Approximate spectral approach to transport properties of waves in heterogeneous media

We discuss the long-time transport properties of linear wave equations in heterogeneous media that are small random perturbations of periodic media. Although periodic Bloch waves cannot be simply deformed into an exact diagonalisation of the perturbed operator, we construct a natural approximate diagonalisation that leads to a very precise description of the solution as long as the transport remains ballistic. In the case of a quasiperiodic perturbation, this approach establishes ballistic transport up to exponential time scales.