Telegraph random evolutions on a circle.

We consider the random evolution described by the motion of a particle on a circle with angular velocity $ \pm c $ changing direction 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 and we obtain the asymptotic behavior of the wrapped process 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.