A generalized telegraph process with velocity driven by random trials

TL;DR

This paper introduces a generalized telegraph process driven by random trials, analyzing its probability law and mean velocity, with specific focus on Bernoulli and Pólya urn trial schemes and their impact on the process's behavior.

Contribution

It extends the telegraph process framework by incorporating random trial-based velocity changes, providing new analytical results for different trial schemes and intertime distributions.

Findings

Process exhibits logistic stationary density in Bernoulli case

Mean velocity depends on trial outcomes and intertime distributions

Different trial schemes lead to distinct process behaviors

Abstract

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical P\'olya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with non-zero mean), and (ii) the case of P\'olya trials and intertimes having first Gamma and then exponential distributions with constant rates.