An alternating motion with stops and the related planar cyclic motion with four directions

In this paper we study a planar random motion (X(t), Y(t)), t > 0, with orthogonal directions taken cyclically at Poisson paced times. The process is split into one-dimensional motions with alternating displacements interrupted by exponentially distributed stops. The distributions of X = X (t) (conditional and nonconditional) are obtained by means of order statistics and the connection with the telegrapher's process is derived and discussed. We are able to prove that the distributions involved in our analysis are solutions of a certain differential system and of the related fourth-order hyperbolic equation.