Multiscale functional inequalities in probability: Concentration properties

TL;DR

This paper explores multiscale functional inequalities for random fields, demonstrating their connection to decorrelation properties and their role in establishing concentration results crucial for stochastic homogenization and regularity theories.

Contribution

It introduces multiscale functional inequalities for random fields and links them to decorrelation properties, enabling advanced concentration and homogenization results.

Findings

Multiscale inequalities hold for heterogeneous material models.

They imply strong concentration properties for nonlinear functions.

These results support stochastic homogenization and regularity theory.

Abstract

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 $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincar\'e, 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.