Reflecting Random Flights

We consider random flights in Rd reflecting on the surface of a sphere Sd−1 R , with center at the origin and with radius R, where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere Sd−1 1 . We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to n ≥ 1. We show that these distributions involve functions which are solutions of the Euler–Poisson–Darboux equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing n by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.