System of interacting particles with Markovian switching

Abstract: Most of the published articles on random motions have been devoted to the study of the telegraph process or its generalizations that describe the random motion in $R^n$ of a single particle in a Markov or semi-Markov medium. However, up to our best knowledge there are no published papers dealing with the interaction of two or more particles which move according to the telegraph processes. In this paper, we construct the system of telegraph processes with interactions, which can be interpreted as a model of ideal gas. In this model, we investigate the free path times of a family of particles before they are collided with any other particle. We also study the distribution of particles, which described by telegraph processes with hard collisions and reflecting boundaries, and investigate its limiting properties.