Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators

We consider a particle of mass 1/beta submitted to the action of an harmoni c oscillator. If we add a white-noise external force, it is well known that the trajectories of the particle, for beta tending to infinity, converge t o an Ornstein-Uhlenbrck process. Using the number of crossings of the parti cle with a fixed level u, we construct a consistent estimator of the Ornste in-Uhlenbeck local time, giving an estimate of the speed of this convergenc e.