The computer simulations of fluctuational dynamics of an annular system governed by the sine-Gordon model with a white noise source are performed. It is demonstrated that the mean escape time (MET) of a phase string for an annular structure can be much larger than for a linear one and has a strongly pro<?format ?>nounced maximum as a function of system’s length. The location of the MET maximum roughly equals the size of the kink-antikink pair, which leads to evidence of a spatial crossover between two dynamical regimes: when the phase string escapes over the potential barrier as a whole and when the creation of kink-antikink pairs is the main mechanism of the escape process. For large lengths and in the limit of small noise intensity , for both MET and inverse concentration of kinks, we observe the same dependence versus the kink energy : ∼exp⁡(2Ek/γ) for the annular structure and ∼exp⁡(Ek/γ) for the linear one.