Asymptotics of damped periodic motions with random initial speed

Abstract

We consider motion on the circle, possibly with friction and external forces, the initial velocity being a large random variable. We prove that under various assumptions the probability law of the stopping position of the motion converges to a distribution depending only on the motion equation. Here the time of stopping is either a constant or the first time instant at which the velocity vanishes, and the initial velocity is of the form all alpha U beta, where U is a fixed random variable and alpha and/or beta tend to infinity.