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Stochastic homogenization of coarse-grained elliptic equations

Aidan Lau

TL;DR

The article develops quenched stochastic homogenization for divergence-form elliptic equations with stationary ergodic coefficients that may be degenerate or singular, by introducing coarse-grained ellipticity and a decomposition a = s + k. It constructs coarse-grained matrices from a variational energy $J(U,p,q)$, proves their convergence via an ergodic theorem, and shows that the homogenization error vanishes under a strong coarse-grained ellipticity condition, yielding almost sure homogenization for Dirichlet problems. A key contribution is a deterministic, scale-aware error bound in terms of coarse-grained constants, together with Besov-type and $L^p$ integrability criteria that guarantee the ellipticity hypothesis. The results unify and extend prior work by AK.HC and related authors, providing an alternative coarse-graining criterion and applicable homogenization theory for irregular coefficient fields with rigorous quenched guarantees and energy-based control. This framework advances the understanding of large-scale regularity and homogenization for complex random media and informs applications where ellipticity may fail pointwise but persists in a coarse-grained sense.

Abstract

We prove quenched stochastic homogenization for divergence-form elliptic equations, under the assumption that the coefficients are stationary, ergodic, integrable, and satisfy a coarse-grained ellipticity assumption. The ellipticity assumption requires that the coefficients remain bounded in a negative regularity sense on large scales. As a corollary, we recover a sufficient joint integrability condition on the symmetric and skew-symmetric parts of the coefficient field.

Stochastic homogenization of coarse-grained elliptic equations

TL;DR

The article develops quenched stochastic homogenization for divergence-form elliptic equations with stationary ergodic coefficients that may be degenerate or singular, by introducing coarse-grained ellipticity and a decomposition a = s + k. It constructs coarse-grained matrices from a variational energy , proves their convergence via an ergodic theorem, and shows that the homogenization error vanishes under a strong coarse-grained ellipticity condition, yielding almost sure homogenization for Dirichlet problems. A key contribution is a deterministic, scale-aware error bound in terms of coarse-grained constants, together with Besov-type and integrability criteria that guarantee the ellipticity hypothesis. The results unify and extend prior work by AK.HC and related authors, providing an alternative coarse-graining criterion and applicable homogenization theory for irregular coefficient fields with rigorous quenched guarantees and energy-based control. This framework advances the understanding of large-scale regularity and homogenization for complex random media and informs applications where ellipticity may fail pointwise but persists in a coarse-grained sense.

Abstract

We prove quenched stochastic homogenization for divergence-form elliptic equations, under the assumption that the coefficients are stationary, ergodic, integrable, and satisfy a coarse-grained ellipticity assumption. The ellipticity assumption requires that the coefficients remain bounded in a negative regularity sense on large scales. As a corollary, we recover a sufficient joint integrability condition on the symmetric and skew-symmetric parts of the coefficient field.

Paper Structure

This paper contains 12 sections, 25 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Statement of assumptions and examples
  4. Coarse-grained matrices
  5. Definition of the coarse-grained matrices
  6. Coarse-grained energy estimates
  7. Convergence of the coarse-grained matrices
  8. Ergodic theorem
  9. Convergence of the homogenization error
  10. Homogenization of the Dirichlet problem
  11. Functional inequalities
  12. Proof of Example \ref{['ex.field']}

Key Result

Theorem 1.1

Assume $\mathbb{P}$ satisfies $\mathbb{Z}^d$-stationarity a.stationarity, ergodicity a.ergodicity, the minimal moment condition a.finite.E, and the coarse-grained ellipticity condition a.ellipticity. Let $s$ and $t$ be as in a.ellipticity. There exists a matrix $\accentset{}{\mathbf{a}}\in\mathbb{R} such that, for every bounded domain $U\subseteq \mathbb{R}^d$ which is either $C^{1,1}$ or convex a

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Example 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • ...and 36 more