Bose-Einstein-type statistics, order statistics and planar random motions with three directions

In this paper we study different types of planar random motions (performed with constant velocity) with three directions, defined by the vectors d(j) = (cos (2pij/3), sin (2pij/3)) for j = 0, 1, 2, changing at Poisson-paced times. We examine the cyclic motion (where the change of direction is deterministic), the completely uniform motion (where at each Poisson event each direction can be taken with probability 1/3) and the symmetrically deviating case (where the particle can choose all directions except that taken before the Poisson event). For each of the above random motions we derive the explicit distribution of the position of the particle, by using an approach based on order statistics. We prove that the densities obtained are solutions of the partial differential equations governing the processes. We are also able to give the explicit distributions on the boundary and, for the case of the symmetrically deviating motion, we can write it as the distribution of a telegraph process. For the symmetrically deviating motion we use a generalization of the Bose-Einstein statistics in order to determine the distribution of the triple (N-0, N-1, N-2) (conditional on N(t) = k, with N-0 + N-1 + N-2 = N(t) + 1, where N(t) is the number of Poisson events in [0, t]), where N-j denotes the number of times the direction d(j) (j = 0, 1, 2) is taken. Possible extensions to four directions or more are briefly considered.