Fluctuation estimates for the multi-cell formula in stochastic homogenization of partitions

TL;DR

This paper provides quantitative fluctuation estimates for the multi-cell formula in stochastic homogenization of partitions, utilizing multiscale inequalities to control surface integrand fluctuations and simplifying the cell formula structure.

Contribution

It introduces a multiscale functional inequality approach to quantify fluctuations and simplifies the cell formula in stochastic homogenization of partitions.

Findings

Quantitative fluctuation estimates derived for the multi-cell formula.

Control of surface integrand fluctuations via multiscale inequalities.

Simplified cell formula replacing cubes with almost flat hyperrectangles.

Abstract

In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.