Homogenization of periodic diffusion with small jumps

TL;DR

This paper investigates the homogenization of a diffusion process with small jumps, showing that under certain conditions, the complex process simplifies to a Brownian motion in the limit, extending classical results.

Contribution

It generalizes classical homogenization results by including processes with small jumps and proves convergence to Brownian motion under new conditions.

Findings

Homogenized process is a Brownian motion.

Results extend classical diffusion homogenization.

Applicable to media with small jumps.

Abstract

In this paper, we study the homogenization of a diffusion process with jumps, that is, Feller process generated by an integro-differential operator. This problem is closely related to the problem of homogenization of boundary value problems arising in studying the behavior of heterogeneous media. Under the assumptions that the corresponding generator has vanishing drift coefficient, rapidly periodically oscillating diffusion and jump coefficients, that it admits only "small jumps" (that is, the jump kernel has finite second moment) and under certain additional regularity conditions, we prove that the homogenized process is a Brownian motion. The presented results generalize the classical and well-known results related to the homogenization of a diffusion process.