We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.

中文翻译：

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随机同质化波动路径结构的稳健性

我们考虑具有随机系数的发散形式的线性椭圆系统，并研究场的大规模平均值的随机波动和解算子的通量。在随机电导模型的背景下，我们在之前的工作中开发了基于同质化换向器概念的涨落理论：我们证明了这个特殊量的两尺度展开在平均时在涨落尺度的领先阶上是准确的在大尺度上（相对于解算子的两个尺度展开单独进行），并且解算子的场和通量的大尺度波动可以从换向器的波动中恢复。这意味着校正器的换向器的大规模波动将所有其他大规模波动驱动到领先阶，我们称之为随机同质化波动的路径结构。在目前的贡献中，我们将这个结果扩展到两个方向：我们处理连续椭圆（可能是非对称）系统并允许强相关系数场（具有协方差函数的类高斯，可以在无穷远处显示任意缓慢的代数衰减）。我们的主要结果表明，在这种一般情况下，同质化换向器的两尺度扩展在大尺度上平均时仍然精确到领先阶，这说明了波动路径结构的鲁棒性。我们处理连续椭圆（可能是非对称）系统并允许强相关系数场（具有协方差函数的类高斯场，可以在无穷远处显示任意缓慢的代数衰减）。我们的主要结果表明，在这种一般情况下，同质化换向器的两尺度扩展在大尺度上平均时仍然精确到领先阶，这说明了波动路径结构的鲁棒性。我们处理连续椭圆（可能是非对称）系统并允许强相关系数场（具有协方差函数的类高斯场，可以在无穷远处显示任意缓慢的代数衰减）。我们的主要结果表明，在这种一般情况下，同质化换向器的两尺度扩展在大尺度上平均时仍然精确到领先阶，这说明了波动路径结构的鲁棒性。