Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances
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Authors
Andres, Sebastian
Neukamm, Stefan
Publication Date
2019-06Journal Title
Stochastics and Partial Differential Equations: Analysis and Computations
ISSN
2194-0401
Publisher
Springer Science and Business Media LLC
Volume
7
Issue
2
Pages
240-296
Language
en
Type
Article
This Version
VoR
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Andres, S., & Neukamm, S. (2019). Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances. Stochastics and Partial Differential Equations: Analysis and Computations, 7 (2), 240-296. https://doi.org/10.1007/s40072-018-0127-8
Abstract
We study the random conductance model on the lattice $Z^d$, 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 $d\geq 3$ quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed $t^{-\frac 1 5+\varepsilon}$ for $d\geq 4$ 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 $d\geq 3$ 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.
Identifiers
External DOI: https://doi.org/10.1007/s40072-018-0127-8
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287977
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