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 discussed 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 results.

我们考虑由在圆周上移动的粒子交替角速度的运动所描述的随机演化 $\pm C$并在泊松随机时间改变旋转，导致在圆上的电报过程。我们研究它生成的半群的解析性质及其概率分布。还根据圆形布朗运动讨论了包裹过程的渐近行为。此外，可以推导出具有随机方向变化的谐振子的随机模型，并且我们给出了该过程的扩散近似。此外，我们介绍了圆形电报模型在非对称情况下和非马尔可夫等待时间的一些扩展。在最后一种情况下，我们还提供了一些渐近结果。