On a Telegraph Process with Generalized Mittag-Leffler Waiting times and Velocity Driven by Random trials

Abstract

We study a generalized telegraph process in which the velocity is governed by random trials by considering the specific distribution of waiting times. In this telegraph process, a particle moving on real line may change its direction whenever there is an arrival in a counting process. This direction change is driven by the outcomes of random trials. In the first case, random trials are independent and identically distributed, and waiting times have Mittag-Leffler distribution. In the second case, random trials follow P\'olya urn scheme and the first waiting time is generalized Mittag-Leffler distributed whereas other waiting times have Mittag-Leffler distribution. In both cases, we obtain the discrete component of their probability law. Also, absolutely continuous components of their conditional probability law given initial velocity are derived. The plots of absolutely continuous components of their probability law are compared for different parameters. Conditional on the initial velocity, the distributions of $n$th event time of counting processes associated with these telegraph processes are obtained.