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Goldstein-Kac telegraph equations and random flights in higher dimensions

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

Abstract

In this paper we deal with random motions in dimensions two, three, and five, where the governing equations are telegraph-type equations in these dimensions. Our methodology is first applied to the second-order telegraph equation and we refine well-known results found by other methods. Next, we show that the (2,λ)-Erlang distribution for sojourn times defines the underlying stochastic process for the three-dimensional Goldstein-Kac type telegraph equation and by finding the corresponding fundamental solution of this equation, we have obtained the approximated expression for the transition density of the three-dimensional movement, our results are more complete than previous ones, and this result may have important consequences in applications. We also obtain the 5-dimensional telegraph-type equation by assuming a random motion with an (4,λ)-Erlang distribution for sojourn times, and such equation can be factorized as the product of two telegraph-type equations where one of them is the Goldstein-Kac 5-dimensional telegraph equation. In our analysis the dimension n is related to the (n−1,λ)-Erlang distribution for sojourn times of the random walks.

Suggested Citation

  • Pogorui, Anatoliy A. & Rodríguez-Dagnino, Ramón M., 2019. "Goldstein-Kac telegraph equations and random flights in higher dimensions," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 617-629.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:617-629
    DOI: 10.1016/j.amc.2019.05.045
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    References listed on IDEAS

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    1. Pogorui, Anatoliy A. & Rodríguez-Dagnino, Ramón M., 2013. "Random motion with gamma steps in higher dimensions," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1638-1643.
    2. De Gregorio, Alessandro & Orsingher, Enzo, 2012. "Flying randomly in Rd with Dirichlet displacements," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 676-713.
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