Random motions governed by third-order equations

In this paper we analyse the motion of a particle P whose velocity is represented by a three-valued telegraph process. We prove that the probability law of the process describing the position of P is a solution of a third-order, linear, partial differential equation. We obtain probability distributions of some generalised versions of the process of random signals, as well as other probabilistic features of the related process. Finally, accelerated motions of P (where acceleration follows the classical telegraph process) are also analysed.