An optimal variance estimate in stochastic homogenization of discrete elliptic equations
01 May 2011-Annals of Probability (Institute of Mathematical Statistics)-Vol. 39, Iss: 3, pp 779-856
TL;DR: In this article, the authors consider a discrete elliptic equation with random coefficients and show that the average of the energy density of a stationary random field can be recovered by a system average.
Abstract: We 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 % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+
abla\phi)\cdot A(\xi+
abla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ''corrector problem'' % $$ -
abla\cdot A(\xi+
abla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (''by ergodicity'') that the above ensemble average of the energy density $e=(\xi+
abla\phi)\cdot A(\xi+
abla\phi)$, 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 $|
abla\eta_L|\lesssim L^{-1}$. 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 $A_{hom}$.
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TL;DR: In this article, a Spectral Gap Estimate w.r.t. (SGE) was proposed for the corrector problem, which can be viewed as a degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields.
Abstract: We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\mathbb Z^d$$
with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions $$d>2$$
) and prove new optimal estimates for regularized versions of the corrector (in dimensions $$d\ge 2$$
). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.
220 citations
TL;DR: The scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of gradient fields with non-convex interactions, and the effective resistance as well as the scaling limit for certain models are discussed in this paper.
Abstract: Recent progress on the understanding of the Random Conductance Model is
reviewed and commented. A particular emphasis is on the results on the
scaling limit of the random walk among random conductances for almost
every realization of the environment, observations on the behavior of
the effective resistance as well as the scaling limit of certain models
of gradient fields with non-convex interactions. The text is an
expanded version of the lecture notes for a course delivered at the
2011 Cornell Summer School on Probability.
210 citations
TL;DR: In this article, the authors considered a discrete elliptic equation with random coefficients and showed that the average of the energy density of the lattice can be recovered by a system average.
Abstract: We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ 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 $A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$ is characterized by $\xi\cdot A_{\mathrm {hom}}\xi=\langle(\xi+
abla\phi )\cdot A(\xi+
abla\phi)\rangle$ for any direction $\xi\in\mathbb {R}^d$, where the random field $\phi$ (the ``corrector'') is the unique solution of $-
abla^*\cdot A(\xi+
abla\phi)=0$ such that $\phi(0)=0$, $
abla\phi$ is stationary and $\langle
abla\phi\rangle=0$, $\langle\cdot\rangle$ denoting the ensemble average (or expectation). It is known (``by ergodicity'') that the above ensemble average of the energy density $\mathcal {E}=(\xi+
abla\phi)\cdot A(\xi+
abla\phi)$, 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., $\sum\eta_L=1$, $\operatorname {supp}(\eta_L)\subset\{|x|\le L\}$] has to be smooth in the sense that $|
abla\eta_L|\lesssim L^{-1-d}$. 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 $a_{\mathrm {hom}}$.
189 citations
Book•
13 Jun 2020TL;DR: In this article, the authors present a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form.
Abstract: This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field The last chapter contains new results on the homogenization of the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion
183 citations
Posted Content•
TL;DR: In this article, the authors extend the notion of large-scale Lipschitz estimates for elliptic operators with random coefficients to the case of Gaussian-type coefficient fields with arbitrary slow-decaying correlations.
Abstract: Since the seminal results by Avellaneda \& Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong \& Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the \emph{intrinsic large-scale} regularity of Avellaneda \& Lin (namely, intrinsic large-scale Schauder and Caldereron-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field $r_*$ which we call the minimal radius. This regularity theory is \textit{qualitative} in the sense that $r_*$ is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is \textit{quantifiable} in the sense that $r_*$ has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing \emph{optimal} moment bounds on $r_*$ for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.
133 citations