Higher-order pathwise theory of fluctuations in stochastic homogenization

Abstract

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $\frac{d}{2}$, where $d$ is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the homogenization commutator, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón-Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

57 pages

References in corpus (4)

• Quantitative stochastic homogenization and large-scale regularity
• Quantitative estimates in stochastic homogenization for correlated coefficient fields
• Robustness of the pathwise structure of fluctuations in stochastic homogenization
• On a homogenization problem

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• A diagram-free approach to the stochastic estimates in regularity structures
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• Quantitative nonlinear homogenization: control of oscillations
• Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains
• Eigenvalue fluctuations for random elliptic operators in homogenization regime
• Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media
• Tiny fluctuations of the averaging process around its degenerate steady state
• Corrector estimates for higher-order linearizations in stochastic homogenization of nonlinear uniformly elliptic equations