The Annals of Probability

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Antoine Gloria and Felix Otto

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We consider a discrete elliptic equation on the d-dimensional lattice ℤd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized” matrix Ahom=ahom Id is characterized by ξAhomξ=〈(ξ+∇ϕ)⋅A(ξ+∇ϕ)〉 for any direction ξ∈ℝd, where the random field ϕ (the “corrector”) is the unique solution of −∇A(ξ+∇ϕ)=0 such that ϕ(0)=0, ∇ϕ is stationary and 〈∇ϕ〉=0, 〈⋅〉 denoting the ensemble average (or expectation).

It is known (“by ergodicity”) that the above ensemble average of the energy density $\mathcal {E}=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $\mathcal {E}$ on length scales L satisfies the optimal estimate, that is, $\operatorname{var}[\sum \mathcal {E}\eta_{L}]\lesssim L^{-d}$, where the averaging function [i.e., ∑ηL=1, supp(ηL)⊂{|x|≤L}] has to be smooth in the sense that |∇ηL|≲L−1−d. In two space dimensions (i.e., d=2), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density $\mathcal {E}$ decay in L as if $\mathcal {E}$ would be independent from edge to edge (which it is not for d>1).

This result is of practical significance, since it allows to estimate the dominant error when numerically computing ahom.

Article information

Ann. Probab., Volume 39, Number 3 (2011), 779-856.

First available in Project Euclid: 16 March 2011

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Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 39A70: Difference operators [See also 47B39] 60H25: Random operators and equations [See also 47B80] 60F99: None of the above, but in this section

Stochastic homogenization variance estimate difference operator


Gloria, Antoine; Otto, Felix. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011), no. 3, 779--856. doi:10.1214/10-AOP571.

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