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A Cyclic Random Motion in $$\mathbb {R}^3$$ R 3 Driven by Geometric Counting Processes

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  • Antonella Iuliano

    (University of Basilicata)

  • Gabriella Verasani

    (University of Basilicata)

Abstract

We consider the random motion of a particle that moves with constant velocity in $$\mathbb {R}^3$$ R 3 . The particle can move along four different directions that are attained cyclically. It follows that the support of the stochastic process describing the particle’s position at a fixed time is a tetrahedron. We assume that the sequence of sojourn times along each direction follows a Geometric Counting Process (GCP). When the initial condition is fixed, we obtain the explicit form of the probability law of the process, for the particle’s position. We also investigate the limiting behavior of the related probability density when the intensities of the four GCPs tend to infinity. Furthermore, we show that the process does not admit a stationary density. Finally, we introduce the first-passage-time problem for the first component of the process through a constant positive boundary providing the bases for future developments.

Suggested Citation

  • Antonella Iuliano & Gabriella Verasani, 2024. "A Cyclic Random Motion in $$\mathbb {R}^3$$ R 3 Driven by Geometric Counting Processes," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-23, June.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:2:d:10.1007_s11009-024-10083-0
    DOI: 10.1007/s11009-024-10083-0
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    References listed on IDEAS

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    3. 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.
    4. Antonio Di Crescenzo & Franco Pellerey, 2019. "Some Results and Applications of Geometric Counting Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 203-233, March.
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