Annealed estimates on the Green functions and uncertainty quantification

We prove Lipschitz bounds for linear elliptic equations in divergence form whose measurable coefficients are random stationary and satisfy a logarithmic Sobolev inequality, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. This improves the celebrated De Giorgi–Nash–Moser theory in the large (that is, away from the singularity) for this class of coefficients. This regularity result is obtained as a corollary of optimal decay estimates on the derivative and mixed second derivative of the elliptic Green functions on RdRd. As another application of these decay estimates we derive optimal estimates on the fluctuations of solutions of linear elliptic PDEs with “noisy” diffusion coefficients.