In this paper, the diffusion and persistence of synchronous full annular rub in a rotor/stator system under weak random disturbances are studied. First, the model of the rotor/stator rubbing system and the global response characteristics of the corresponding deterministic system are introduced. Then the diffusion features of the synchronous full annular rub response under small random perturbation are quantified and visualized using the stochastic sensitivity function for the discrete-time systems. The shape of the density distribution of the rotor orbit under different noise intensities and rotation speeds are compared and discussed. Finally, using the confidence ellipsoid of the response, the mechanism of noise-induced escape from the synchronous full annular rub to a large amplitude response of dry friction backward whirl is revealed, and the persistence of full annular rub response responses under different conditions is discussed and analyzed in detail. The results may be helpful in studying complex phenomena in random-disturbed rotor/stator rubbing systems.