Journal of Statistical Physics

, Volume 147, Issue 6, pp 1216–1225 | Cite as

Random Motion with Uniformly Distributed Directions and Random Velocity

  • Anatoliy A. Pogorui
  • Ramón M. Rodríguez-Dagnino


In this paper we deal with uniformly distributed direction of motion or isotropic motion at random speed or velocity where the direction alternations occur according to the renewal epochs of a general distribution. We derive the renewal equation for the characteristic function of the transition density of the multidimensional motion. Then, by using the renewal equation, we study the behavior of the transition density near the sphere of its singularity for one-, two-, three-, and four-dimensional cases. To illustrate our solution methodology we present detailed calculations of many solvable examples.


Random evolutions Semi-Markov processes General distributions Random velocity 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Anatoliy A. Pogorui
    • 1
  • Ramón M. Rodríguez-Dagnino
    • 2
  1. 1.Department of MathematicsZhytomyr State UniversityZhytomyrUkraine
  2. 2.Electrical and Computer EngineeringTecnológico de MonterreyMonterreyMexico

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