On some finite-velocity random motions with underlying geometric counting processes

We analyse the probability law of a stochastic process which describes the location of a particle performing a finite-velocity random motion whose velocities alternate cyclically. We consider two cases, in which the state-space of the process is (i) R × {v1,v2}, with velocities v1 > v2, and (ii) R^2 × {v1,v2,v3}, where the particle moves along three different directions with possibly unequal velocities. Assuming that the random inter-times between consecutive changes of directions are governed by a geometric counting process, we first construct the stochastic models of the particle motion. Then we investigate various features of the considered processes and obtain the formal expression of their probability laws. In particular, in the case (ii) we study a planar random motion with three specific directions and determine the exact transition probability density functions of the process when the initial velocity is fixed.