Annealed estimates on the Green function

By Daniel Marahrens, Felix Otto. 2013

95 citations recorded by the source corpus.

We consider a random, uniformly elliptic coefficient field $$a(x)$$a(x) on the $$d$$d-dimensional integer lattice $$\mathbb {Z}^d$$Zd. We are interested in the spatial decay of the quenched elliptic Green function $$G(a;x,y)$$G(a;x,y). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $$\langle \cdot \rangle $$⟨·⟩. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $$\langle |\nabla _x G(x,y)|^p\rangle $$⟨|∇xG(x,y)|p⟩ and $$\langle |\nabla _x\nabla _y G(x,y)|^p\rangle $$⟨|∇x∇yG(x,y)|p⟩, have the same decay rates in $$|x-y|\gg 1$$|x-y|≫1 as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358–390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $$G$$G, that is, $$\langle |\nabla _x G(x,y)|^2\rangle $$⟨|∇xG(x,y)|2⟩ and $$\langle |\nabla _x\nabla _y G(x,y)|\rangle $$⟨|∇x∇yG(x,y)|⟩. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f0a4ac303afe01062b29cd5836821fca22c1d5cd

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In Applied Mathematics

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Citations in the corpus

Quantitative stochastic homogenization of convex integral functionals (2014, background): “unds for the error in homogenization. These results were proved for discrete elliptic equations, but have been extended to the continuum setting in [17]. See also Mourrat [20, 21], Marahrens and Otto [19], Conlon and Spencer [7] as well as earlier works of Yurinskii [28], Naddaf and Spencer [24], Bo”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “r assumptions (scalar equation, - nite range of dependence, symmetric coecients) using an ingenious combination of sub-additivity methods, duality, and a concentration argument. The improvement over [28] is twofold: in terms of regularity (C1;0 versus C0;1 | where C0;1 means C0; for all <1) and i”
The additive structure of elliptic homogenization (2016, methodology): “Before the regularity theory was introduced in [4], the best estimates available (see [16, 28]) were finite moment bounds.”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “Gloria and Otto used this method to derive estimates on the first-order correctors which are optimal in terms of the ratio of length scales (although not optimal in terms of stochastic integrability).”
Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, background): “the use of sensitivity estimates and the spectral gap assumption to derive moment bounds on the gradient of the corrector and the Green’s functions, which culminated in the work of Marahrens and Otto [19]. These gradient estimates, following Naddaf and Spencer [20], give control of the sensitivity o”
Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, unknown): “We refer the reader to the companion article [21] for a thorough discussion of the literature, and in particular the works [30, 17] based on functional inequalities, and the work of [5, 4] based on finite-range or α-mixing assumptions.”
A higher-order large-scale regularity theory for random elliptic operators (2015, background): “Motivated by error estimates in stochastic homogenization, the topic of a regularity theory for random elliptic operators was independently addressed in a more quantitative way by Marahrens and Otto [14].”