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Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case

Antoine Gloria, Siguang Qi

Abstract

The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits). The aim of this article is threefold. First we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension $d=1$, establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to by-pass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimension (provided results are correctly reformulated) to which we give precise entries to the recent literature.

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case

Abstract

The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits). The aim of this article is threefold. First we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension , establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to by-pass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimension (provided results are correctly reformulated) to which we give precise entries to the recent literature.
Paper Structure (24 sections, 12 theorems, 172 equations)

This paper contains 24 sections, 12 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Position of the problem
  3. Aim and outline of the article
  4. Malliavin calculus in practice
  5. Stationarity and the ergodic theorem
  6. The Malliavin derivative
  7. Control of the distance to a constant: Moment bounds
  8. Formula for covariances
  9. Control of the distance to a normal random variable
  10. Qualitative homogenization via explicit formulas
  11. Random coefficients
  12. Qualitative homogenization
  13. Quantification of oscillations
  14. Quantification of fluctuations: Scaling law of observables
  15. Scaling of fluctuations
  16. ...and 9 more sections

Key Result

Lemma 3.1

For all $p\in \mathbb N$ we have $\mathbb{E}\left[ a^p \right]^\frac{1}{p}\,=\,\mathbb{E}\left[ a^{-p} \right]^\frac{1}{p} \,=\, \exp(\frac{\mathcal{C}(0)}{2}p)$.

Theorems & Definitions (19)

  • Lemma 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['0:1D-thm-qual']}
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['0:1D-thm-qual2']}
  • Theorem 4.1
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem:Minko']}
  • proof : Proof of Theorem \ref{['0:1D-thm-quant1']}
  • Theorem 5.1
  • ...and 9 more