Short time scales in the Kramers problem: A stepwise growth of the escape flux

We prove rigorously and demonstrate in simulations that. for a potential system staying initially at the bottom of a well, the escape flux over the barrier grows on times of the order of a period of eigenoscillation in a stepwise manner, provided that friction is: small or moderate. If the initial state is not at the bottom of the well, then, typically, some of the steps transform into oscillations. The stepwise/oscillatory evolution at short times appears to be a generic feature of a noise-induced flux.