Periodic homogenization of non-local operators with a convolution type kernel

TL;DR

This paper investigates the homogenization of non-local convolution-type operators in periodic media, demonstrating their convergence to a second-order elliptic operator and establishing related invariance principles for associated Markov processes.

Contribution

It provides a rigorous analysis of the homogenization process for non-local operators with convolution kernels, including resolvent convergence, semigroup convergence, and invariance principles.

Findings

Rescaled operators converge to a second-order elliptic operator with constant coefficients.

Semigroups associated with the operators converge in $L^2$ and continuous function spaces.

Invariance principle holds for the related family of Markov processes.

Abstract

The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in $L^2$ space and the space of continuous functions, and show that for the related family of Markov processes the invariance principle holds.