  Motivated by the long-time transport properties of quantum waves in weakly
disordered media, the present work puts random Schr\"odinger operators into a
new spectral perspective. Based on a stationary random version of a Floquet
type fibration, we reduce the description of the quantum dynamics to a fibered
family of abstract spectral perturbation problems on the underlying probability
space. We state a natural resonance conjecture for these fibered operators: in
contrast with periodic and quasiperiodic settings, this would entail that Bloch
waves do not exist as extended states, but rather as resonant modes, and this
would justify the expected exponential decay of time correlations. Although
this resonance conjecture remains open, we develop new tools for spectral
analysis on the probability space, and in particular we show how ideas from
Malliavin calculus lead to rigorous Mourre type results: we obtain an
approximate dynamical resonance result and the first spectral proof of the
decay of time correlations on the kinetic timescale. This spectral approach
suggests a whole new way of circumventing perturbative expansions and
renormalization techniques.
