  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).
