Planar Random Motions in a Vortex

Enzo Orsingher () and Manfred Marvin Marchione ()
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Enzo Orsingher: Sapienza Università di Roma
Manfred Marvin Marchione: Sapienza Università di Roma

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-42

Abstract: 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.

Keywords: Telegraph process; Bessel functions; Hyperbolic equations; 60K50; 60J25; 60K99 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01378-6

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