Abstract
Abstract This paper deals with random motion at non-constant speed with uniformly distributed directions where the direction alternations occur according to renewal epochs of general distribution. We derive the renewal equation for the characteristic function of the transition density of the multidimensional motion. By using the renewal equation, we study the behavior of the transition density near the sphere of its singularity for the two- and four-dimensional cases and variable velocity, and the three-dimensional case for constant velocity. As examples, we have derived the distribution for one-, two- and three-dimensional random motion.
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