Characteristic types of evolution of noise-induced escape flux at short time scales

Unlike the quasi-stationary escape flux, the flux on time scales preceding the quasi-stationary stage may significantly depend on the initial state of the system. We analyze three characteristic initial states: (i) the stable state of the noise-free system, i.e. the bottom of the potential well; (ii) the non-bottom state with given coordinate and velocity; (iii) a thermalized state. We prove rigorously and demonstrate in simulations that, on the time scale of a period of eigenoscillation, the flux grows stepwise for cases (i) and (iii), and in an oscillatory manner for case (ii). Different steps/oscillations correspond to different topologies of the most probable escape path.