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Green function and invariant measure estimates for nondivergence form elliptic homogenization

Scott Armstrong, Benjamin Fehrman, Jessica Lin

TL;DR

This work develops a quantitative theory of stochastic homogenization for elliptic equations in nondivergence form, focusing on large-scale behavior of solutions, the parabolic Green function, and the environmental process in a random medium with finite range dependence. By linking the parabolic Green function to the stationary invariant measure and exploiting deterministic regularity plus a robust homogenization framework, the authors obtain a quenched local central limit theorem with explicit rates, a large-scale $C^{0,1}$-type regularity for the invariant measure density, and quantitative ergodicity for the environment from the particle’s viewpoint. The results hinge on sharp algebraic decay rates and optimal stochastic integrability, along with deterministic Cauchy–Dirichlet error bounds and a carefully constructed minimal scale $\mathcal{Y}$. Collectively, these advances bridge PDE techniques and stochastic processes to yield precise, environment-dependent rates for homogenization in nondivergence form, setting the stage for a renormalization-based approach in this setting.

Abstract

We prove quantitative estimates on the the parabolic Green function and the stationary invariant measure in the context of stochasic homogenization of elliptic equations in nondivergence form. We consequently obtain a quenched, local CLT for the corresponding diffusion process and a quantitative ergodicity estimate for the environmental process. Each of these results are characterized by deterministic (in terms of the environment) estimates which are valid above a random, ``minimal'' length scale, the stochastic moments of which we estimate sharply.

Green function and invariant measure estimates for nondivergence form elliptic homogenization

TL;DR

This work develops a quantitative theory of stochastic homogenization for elliptic equations in nondivergence form, focusing on large-scale behavior of solutions, the parabolic Green function, and the environmental process in a random medium with finite range dependence. By linking the parabolic Green function to the stationary invariant measure and exploiting deterministic regularity plus a robust homogenization framework, the authors obtain a quenched local central limit theorem with explicit rates, a large-scale -type regularity for the invariant measure density, and quantitative ergodicity for the environment from the particle’s viewpoint. The results hinge on sharp algebraic decay rates and optimal stochastic integrability, along with deterministic Cauchy–Dirichlet error bounds and a carefully constructed minimal scale . Collectively, these advances bridge PDE techniques and stochastic processes to yield precise, environment-dependent rates for homogenization in nondivergence form, setting the stage for a renormalization-based approach in this setting.

Abstract

We prove quantitative estimates on the the parabolic Green function and the stationary invariant measure in the context of stochasic homogenization of elliptic equations in nondivergence form. We consequently obtain a quenched, local CLT for the corresponding diffusion process and a quantitative ergodicity estimate for the environmental process. Each of these results are characterized by deterministic (in terms of the environment) estimates which are valid above a random, ``minimal'' length scale, the stochastic moments of which we estimate sharply.
Paper Structure (8 sections, 13 theorems, 91 equations)

This paper contains 8 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Motivation and informal summary of the main results
  3. Statement of the main results
  4. Relation to previous works in the literature
  5. Outline of the Paper.
  6. Some deterministic estimates for nondivergence form equations
  7. Some quantitative homogenization estimates
  8. Estimates for the Cauchy-Dirichlet problem

Key Result

Theorem 1.1

Let $p\in (0,d)$. There exist $\gamma(d,\lambda,\Lambda, p)\in(0,1]$, $C(d,\lambda,\Lambda)\in [1,\infty)$, and a random variable $\mathcal{Y}$ satisfying such that, for every $R\geq \mathcal{Y}$ and solution $v$ of the initial-value problem with initial data satisfying, for $M\geq 0$, the bound there exists a random variable $c[v_0]$ satisfying, for every $t\geq R^2$, If we assume moreover th

Theorems & Definitions (20)

  • Theorem 1.1: Quantitative local CLT
  • Theorem 1.2: Parabolic Green function estimates
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5: Large-scale $C^{0,1}$-regularity of invariant measures
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • ...and 10 more