Annealed estimates on the Green functions and uncertainty quantification

Antoine Gloria (1, 2) , Daniel Marahrens (3)

1 ULB - Département de Mathématique [Bruxelles]
2 MEPHYSTO - Quantitative methods for stochastic models in physics
3 MPI-MiS - Max-Planck-Institut für Mathematik in den Naturwissenschaften

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).

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Antoine Gloria, Daniel Marahrens. Annealed estimates on the Green functions and uncertainty quantification. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2016, 33 (6), pp.1153--1197. ⟨10.1016/j.anihpc.2015.04.001⟩. ⟨hal-01093386⟩