Random flights governed by Klein–Gordon-type partial differential equations

In this paper we study random flights in RdRd with displacements possessing Dirichlet distributions of two different types and uniformly oriented. The randomization of the number of displacements has the form of a generalized Poisson process whose parameters depend on the dimension dd. We prove that the distributions of the points Xd(t) and Yd(t), t≥0t≥0, performing the random flights (with the first and the second form of Dirichlet intertimes) are related to Klein–Gordon-type p.d.e.’s. Our analysis is based on McBride theory of integer powers of hyper-Bessel operators. A special attention is devoted to the three-dimensional case where we are able to obtain the explicit form of the equations governing the law of Xd(t) and Yd(t). In particular we show that the distribution of Yd(t) satisfies a telegraph-type equation with time-varying coefficients.