Telegraph random evolutions on a circle

TL;DR

This paper analyzes a circular telegraph process where a particle switches direction randomly, exploring its mathematical properties, asymptotic behavior, and extensions to non-Markovian cases, with applications to harmonic oscillators.

Contribution

It introduces a detailed analysis of the circular telegraph process, including its semigroup, distribution, asymptotics, and extensions to asymmetric and non-Markovian models.

Findings

Derived the process's probability distribution and semigroup properties.

Established asymptotic behavior related to circular Brownian motion.

Extended the model to non-Markovian waiting times with asymptotic insights.

Abstract

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.