We prove rigorously and demonstrate in simulations that. for a potential sy
stem staying initially at the bottom of a well, the escape flux over the ba
rrier grows on times of the order of a period of eigenoscillation in a step
wise manner, provided that friction is: small or moderate. If the initial s
tate is not at the bottom of the well, then, typically, some of the steps t
ransform into oscillations. The stepwise/oscillatory evolution at short tim
es appears to be a generic feature of a noise-induced flux.