The quantum kicked rotor is studied in a regime of high amplitude noise. A
transition to diffusive behavior is observed as dynamical localization, cha
racterized by suppressed diffusion and exponential momentum distributions,
is completely destroyed by noise. With increasing noise amplitude, further
transition to classical behavior is shown through an accurate quantitative
analysis, which demonstrates that both the energy growth and the momentum d
istributions are reaching their classical limits. The importance of short-t
ime correlations in the recovery of classically chaotic behavior is discuss
ed.