Annealed estimates on the Green functions and uncertainty quantification

TL;DR

This paper establishes optimal decay estimates for Green functions in elliptic PDEs with random or periodic coefficients, leading to improved uncertainty quantification and regularity results.

Contribution

It extends decay estimates to the continuum setting for random and periodic coefficients, providing new tools for uncertainty quantification in elliptic PDEs.

Findings

Optimal decay estimates for Green functions derivatives.

Enhanced fluctuation bounds for solutions with noisy coefficients.

Hölder regularity approaching 1 for elliptic equations with these coefficients.

Abstract

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality and for periodic coefficients, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. As a main application we obtain optimal estimates on the fluctuations of solutions of linear elliptic PDEs with "noisy" diffusion coefficients, an uncertainty quantification result. As a direct corollary of the decay estimates we also prove that for these classes of coefficients the H\"older exponent of the celebrated De Giorgi-Nash-Moser theory can be taken arbitrarily close to 1 in the large (that is, away from the singularity).