- Localization in the quantum kicked rotator (QKR) problem leads to nontrivial dynamical correlations that are absent in the classical limit. These correlations are related to the spectral properties of the model. The effect of noise on coherence is studied. Also, its effect on the dynamical correlations is investigated, and thus one is able to determine analytically the diffusion that is induced. If the noise is white, then coherence time is the same as that for the undriven system. For colored noise it is essential to take into account the nature of the dynamical correlations. It follows that a Markovian treatment of the dynamics for a system that is coupled to a low-temperature heat bath is not valid even if the system is classically chaotic. The formalism applies also to the problem of QKR with kicks that are not strictly periodic in time, and also to the quantum kicked particle problem, where a nonperturbative mechanism for destruction of coherence has been found.