Ukrainian Mathematical Journal

, 63:665 | Cite as

Multidimensional random motion with uniformly distributed changes of direction and erlang steps

  • A. A. Pogorui
  • R. M. Rodríguez-Dagnino

We study transport processes in ℝ n , n ≥ 1; that have nonexponentially distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection to a fixed line, and, using this idea, we avoid many difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang-distributed sojourn times.


Probability Density Function Cumulative Distribution Function Random Vector Sojourn Time Random Motion 
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  1. 1.
    A. Di Crescenzo, “On random motions with velocities alternating at Erlang-distributed random times,” Adv. Appl. Probab., 61, 690–701 (2001).Google Scholar
  2. 2.
    A. A. Pogorui and R. M. Rodríguez-Dagnino, “One-dimensional semi-Markov evolutions with general Erlang sojourn times,” Rand. Oper. Stochast. Equat., 13, 1720–1724 (2005).Google Scholar
  3. 3.
    M. Pinsky, “Isotropic transport process on a Riemann manifold,” Trans. Amer. Math. Soc., 218, 353–360 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    E. Orsingher and A. De Gregorio, “Random flights in higher spaces,” J. Theor. Probab., 20, 769–806 (2007).zbMATHCrossRefGoogle Scholar
  5. 5.
    A. D. Kolesnik, “Random motions at finite speed in higher dimensions,” J. Statist. Phys., 131, 1039–1065 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    W. Stadje, “Exact solution for non-correlated random walk models,” J. Statist. Phys., 56, 415–435 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Le Caër, “A Pearson–Dirichlet random walk,” J. Statist. Phys., 40, 728–751 (2010).CrossRefGoogle Scholar
  8. 8.
    A. A. Pohorui, “Fading evolutions in multidimensional spaces,” Ukr. Mat. Zh., 62, No. 11, 1577–1582 (2010); English translation: Ukr. Math. J., 62, No. 11, 1828–1834 (2011).Google Scholar
  9. 9.
    V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific (2005).Google Scholar
  10. 10.
    S. Bochner and K. Chandrasekharan, “Fourier transforms,” Ann. Math. Stud., No. 19 (1949).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • A. A. Pogorui
    • 1
  • R. M. Rodríguez-Dagnino
    • 2
  1. 1.Zhytomyr State UniversityZhytomyrUkraine
  2. 2.Monterrey Institute of Technology and Higher EducationMonterreyMexico

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