Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the homogenization commutator and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works.

中文翻译：

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随机均质化中的波动结构

四个量对于散度形式的椭圆系统的均质化及其应用是基本的：解算符的场和通量（应用于一般确定性右侧），以及校正器的场和通量。同质化是对这些物体的大尺度特性的研究。在随机系数的情况下，这些量会波动并且它们的波动是先验无关的。根据系数场的规律，特别是其相关性在大尺度上的衰减，这些波动可能会显示不同的尺度和不同的限制规律（如果有的话）。在这个贡献中，我们确定了另一个关键的内在量，由 H 收敛驱动，我们将其称为均质化换向器，并且与 Armstrong 和 Smart 首次考虑的变分量有关。在随机电导模型的简化设置中，我们展示了我们认为的一般原则，即均质化换向器以领先的顺序驱动其他四个量中的每一个的波动，其概率为强范数，表示为一种合适的两尺度扩展形式，揭示了随机同质化波动的路径结构。此外，我们表明（重新缩放的）同质化换向器在法律上收敛到高斯白噪声，我们分析了可以从代表性体积元素方法中提取表征后者的协方差张量的精度。这组结果构成了随机同质化波动的新理论，该理论适用于任何维度并产生最佳速率。还讨论了（非对称）连续体设置的扩展，其细节被推迟到即将到来的工作中。