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Optimal convergence rates in stochastic homogenization in a balanced random environment

Xiaoqin Guo, Hung V. Tran

TL;DR

The paper develops a comprehensive quantitative theory of stochastic homogenization for non-divergence form operators in balanced i.i.d. random environments on Z^d. By combining large-scale regularity techniques, sensitivity analysis of the invariant measure, and a novel local corrector construction, it delivers near-optimal rates for homogenization (O(R^{-1}) when d≥3) and establishes robust ergodicity and QCLT results with explicit rates. It also proves the existence, uniqueness, and stationarity of global correctors across dimensions, including delicate treatment in low dimensions via local constructions. The results advance understanding of how random environments influence diffusive behavior and offer precise probabilistic controls that improve upon prior qualitative homogenization results. The methods unify two-scale expansions, deterministic large-scale regularity, and martingale-based probabilistic tools to achieve sharp quantitative outcomes for Dirichlet problems and CLTs in non-divergence form RWRE settings.

Abstract

We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for $d\geq 3$ and nearly optimal when $d=2$. Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk.

Optimal convergence rates in stochastic homogenization in a balanced random environment

TL;DR

The paper develops a comprehensive quantitative theory of stochastic homogenization for non-divergence form operators in balanced i.i.d. random environments on Z^d. By combining large-scale regularity techniques, sensitivity analysis of the invariant measure, and a novel local corrector construction, it delivers near-optimal rates for homogenization (O(R^{-1}) when d≥3) and establishes robust ergodicity and QCLT results with explicit rates. It also proves the existence, uniqueness, and stationarity of global correctors across dimensions, including delicate treatment in low dimensions via local constructions. The results advance understanding of how random environments influence diffusive behavior and offer precise probabilistic controls that improve upon prior qualitative homogenization results. The methods unify two-scale expansions, deterministic large-scale regularity, and martingale-based probabilistic tools to achieve sharp quantitative outcomes for Dirichlet problems and CLTs in non-divergence form RWRE settings.

Abstract

We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice for and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for and nearly optimal when . Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions . Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk.
Paper Structure (29 sections, 34 theorems, 295 equations)

This paper contains 29 sections, 34 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Settings
  3. Main assumptions
  4. Earlier results in the literature
  5. Main results
  6. Large scale TEXT and TEXT estimates
  7. Some regularity properties of deterministic functions
  8. Large scale regularity
  9. Mixing properties of the invariant measure
  10. A sensitivity estimate of the invariant measure
  11. Rate of convergence for the average of the invariant measure: Proof of Theorem \ref{['thm:ave-rho-quant']}
  12. Correlation structure of the field of the invariant measure
  13. Quantification of the diffusive behavior
  14. Estimates of the approximate corrector
  15. Quantification of the ergodicity of the environmental process: Proof of Theorem \ref{['thm:quant-ergo']}
  16. ...and 14 more sections

Key Result

Theorem A

Assume (A2) and that law $\mathbb{P}$ of the environment is ergodic under spatial shifts $\{\theta_x:x\in\mathbb{Z}^d\}$. Then

Theorems & Definitions (79)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem A
  • Theorem B
  • Remark 4
  • Proposition C
  • Definition 5
  • Theorem D
  • Theorem E
  • ...and 69 more