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This paper is devoted to the analysis of random motions on the line and inthe space R^d (d > 1) performed at finite velocity and governed by anon-homogeneous Poisson process with rate \lambda(t). The explicitdistributions p(x,t) of the position of the randomly moving particles areobtained solving initial-value problems for the Euler- Poisson-Darboux equationwhen \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 someRiccati differential equations emerge and the explicit distributions areobtained for d = 1. We also examine planar random motions with randomvelocities by projecting random flights in R^d onto the plane. Finally the caseof planar motions with four orthogonal directions is considered and thecorresponding higher-order equations with time-varying coefficients obtained.

Random flights related to the Euler-Poisson-Darboux equation