Exact transient analysis of a planar random motion with three directions

Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.