[Paper Review] Annealed estimates on the Green function

Daniel Marahrens, Félix Otto|arXiv (Cornell University)|Apr 16, 2013

Advanced Mathematical Modeling in Engineering14 references9 citations

TL;DR

This paper establishes annealed estimates for the Green's function in discrete, random, uniformly elliptic equations on the d-dimensional lattice. Under a logarithmic Sobolev inequality assumption on the coefficient field ensemble, it proves that all stochastic moments of the first and second mixed derivatives of the quenched Green function decay at the same rate as in the constant-coefficient case, extending prior results and enabling optimal homogenization error estimates even for high ellipticity contrast.

ABSTRACT

We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. 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 $\langle\cdot angle$. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $\langle| abla_x G(x,y)|^p angle$ and $\langle| abla_x abla_y G(x,y)|^p angle$, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel \cite{DeuschelDelmotte}, which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle| abla_x G(x,y)|^2 angle$ and $\langle| abla_x abla_y G(x,y)| angle$. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

Motivation & Objective

• To derive annealed moment estimates for the first and second mixed derivatives of the quenched Green function in discrete, random, uniformly elliptic equations.
• To extend the results of Delmotte and Deuschel by establishing optimal decay rates for all p-th moments of the gradient and Hessian of the Green function.
• To provide optimal estimates on the random part of the homogenization error in stochastic homogenization, even for large ellipticity contrast.
• To justify the use of logarithmic Sobolev inequalities as a statistical assumption ensuring sufficient mixing of the coefficient field.
• To establish an annealed Hölder estimate in the spirit of De Giorgi, extending regularity theory to the quenched setting.

Proposed method

• The analysis is conducted on the d-dimensional integer lattice Z^d with uniformly elliptic, bounded coefficient fields a(e) ∈ [λ,1] for e ∈ Ed.
• The Green function G(x,y) is defined as the solution to ∇*(a∇G)(·,y) = δ(·−x), with distributional characterization ∑ₑ ∇ζ(e) a(e) ∇G(e,y) = ζ(y).
• The main technical tool is a logarithmic Sobolev inequality (LSI) for the ensemble of coefficient fields, which holds under fast decay of spatial correlations and is satisfied by i.i.d. coefficient fields.
• The proof builds on the work of Delmotte and Deuschel, extending their optimal control of second moments of first derivatives and first moments of second derivatives to all p-th moments.
• A key step involves tensorization of the LSI over edges, using a sequential averaging argument over an enumeration of edges to control the variance of local functionals.
• The method relies on a telescoping sum representation of the relative entropy and bounds on oscillations of square roots of local averages.

Experimental results

Research questions

• RQ1 Do all p-th moments of the first and second mixed derivatives of the quenched Green function decay at the same rate as in the constant-coefficient case under suitable ergodicity assumptions?
• RQ2 Can the annealed estimates for the Green function gradient and Hessian be extended beyond second and first moments, respectively, to all p ≥ 1?
• RQ3 What is the role of the logarithmic Sobolev inequality in ensuring optimal moment decay for the Green function derivatives?
• RQ4 How do these improved moment estimates impact the quantification of the random part of the homogenization error in stochastic homogenization?
• RQ5 Can an annealed Hölder regularity estimate be derived for the quenched Green function using the same framework?

Key findings

• All stochastic moments of the first derivative |∇ₓG(x,y)|^p have the same decay rate in |x−y| ≫ 1 as in the constant-coefficient case, under the LSI assumption.
• All stochastic moments of the second mixed derivative |∇ₓ∇ᵧG(x,y)|^p exhibit the same decay as in the constant-coefficient case, extending prior results that only controlled second and first moments.
• The logarithmic Sobolev inequality ensures sufficient mixing of the coefficient field, which is essential for controlling the annealed moments of the Green function derivatives.
• The results enable optimal estimates on the random part of the homogenization error, even for large ellipticity contrast, which was previously unattainable with existing methods.
• An annealed Hölder estimate is established in the spirit of De Giorgi, showing that the quenched Green function satisfies a regularity property in an averaged (annealed) sense.
• The proof technique, based on tensorization of the LSI and telescoping entropy decompositions, provides a robust framework for extending moment estimates beyond the second order.