Long-time homogenization of the wave equation.

In this talk I'll present recent results on the long-time homogenization of the wave equation in random media. To this aim I'll introduce the notion of Taylor-Bloch waves, at the basis of an approximate spectral theory at low frequencies. For periodic and quasiperiodic coefficients, this allows one to define a family of higher-order homogenized operators which describe the behavior of the solution on arbitrarily large time frames (and encompasses the standard dispersive approximation). I will then turn to the random case, give a short review on quantitative results in the elliptic case, and address the long-time homogenization in this setting. If time allows I'll give the counterpart of these results for the Schrödinger equation with random potential.
This is joint work with Antoine Benoit (ULB).