Stochastic homogenisation of high-contrast media

TL;DR

This paper develops a probabilistic multiscale framework for high-contrast heterogeneous media, extending two-scale convergence methods to analyze spectral convergence and effective dispersion properties.

Contribution

It introduces a stochastic homogenisation approach for high-contrast media, generalizing existing methods to include spectral convergence and effective dispersion analysis.

Findings

Spectra of original problems converge to the homogenised limit spectrum.

A multiscale limit satisfies coupled macroscopic and microscopic equations.

The framework introduces an analogue of the Zhikov function for effective dispersion.

Abstract

Using a suitable stochastic version of the compactness argument of [V. V. Zhikov, 2000. On an extension of the method of two-scale convergence and its applications. Sb. Math., 191(7--8), 973--1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled "macroscopic" and "microscopic" components of the field, giving rise to an analogue of the "Zhikov function", which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.