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A planar random motion with an infinite number of directions controlled by the damped wave equation

Part of: Special processes Markov processes Nonparametric inference Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Alexander D. Kolesnik  and
Enzo Orsingher
Alexander D. Kolesnik*
Affiliation:
Academy of Sciences of Moldova
Enzo Orsingher*
Affiliation:
University of Rome ‘La Sapienza’
*
∗Postal address: Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academy Street 5, Kishinev, MD-2028, Moldova. Email address: [email protected]
∗∗Postal address: Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Roma, Italy. Email address: [email protected]
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Abstract

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We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction θ with uniform law in [0, 2π). This model represents the natural two-dimensional counterpart of the well-known Goldstein–Kac telegraph process. For the particle's position (X(t), Y(t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f(x, y, t) of (X(t), Y(t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity λ of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.

Keywords

Random motionfinite speeduniformly distributed directiontelegraph processdamped wave equationmembrane vibrationBessel functionplanar Brownian motionSonine formula

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 62G30: Order statistics; empirical distribution functions 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J60: Diffusion processes 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1168 - 1182
Copyright
© Applied Probability Trust 2005 

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