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Elliptic homogenization from qualitative to quantitative

Scott Armstrong, Tuomo Kuusi

TL;DR

The work provides a thorough, self-contained treatment of elliptic homogenization for random coefficient fields, bridging qualitative and quantitative theories. It develops the classical two-scale approach in periodic settings and portends quantitative stochastic homogenization through coarse-graining and renormalization under a unifying mixing condition, CFS. A key contribution is establishing optimal, scale- and moment-sensitive estimates for first-order correctors, including sharp results for Gaussian fields with power-like decay of correlations, and offering alternative proofs via nonlinear concentration inequalities. The exposition also connects PDE homogenization to diffusion-limit results in probability, including invariance principles for diffusions in random media and parabolic Green function convergence, while outlining a flexible framework that accommodates non-symmetric coefficients and future numerical/nodal-method developments.

Abstract

We give a self-contained introduction to the theory of elliptic homogenization for random coefficient fields, starting from classical qualitative homogenization. The presentation also contains new results, such as optimal estimates (both in terms of stochastic moments and scaling of the error) for coefficient fields which are local functions of Gaussian random fields.

Elliptic homogenization from qualitative to quantitative

TL;DR

The work provides a thorough, self-contained treatment of elliptic homogenization for random coefficient fields, bridging qualitative and quantitative theories. It develops the classical two-scale approach in periodic settings and portends quantitative stochastic homogenization through coarse-graining and renormalization under a unifying mixing condition, CFS. A key contribution is establishing optimal, scale- and moment-sensitive estimates for first-order correctors, including sharp results for Gaussian fields with power-like decay of correlations, and offering alternative proofs via nonlinear concentration inequalities. The exposition also connects PDE homogenization to diffusion-limit results in probability, including invariance principles for diffusions in random media and parabolic Green function convergence, while outlining a flexible framework that accommodates non-symmetric coefficients and future numerical/nodal-method developments.

Abstract

We give a self-contained introduction to the theory of elliptic homogenization for random coefficient fields, starting from classical qualitative homogenization. The presentation also contains new results, such as optimal estimates (both in terms of stochastic moments and scaling of the error) for coefficient fields which are local functions of Gaussian random fields.
Paper Structure (8 sections, 6 theorems, 103 equations)

This paper contains 8 sections, 6 theorems, 103 equations.

Table of Contents

  1. Preface
  2. Qualitative homogenization
  3. Periodic homogenization: a warmup
  4. Heuristic derivation of the correctors
  5. Two-scale expansion computations
  6. Homogenization of boundary-value problems
  7. Ergodic theorems in Rd and Zd
  8. A multiparameter version of the Wiener ergodic theorem

Key Result

Lemma 1.1

Let $U\subseteq \mathbb{R}^d$ be open, $\varepsilon \in \bigl(0,\tfrac12\bigr]$ and $u\in W^{2,\infty}(U)$. Define the function $w^\varepsilon \in H^1_{\mathrm{loc}}(U)$ by Then, we have the identities and

Theorems & Definitions (12)

  • Lemma 1.1: Basic two-scale expansion computation
  • proof
  • Corollary 1.2
  • proof
  • Lemma 1.3: Quantitative homogenization for Dirichlet problem
  • proof
  • Corollary 1.4: Homogenization of the Dirichlet problem
  • proof
  • Remark 1.5: Relaxing the regularity for $u\in H^2_{\mathrm{loc}}(\mathbb{R}^d)$
  • Lemma 1.6: Neumann problem
  • ...and 2 more