Large Deviation Theory for a Homogenized and "Corrected" Elliptic ODE

TL;DR

This paper develops a large deviation framework for a one-dimensional elliptic PDE with random oscillatory coefficients, providing homogenized solutions, Gaussian correctors, and insights into the limits of these approximations for uncertainty quantification.

Contribution

It introduces a pointwise large deviation principle for the full solution and compares it with Gaussian correctors, advancing understanding of their limitations in stochastic homogenization.

Findings

Derivation of a homogenized solution for the elliptic problem

Establishment of a pointwise large deviation principle for the full solution

Analysis showing Gaussian correctors do not generally capture large deviation behavior

Abstract

We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient. We derive a homogenized solution and a so-called Gaussian corrector. We also prove a "pointwise" large deviation principle (LDP) for the full solution and approximate this LDP with a more tractable form. These results allow one to access the limits of Gaussian correctors. In general, the corrector does not capture the large deviation behavior. Applications to uncertainty quantification are considered.