On some finite-velocity random motions driven by the geometric counting process

We study a stochastic process which describes the dynamics 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) $\mathbb{R} \times \{\vec{v}_1, \vec{v}_2\}$, with velocities $v_1 > v_2$, and (ii) $\mathbb{R}^2 \times \{\vec{v}_1, \vec{v}_2, \vec{v}_3\}$, where the particle moves along three different directions with possibly unequal velocities. Assuming that the random intertimes between consecutive changes of directions are governed by geometric counting processes, 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 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. In both cases we also investigate the behavior of the probability distributions of the motion when the intensities of the underlying geometric counting processes tend to infinity. The asymptotic probability law of the particle is found to be a uniform distribution in case (i) and a three-peaked distribution in case (ii).