Hostname: page-component-5cf477f64f-qls9x Total loading time: 0 Render date: 2025-03-29T11:56:50.876Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Nonparametric inference Special processes 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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Bartlett, M. (1978). A note on random walks at constant speed. Adv. Appl. Prob. 10, 704–707.Google Scholar
Bowman, F. (1958). Introduction to Bessel Functions. Dover, New York.Google Scholar
Brooks, E. (1999). Probabilistic methods for a linear reaction-hyperbolic system with constant coefficients. Ann. Appl. Prob. 9, 719–731.Google Scholar
Di Crescenzo, A. (2002). Exact transient analysis of a planar motion with three directions. Stoch. Stoch. Reports 72, 175–189.Google Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980). Tables of Integrals, Series and Products. Academic Press, New York.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.Google Scholar
Kolesnik, A. D. (1989). A model of Markovian random evolution on a plane. In Analytical Methods for Studying the Evolution of Stochastic Systems, Akad. Nauk. Ukrain. SSR, Inst. Math., Kiev, pp. 55–61 (in Russian).Google Scholar
Kolesnik, A. D. (2001). Weak convergence of a planar random evolution to the Wiener process. J. Theoret. Prob. 14, 485–494.Google Scholar
Kolesnik, A. D. (2003). Weak convergence of the distributions of Markovian random evolutions in two and three dimensions. Bull. Acad. Sci. Moldova 3, 41–52.Google Scholar
Kolesnik, A. D. and Orsingher, E. (2002). Analysis of a finite-velocity planar random motion with reflection. Theory Prob. Appl. 46, 132–140.Google Scholar
Kolesnik, A. D. and Turbin, A. F. (1998). The equation of symmetric Markovian random evolution in a plane. Stoch. Process. Appl. 75, 67–87.Google Scholar
Leorato, S. and Orsingher, E. (2004). Bose–Einstein-type statistics, order statistics and planar random motions with three directions. Adv. Appl. Prob. 36, 937–970.Google Scholar
Leorato, S., Orsingher, E. and Scavino, M. (2003). An alternating motion with stops and the related planar, cyclic motion with four directions. Adv. Appl. Prob. 35, 1153–1168.Google Scholar
Masoliver, J., Porrà, J. M. and Weiss, G. H. (1993). Some two and three-dimensional persistent random walks. Physica A. 193, 469–482.CrossRefGoogle Scholar
Orsingher, E. (1986). A planar random motion governed by the two-dimensional telegraph equation. J. Appl. Prob. 23, 385–397.Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.Google Scholar
Orsingher, E. (2000). Exact Joint distribution in a model of planar random motion. Stoch. Stoch. Reports 69, 1–10.Google Scholar
Pinsky, M. (1976). Isotropic transport process on a Riemannian manifold. Trans. Amer. Math. Soc. 218, 353–360.Google Scholar
Pinsky, M. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.Google Scholar
Stadje, W. (1987). The exact probability distribution of a two-dimensional random walk. J. Statist. Phys. 46, 207–216.Google Scholar
Stadje, W. (1989). Exact probability distributions for non-correlated random walk models. J. Statist. Phys. 56, 415–435.Google Scholar
Tolubinsky, E. V. (1969). The Theory of Transfer Processes. Naukova Dumka, Kiev.Google Scholar