Abstract
Consider an ergodic stationary random field A on the ambient space ℝ d . In order to establish concentration properties for nonlinear functions Z(A), it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.
Highlights
Que nous développons ici principalement en vue de leur utilisation en homogénéisation stochastique quantitative, ont un intérêt propre. This contribution focuses on functional inequalities in the probability space and constitutes the first and main part of a series of three articles where we introduce multiscale functional inequalities, which are multiscale weighted versions of standard functional inequalities (Poincaré, covariance, and logarithmic Sobolev inequalities)
One of the main achievements of the present contribution is the proof that most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including some important examples from stochastic geometry, do satisfy such multiscale functional inequalities whereas they do not satisfy their standard versions
Such standard functional inequalities (1.3)–(1.4) are very restrictive: the random field essentially either has to be Gaussian with integrable covariance (in which case (1.3) holds) or has to display a product structure (e.g. Poisson point process, in which case (1.4) holds). This rules out most models of interest for heterogeneous materials considered in the applied sciences [Tor02] and is the starting point for the present series of articles on functional inequalities, which aims at closing this gap
Summary
This contribution focuses on functional inequalities in the probability space and constitutes the first and main part of a series of three articles (together with [DG18a, DG18b]) where we introduce multiscale functional inequalities, which are multiscale weighted versions of standard functional inequalities (Poincaré, covariance, and logarithmic Sobolev inequalities). One of the main achievements of the present contribution is the proof that most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including some important examples from stochastic geometry (the random parking process and Poisson random tessellations), do satisfy such multiscale functional inequalities whereas they do not satisfy their standard versions. As shown in the companion article [DG18a], these weaker inequalities still imply fine concentration properties and they can be used as convenient quantitative mixing assumptions in stochastic homogenization, which was our original motivation for this work (see Section 1.3 below for details)
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