Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization

TL;DR

This paper studies the asymptotic behavior of random oscillatory integrals with long-range dependence and applies these results to solve the corrector problem in one-dimensional stochastic homogenization, demonstrating convergence to Hermite process-based integrals.

Contribution

It introduces a novel analysis of oscillatory integrals with long-range dependence and applies it to establish convergence in stochastic homogenization of elliptic equations.

Findings

Convergence of oscillatory integrals to Hermite process-based stochastic integrals.

Solution to the corrector problem in one-dimensional random homogenization.

Extension of homogenization theory to coefficients with long-range dependence.

Abstract

This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of one-dimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.