If a localized quantum state in a tight-binding model with structural aperiodicity is subject to noisy evolution, then it is generally expected to result in diffusion and delocalization. In this work, it is shown that the localized phase of the kicked Aubry-Andre-Harper (AAH) model is robust to the effects of noisy evolution, for long times, provided that some kick is delivered once every time period. However, if strong noisy perturbations are applied by randomly missing kicks, a sharp dynamical transition from a ballistic growth phase at initial times to a diffusive growth phase for longer times is observed. Such sharp transitions are seen even in translationally invariant models. These transitions are related to the existence of flat bands, and using a 2-band model we obtain analytical support for these observations. The diffusive evolution at long times has a mechanism similar to that of a random walk. The time scale at which the sharp transition takes place is related to the characteristics of noise. Remarkably, the wavepacket evolution scales with the noise parameters. Further, using kick sequence modulated by a coin toss, it is argued that the correlations in the noise are crucial to the observed sharp transitions.
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