Weighted function inequalities: Concentration properties

Mitia Duerinckx 1, 2 Antoine Gloria 1
2 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : Consider an ergodic stationary random field A on the ambient space R d. We are interested in the concentration of measure phenomenon for nonlinear functions X(A) in terms of assumptions on A. In mathematical physics, this phenomenon is often associated with functional inequalities like spectral gap or logarithmic Sobolev inequality. These inequalities are however only known to hold for a restricted class of laws (like product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce a more general class of functional inequalities (which we call weighted functional inequalities) that strictly contains standard functional inequalities, and we study their concentration properties. As an application, we prove specific concentration results for averages of approximately local functions of the field A, which constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto. In a companion article, we develop a constructive approach to weighted functional inequalities based on product structures in higher-dimensional spaces, which allows us to treat all the examples of heterogeneous materials encountered in stochastic homogenization in the applied sciences.
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Mitia Duerinckx, Antoine Gloria. Weighted function inequalities: Concentration properties. 2017. ⟨hal-01633040⟩

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