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

In this paper we study random flights in R d 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 d ( t ) and Y d ( 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 d ( t ) and Y d ( t ) . In particular we show that the distribution of Y d ( t ) satisfies a telegraph-type equation with time-varying coefficients.