An optimal variance estimate in stochastic homogenization of discrete elliptic equations

@article{Gloria2011AnOV,
title={An optimal variance estimate in stochastic homogenization of discrete elliptic equations},
author={Antoine Gloria and Felix Otto},
journal={Annals of Probability},
year={2011},
volume={39},
pages={779-856}
}
• Published 7 April 2011
• Mathematics
• Annals of Probability
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 % $$\xi\cdot A_{hom}\xi… 217 Citations An optimal error estimate in stochastic homogenization of discrete elliptic equations • Mathematics • 2012 We consider a discrete elliptic equation with random coefficients A, which (to fix ideas) are identically distributed and independent from grid point to grid point x\in\mathbb{Z}^d. On scales Homogenization of iterated singular integrals with applications to random quasiconformal maps. • Mathematics • 2020 We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (F_j)_{j Quantitative Homogenization and Convergence of Moving Averages It is shown that local averages can result in faster convergence: for example, if a(x) = a(1-x), then for x \in (\varepsilon, 1-\varpsilon) , then subtracting an explicitly given linear function (depending on a(\cdot),f results in the same bound. The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint • Mathematics Archive for Rational Mechanics and Analysis • 2021 We consider the stochastic heat equation \partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u, driven by a smooth space-time stationary Gaussian random field V(s,y), in dimensions d\geq Boundary Estimates in Elliptic Homogenization For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: Lp Estimates • Mathematics, Computer Science • 2015 The Lp convergence results for solutions of the above system and for the non oscillating operator A_\varepsilon(x) = A(x)} Aε(x)=A(x), with the following convergence rate for all. Annealed estimates on the Green function • Mathematics • 2013 We consider a random, uniformly elliptic coefficient field$$a(x)$$a(x) on the$$d$$d-dimensional integer lattice$$\mathbb {Z}^d$$Zd. We are interested in the spatial decay of the quenched elliptic Normal approximation for a random elliptic equation We consider solutions of an elliptic partial differential equation in$$\mathbb{R }^d$$Rd with a stationary, random conductivity coefficient that is also periodic with period$$LL. Boundary
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References

SHOWING 1-10 OF 29 REFERENCES
Elliptic Partial Differential Equations of Second Order
• Mathematics
• 1997
We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations
Estimates On the Variance of Some Homogenization Problems
• Mathematics
• 1998
We use`p estimates together with Brascamp-Lieb inequalities to obtain bounds on the variance of the solution u" to the elliptic equation r" a(x="; ')r"u" = r" f . The variance is shown to be O("r)
Averaging of symmetric diffusion in random medium
Let a~j(y, ~)), y ~ B d, ~ , be random fields, homogeneous with respect to integral shifts, let a~j = aj~, i, ]= i~ d and let, with probability I, almost everywhere in R d, the condition hold: A_J~l~
On elliptic partial differential equations
This series of lectures will touch on a number of topics in the theory of elliptic differential equations. In Lecture I we discuss the fundamental solution for equations with constant coefficients.
AVERAGING OF DIFFERENCE SCHEMES
The author considers a natural class of difference equations whose coefficients have micro-inhomogeneities. A general compactness theorem is established, asserting that the solutions of these
An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions
Variance decay for functionals of the environment viewed by the particle
For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that