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 by solving initial value problems for the Euler - Poisson - Darboux equation when lambda(t) = alpha/t, alpha > 0. We consider also the case where lambda(t) = lambda coth lambda t and lambda(t) = lambda tank 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 obtained 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.