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A Regularity Theory for Random Elliptic Operators
Milan Journal of Mathematics ( IF 1.7 ) Pub Date : 2020-03-27 , DOI: 10.1007/s00032-020-00309-4
Antoine Gloria , Stefan Neukamm , Felix Otto

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic large-scale regularity of Avellaneda & Lin (namely, intrinsic large-scale Schauder and Calderón-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field r* which we call the minimal radius. This regularity theory is qualitative in the sense that r* is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is quantifiable in the sense thatr* has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing optimal moment bounds on r* for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.

中文翻译:

随机椭圆算子的正则性理论

由于Avellaneda&Lin的开创性结果,众所周知,具有周期系数的椭圆算子在大尺度上具有与拉普拉斯算子相同的规则性理论。在最近的启发性工作中,Armstrong&Smart用满足有限假设范围的随机系数证明了此类算子的大规模Lipschitz估计。在当前的贡献中,我们将Avellaneda和Lin的固有大尺度正则性(即固有大尺度Schauder和Calderón-Zygmund估计)扩展到具有随机系数的椭圆系统。这种改进的规律性开始的尺度以一个固定磁场r *为特征,我们称之为最小半径。从某种意义上说,这种规律性理论是定性在单纯的遍历性下,r *几乎肯定是有限的(这产生了一个新的Liouville定理),并且在系数满足定量混合假设的前提下,r *具有很高的随机可积性,因此可以量化。我们通过为满足多尺度函数不等式的一类系数场,特别是对于具有任意慢衰变相关性的高斯型系数场,在r *上建立最优矩边界来说明这一点。
更新日期:2020-03-27
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