Random Oper. Stoch. Equ. 2018; 26(4): 201ś209 Research Article A. A. Pogorui* and R. M. Rodríguez-Dagnino Interaction of particles governed by generalized integrated telegraph processes https://doi.org/10.1515/rose-2018-0018 Received September 24, 2017; revised March 18, 2018; accepted June 30, 2018 Abstract: In this paper we introduce the analysis of the interaction of two or more particles, where each of them is governed by a generalized integrated telegraph process. We present systematically the distribution of the meeting instant of two telegraph particles on the line (or collision phenomenon) that started their motion at the same time but from diferent locations. We also investigate the distribution of a switching time such that there exists the őrst moment of the őrst instant of two particles collision. Asymptotic results of particle collisions for a system of interacting particles with Markov switching, as time goes to inőnity, are considered. Keywords: Random evolutions, Arratia ŕow, switching times, telegraph random process, semi-Markov processes MSC 2010: Primary 60K35; secondary 60K99, 60K15 || Communicated by: Vyacheslav L. Girko 1 Introduction There have been many published contributions on telegraph random processes and some variations of them, however, most of these contributions are devoted to describe the random motion of a single particle in a Markov or a semi-Markov medium, see [3] and [18] for two recent works in this direction. In this paper we deal with the interaction or collision of two or more particles that switch their position according to a semi-Markov process. The resulting process is a generalization of the well-known telegraph process, and it is called the generalized integrated telegraph process, since it has semi-Markov switching times. We construct a set of processes with interactions, where the particles move according to telegraph pro- cesses. Our model is diferent from the model presented in [2, Chapter 6], where the particles move in straight lines with constant velocity up to collision. The motion in such a case can be seen as trajectories of a billiard in the n-dimensional simplex. Our model cannot be reduced to a billiard type system. Another similar model of interacting particles was studied in [10], as one of the representatives of the Arratia ŕow described in [1]. In such a difusion model the particles stick together under collision and change the difusion coefcient at the moment of coalescing. Coalescence is an important phenomenon that has been considered in several applications. For instance, customer coalescence in queuing networks [7], to model surface cracks in materials [8], and, as it has been remarked by Marcus, for the growth of cloud droplets into raindrops, the formation of molecular polymers, and the growth of particles in a colloidal suspension [14], just to mention a few of them. *Corresponding author: A. A. Pogorui, Department of Applied Mathematics, Zhytomyr State University, Valyka Berdychivska St. 40, 10008, Zhytomyr, Ukraine, e-mail: pogor@zu.edu.ua R. M. Rodríguez-Dagnino, Electrical and Computer Engineering, Tecnologico de Monterrey, Av. Eugenio Garza Sada 2501 Sur, C.P. 64849, Monterrey, N.L., Mexico, e-mail: rmrodrig@itesm.mx Brought to you by | Göteborg University - University of Gothenburg Authenticated Download Date | 11/29/18 5:22 PM