Asymptotics of Damped Periodic Motions with Random

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 distribution 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 rst time instant at which the velocity vanishes, and the initial velocity is of the form U + , where U is a xed random variable and and/or tend to innnity. 1. The model We consider a motion on the unit circle S = S 1 which is governed by the diierential equation ' 00 = f(') ? g(' 0) (1.1) '(0) = 0; ' 0 (0) = v; (1.2) where ' = '(t) is the angular variable at time t. We will denote the coordinate on the line covering S by y, so that ' = y mod 2. It is clear that y satisses the same equation as ' with f extended 2-periodically over the real line. Throughout the paper we will assume that the (quasi)potential force f is Lipschitz continuous and that the friction g is everywhere nonnegative, twice continuously diierentiable, and grows at most quadratically, i.e. g(w) Cw 2 for some constant C. The uniquely determined solution of (1.1) { (1.2) will be denoted by ' v (t) or by y v (t) (with (1.2) replaced by y(0) = 0; y 0 (0) = v), respectively. The initial velocity v is assumed to be the realization of a random variable V = U +, where U 0 has a density function p(u) and > 0 and 0 are parameters.