2017/11/08 by Mitia Duerinckx, Duerinckx, Mitia, Antoine Gloria +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1711.03148
24 pages
arxiv created 2019/10/10 · arxiv updated 2019/10/11
In a companion article we have introduced a notion of multiscale functional inequalities for functions X(A) of an ergodic stationary random field A on the ambient space \mathbb Rd. These inequalities are multiscale weighted versions of standard Poincaré, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields A arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions X(A). This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results.