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MiS Preprint

Annealed estimates on the Green's function

Daniel Marahrens and Felix Otto


We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional cubic lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green's function $G(a;x,y)$. Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently rapidly 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's function, that is, $\langle|\nabla_x G(x,y)|^p\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|^p\rangle$, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green's function, respectively.

This result relies on and substantially extends the one by Delmotte and Deuschel, which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle|\nabla_x G(x,y)|^2\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|\rangle$. As an application, we derive optimal estimates on the random part of the homogenization error.

MSC Codes:
35B27, 35J08, 39A70, 60H25
stochastic homogenization, elliptic equations, green's function, annealed estimates

Related publications

2014 Repository Open Access
Daniel Marahrens and Felix Otto

Annealed estimates on the Green function

In: Probability theory and related fields, 163 (2014) 3-4, pp. 527-573