Telegraph random evolutions on a circle

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