The distribution of random motion in semi-Markov media

This paper deals with the random motion with finite speed along uniformly distributed directions, where the direction alternations occur according to renewal epochs of a general distribution. We derive a renewal equation for the characteristic function of a transition density of multidimensional motion. By using the renewal equation, we study the behavior of the transition density near the sphere of its singularity in two- and three-dimensional cases. For $\left(n-1\right)$-Erlang distributed steps of the motion in an $n$-dimensional space ($n\geq 2$), we have obtained the characteristic function as a solution of the renewal equation. As an example, we have derived the distribution for the three-dimensional random motion.