Annealed estimates on the Green functions and uncertainty quantification

Antoine Gloria; Daniel Marahrens

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arXiv · 1409.0569

Annealed estimates on the Green functions and uncertainty quantification

Antoine Gloria·Daniel Marahrens

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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).

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Antoine Gloria, Daniel Marahrens. 2014-09-01. Annealed estimates on the Green functions and uncertainty quantification. https://arxiv.org/abs/1409.0569

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