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

In this paper we study random flights in R-partial derivative 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 D. We prove that the distributions of the points X-partial derivative (t) and Y-partial derivative (t), t >= 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 X-partial derivative (t) and Y-partial derivative (t). In particular we show that the distribution of Y-partial derivative (t) satisfies a telegraph-type equation with time-varying coefficients. (C) 2014 Elsevier B.V. All rights reserved.