  We consider the random evolution described by the motion of a particle moving
on a circle alternating the angular velocities $ \pm c $ and changing rotation
at Poisson random times, resulting in a telegraph process over the circle. We
study the analytic properties of the semigroup it generates as well as its
probability distribution. The asymptotic behavior of the wrapped process is
also studied in terms of circular Brownian motion. Besides, it is possible to
derive a stochastic model for harmonic oscillators with random changes in
direction and we give a diffusive approximation of this process. Furthermore,
we introduce some extensions of the circular telegraph model in the asymmetric
case and for non-Markovian waiting times as well. In this last case, we also
provide some asymptotic considerations.
