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.

我们考虑具有可积相关性的平稳随机系数的发散形式的线性椭圆方程。我们表征了相对阶\（\ frac {d} {2} \）的解决方案的宏观可观观测值的波动，其中d是空间维度；波动结果是高斯的。至于先前对前导阶的工作，这种更高阶的表征依赖于一般解决方案的宏观波动与（更高阶）校正器的宏观波动之间的路径接近性，该波动通过注入（更高阶）二阶展开进入“同质换向器”，从而确定了这一概念的范围。这种高阶泛化为更高级版本的校正器，通量校正器，二阶展开和均化换向器的代数结构提供了更清晰的认识。结果表明，尽管这两种代数并不直接相关，但与该代数为微观空间振荡提供高阶理论一样，它也为宏观随机波动提供了高阶理论。我们专注于基础高斯系综的模型框架，该模型框架允许有效地使用（二阶）Malliavin演算进行随机估计。在技​​术方面，我们为具有随机系数的椭圆算子引入了经退火的Calderón–Zygmund估计，可方便地升级已知的淬灭的大规模估计。