Homogenization of nonlocal equations in randomly evolving media. Diffusion approximation

TL;DR

This paper investigates the homogenization and diffusion approximation of nonlocal evolution equations with periodic spatial and stationary temporal coefficients, revealing their limit behavior converges to a stochastic PDE under certain conditions.

Contribution

It introduces a novel analysis of higher order homogenization for nonlocal equations with mixed periodic and random coefficients, including the convergence to a stochastic PDE.

Findings

Normalized difference converges to a solution of a linear SPDE.

Homogenization results hold under proper mixing assumptions.

Finite moments of the kernel are crucial for the analysis.

Abstract

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.