# Long-time homogenization and asymptotic ballistic transport of classical waves

## TL;DR

This paper develops a new framework using Taylor-Bloch waves and extended correctors to analyze long-time homogenization and ballistic transport of classical waves in non-periodic elliptic operators, including almost periodic and random cases.

## Contribution

It introduces Taylor-Bloch waves and extended correctors to study spectral properties and wave transport in non-periodic media, extending homogenization theory beyond periodic coefficients.

## Key findings

- Quantifies homogenization validity over large times.
- Proves asymptotic ballistic transport at low energies.
- Provides new tools for spectral analysis of non-periodic operators.

## Abstract

Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization.When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associatedoperator are unclear.At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization.The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an "eigendefect"), which we express with the help of a new family of extended correctors.We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large timesfor the wave equation in terms of the spatial growth of these extended correctors.On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions).On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.