A Regularity Theory for Random Elliptic Operators

By Antoine Gloria, Stefan Neukamm, Felix Otto. Milan Journal of Mathematics, 2014

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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 that r * 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4c77aee667eb058f44e47a5e4674570155fe5083

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Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, methodology): “ivity estimates, i. e. on estimating how sensitively the corrector dependson the coeﬃcient ﬁeld. However, as we discussin the next paragraph, they now capitalize on a large-scale regularity theory of [21] for the random operator. When it comes to this regularity theory, as in [24], we substitute (qu”