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Planar Random Motions in a Vortex

Author

Listed:
  • Enzo Orsingher

    (Sapienza Università di Roma)

  • Manfred Marvin Marchione

    (Sapienza Università di Roma)

Abstract

We study a planar random motion $$\big (X(t),Y(t)\big )$$ ( X ( t ) , Y ( t ) ) with orthogonal directions which can turn clockwise, turn counterclockwise, and reverse its direction, each with a different probability. The support of the process is given by a time-varying square and the singular distributions on the boundary and the diagonals of the square are obtained explicitly. In the interior of the support, we study the hydrodynamic limit of the distribution. We then investigate the time T(t) spent by the process moving vertically and the joint distribution of $$\big (T(t),Y(t)\big )$$ ( T ( t ) , Y ( t ) ) . We prove that, in the hydrodynamic limit, the process $$\big (X(t),Y(t)\big )$$ ( X ( t ) , Y ( t ) ) spends half the time moving vertically.

Suggested Citation

  • Enzo Orsingher & Manfred Marvin Marchione, 2025. "Planar Random Motions in a Vortex," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-42, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01378-6
    DOI: 10.1007/s10959-024-01378-6
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    References listed on IDEAS

    as
    1. E. Orsingher & A. Gregorio, 2007. "Random Flights in Higher Spaces," Journal of Theoretical Probability, Springer, vol. 20(4), pages 769-806, December.
    2. Weiss, George H, 2002. "Some applications of persistent random walks and the telegrapher's equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 381-410.
    3. 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.
    4. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
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