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Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances
Stochastics and Partial Differential Equations: Analysis and Computations ( IF 1.5 ) Pub Date : 2018-10-12 , DOI: 10.1007/s40072-018-0127-8
Sebastian Andres , Stefan Neukamm

We study the random conductance model on the lattice \({\mathbb {Z}}^d\), i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension \(d\ge 3\) quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed \(t^{-\frac{1}{5}+\varepsilon }\) for \(d\ge 4\) and \(t^{-\frac{1}{10}+\varepsilon }\) for \(d=3\). Additionally, in the uniformly elliptic case in low dimensions \(d=2,3\) we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for \(d\ge 3\) we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.

中文翻译:

具有退化电导的随机电导模型的Berry-Esseen定理和定量均化

我们研究晶格\({{mathbb {Z}} ^ d \)上的随机电导模型,即考虑具有随机系数的线性,有限差分,散度形式算子以及在随机电导下的相关随机游动。我们允许电导不受限制并退化为椭圆,但是它们需要满足强矩条件和以谱隙估计形式的量化遍历性假设。作为主要结果,我们获得了速度为((t ^ {-\ frac {1} {5} + \)的Berry–Esseen估计形式的随机游动的量化中心极限定理(\ d \ ge 3 \)。 varepsilon} \)表示\(d \ ge 4 \)\(t ^ {-\ frac {1} {10} + \ varepsilon} \)表示\(d = 3 \)。另外,在小尺寸\(d = 2,3 \)的均匀椭圆形情况下,我们通过Mourrat最近获得的定量Berry-Esseen定理提高了速率。作为中心分析成分,对于\(d \ ge 3 \),我们在与环境过程相关的半群上建立接近最优的衰减估计。这些估计值在定量随机均质化中也起着核心作用,并将Gloria,Otto和第二作者的一些最新结果扩展到简并的椭圆形情况。
更新日期:2018-10-12
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