Stochastic homogenization of convolution type operators

TL;DR

This paper proves that convolution type non-local operators in random media homogenize to a second order elliptic differential operator with deterministic coefficients, under certain conditions.

Contribution

It establishes the almost sure homogenization of convolution type non-local operators in ergodic media, identifying the limit as a second order elliptic differential operator.

Findings

Homogenization holds almost surely in ergodic media.

Limit operator is a deterministic second order elliptic differential operator.

Conditions include finite second moment, uniform ellipticity, and symmetry.

Abstract

This paper deals with homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies the uniform ellipticity and certain symmetry conditions, we prove the almost sure homogenization result and show that the limit operator is a second order elliptic differential operator with constant deterministic coefficients.