Random flights in higher spaces

We consider in this paper random flights in R-d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S-1(d). We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n + 1)-fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d = 2 and d = 4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in R-3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d = 2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.