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Random motions in R3 with orthogonal directions

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  • Cinque, Fabrizio
  • Orsingher, Enzo

Abstract

This paper is devoted to the detailed analysis of three-dimensional motions in R3 with orthogonal directions switching at Poisson times and moving with constant speed c>0. The study of the random position at an arbitrary time t>0 on the surface of the support, forming an octahedron Sct, is completely carried out on the edges and faces (Fct). In particular, the motion on the faces Fct is analyzed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on Fct in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position p=p(t,x,y,z) inside Sct implied the derivation of a sixth-order partial differential equation governing p (expressed in terms of the products of three D’Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinate as the motion develops. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformly chosen at each Poisson event. If the switches are governed by a homogeneous Poisson process many explicit results are obtained.

Suggested Citation

  • Cinque, Fabrizio & Orsingher, Enzo, 2023. "Random motions in R3 with orthogonal directions," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 173-200.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:173-200
    DOI: 10.1016/j.spa.2023.04.003
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    References listed on IDEAS

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    1. E. Orsingher & A. Gregorio, 2007. "Random Flights in Higher Spaces," Journal of Theoretical Probability, Springer, vol. 20(4), pages 769-806, December.
    2. Kolesnik, Alexander D., 2018. "Slow diffusion by Markov random flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 186-197.
    3. Cinque, Fabrizio, 2022. "A note on the conditional probabilities of the telegraph process," Statistics & Probability Letters, Elsevier, vol. 185(C).
    4. Cinque, Fabrizio & Orsingher, Enzo, 2021. "On the exact distributions of the maximum of the asymmetric telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 601-633.
    5. Orsingher, Enzo, 1990. "Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 49-66, February.
    6. Kolesnik, Alexander D. & Turbin, Anatoly F., 1998. "The equation of symmetric Markovian random evolution in a plane," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 67-87, June.
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