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.