Random flights related to the Euler-Poisson-Darboux equation

This paper is devoted to the analysis of random motions on the line and in the space R^d (d > 1) performed at finite velocity and governed by a non-homogeneous Poisson process with rate \lambda(t). The explicit distributions p(x,t) of the position of the randomly moving particles are obtained solving initial-value problems for the Euler- Poisson-Darboux equation when \lambda(t) = \alpha/t, t > 0. We consider also the case where \lambda(t) = \lambda coth \lambda t and \lambda(t) = \lambda tanh \lambda t, where some Riccati differential equations emerge and the explicit distributions are obtained for d = 1. We also examine planar random motions with random velocities by projecting random flights in R^d onto the plane. Finally the case of planar motions with four orthogonal directions is considered and the corresponding higher-order equations with time-varying coefficients obtained.