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The problem of noise-induced transitions is often associated with HendrikKramers due to his seminal paper of 1940, where an archetypal example -one-dimensional potential system subject to linear damping and weak white noise- was considered and the quasi-stationary rate of escape over a potentialbarrier was estimated for the ranges of extremely small and moderate-to-largedamping. The gap between these ranges was covered in the 80th by one ofRashba's favourite disciples Vladimir Ivanovich Mel'nikov.  It is natural to pose a question: how does the escape rate achieve thequasi-stationary stage? At least in case of a single potential barrier, theanswer seems to be obvious: the escape rate should smoothly and monotonouslygrow from zero at the initial instant to the quasi-stationary value attime-scales of the order of the time required for the formation of thequasi-stationary distribution within the potential well. Such answer appearedto be confirmed with the analytic work of Vitaly Shneidman in 1997. However ourworks in the end of the 90th and in the beginning of the 2000th incollaboration with one more Rashba's favorite disciple Valentin Ivanovich Shekaand with Riccardo Mannella showed that, at a shorter time-scale, namely that ofthe order of the period of natural oscillations in the potential well, theescape rate growth generically occured stepwise or even in an oscillatorymanner. Analytic results were confirmed with computer simulations.  In the present paper, we review those results and provide a roadmap for thedevelopment of the subject, in particular demonstrating that various recentlyexploited experimental systems are excellent candidates for the observation ofthe above non-trivial theoretical predictions and, moreover, they promiseuseful applications.

Short-time scales in the Kramers problem: past, present, future (review and roadmap dedicated to the 95th birthday of Emmanuel Rashba)