  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.
