An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Abstract

International audienceWe consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\in\mathbb{Z}^d$. On scales large w.\ r.\ t.\ the grid size (i.\ e.\ unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients $A_{hom}$ are characterized by % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ''corrector problem'' % $$ -\nabla\cdot A(\xi+\nabla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (''by ergodicity'') that the above ensemble average of the energy density $e=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, 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 $e$ on length scales $L$ is estimated as follows: % $$ {\rm var}\left[\sum_{x\in\mathbb{Z}^d}\eta_L(x)\,e(x)\right] \;\lesssim\;L^{-d}, $$ % where the averaging function (i.\ e.\ $\sum_{x\in\mathbb{Z}^d}\eta_L(x)=1$, ${\rm supp}\eta_L\subset[-L,L]^d$) has to be smooth in the sense that $|\nabla\eta_L|\lesssim L^{-1}$. In two space dimensions (i.\ e.\ $d=2$), there is a logarithmic correction. \medskip In other words, smooth averages of the energy density $e$ behave like as if $e$ would be independent from grid point to grid point (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the error when numerically computing $A_{hom}$

The additive structure of elliptic homogenization

“…In these works, the space is the discrete lattice Z d , d ≥ 3, and a key assumption is that the probability measure has an underlying product structure which makes available tools such as concentration inequalities, the Chatterjee-Stein [9,10] method of normal approximation and the Helffer-Sjöstrand representation of correlations [26,39,32]. Each of these papers makes essential use of, and refines, the optimal quantitative estimates first proved in [19,20,16].…”

“…The first has its origins in an unpublished paper of Naddaf and Spencer [33], and is based on probabilistic machinery more commonly used in statistical physics [32], namely concentration inequalities, such as spectral gap or logarithmic Sobolev inequalities, which provide a way to quantitatively measure the dependence of the solutions on the coefficients. This approach has been developed extensively by Gloria, Otto and their collaborators [19,20,23,16,15,28], who proved optimal quantitative bounds on the scaling of the first-order correctors (including their sublinear growth and spatial averages of their energy density). In particular, they were the first to obtain estimates for the correctors at the critical scalings, albeit with suboptimal stochastic integrability (typically finite moment bounds) and with somewhat restrictive ergodic assumptions.…”

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

“…We establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for d > 2. This work is based on [5], which is a complete continuum version of [6,7] (with in addition optimal results for d = 2). The main difference with respect to the first part of [5] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.…”

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

“…While in the scalar case the bound (25) holds ( [17], Lemma 2.9), we cannot expect it to be true for every coefficient field a ∈ in the case of systems. The following example of De Giorgi [10] shows indeed that in d > 2, the unbounded function u :…”

“…As in the case d > 2, we prove for the approximate family {G ε,D } ε>0 the bounds (14)- (15) and (18)- (16)- (17). It suffices, by property (48), to fix z = 0.…”