Ergodic theory and application to nonconvex homogenization
type: Item
instance of: scholarly article
title: Ergodic theory and application to nonconvex homogenization
zbMATH Open document ID: 0813.28010
DOI: 10.1007/BF01027096
author: E. Chabi
author: G. Michaille
published in: Set-Valued Analysis
publication date: 1995-05-17T00:00:00Z
review text: Let \(\{\tau_ z: z\in S= k\mathbb{Z}^ d\}\) be a group of \(P\)-preserving transformations in a probability space \((\Sigma, T, P)\). Let \(\mu\) be a random Borel measure, mapping \(\Sigma\) into the set \(M(\mathbb{R}^ d)\) of nonnegative regular Borel measures on \(\mathbb{R}^ d\), and satisfying \(\mu(\tau_ z \omega)(A)= \mu(\omega)(z+ A)\). For a sequence \(\varepsilon_ n> 0\) tending to 0, put \[ \mu_ n(\omega)(A)= \varepsilon^ d_ n \mu(\omega)(A/\varepsilon_ n), \] where \(A\) is a Borel subset of a bounded open set \(\Omega\subset \mathbb{R}^ d\). The main result asserts: If \(\mu_ n\) is almost surely tight, then almost surely \(\mu_ n(\omega)\) converges in the ``narrow'' topology. (Narrow remains undefined, but seems to refer to the vague topology.) The limit is identified. There are applications to stochastic homogenization of nonconvex integral functionals.
Mathematics Subject Classification ID: 28D99
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 627467
zbMATH Keywords: ergodic theory
zbMATH Keywords: \(P\)-preserving transformations
zbMATH Keywords: random Borel measure
zbMATH Keywords: almost surely tight
zbMATH Keywords: stochastic homogenization
zbMATH Keywords: nonconvex integral functionals
reviewed by: Ulrich Krengel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 627467