  Motion of randomly-driven quantum nonlinear pendulum is considered. Utilizing
one-step Poincar\'e map, we demonstrate that classical phase space
corresponding to a single realization of the random perturbation involves
domains of finite-time stability. Statistical analysis of the finite-time
evolution operator (FTEO) is carried out in order to study influence of
finite-time stability on quantum dynamics. It is shown that domains of
finite-time stability give rise to ordered patterns in distributions of FTEO
eigenfunctions. Transition to global chaos is accompanied by smearing of these
patterns; however, some of their traces survive on relatively long timescales.
