogenization results from diffusion processes to Lévy-type processes with small jumps, providing new convergence results.

Findings

Weak convergence to Brownian motion established

Covariance matrix explicitly characterized

Generalizes classical diffusion homogenization results

Abstract

In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called L\'evy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.