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Telegraph processes with random velocities

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

W. Stadje  and
S. Zacks
W. Stadje*
Affiliation:
University of Osnabrück
S. Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de
∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA
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Abstract

We study a one-dimensional telegraph process (M t ) t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of M t and its first two moments are derived. We characterize the level hitting times of M t in terms of integro-differential equations which can be solved in special cases.

Keywords

Telegraph processrandom velocityhitting timeexact distributionintegro-differential equation

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 665 - 678
Copyright
Copyright © Applied Probability Trust 2004 

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References

Beghin, L., Nieddu, L., and Orsingher, E. (2001). Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, 11–25.10.1155/S104895330100003XGoogle Scholar
Berg, H. C., and Brown, D. A. (1979). Chemotaxis in Escherichia coli analyzed by three-dimensional tracking. Nature 239, 500–504.10.1038/239500a0Google Scholar
Boxma, O. J., Perry, D. and van der Duyn Schouten, F. A. (1999). Fluid queues and mountain processes. Prob. Eng. Inf. Sci. 13, 407–427.10.1017/S0269964899134028Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Campbell, G. A., and Foster, R. M. (1967). Fourier Integrals for Practical Applications, 6th edn. Van Nostrand, Princeton, NJ.Google Scholar
Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.10.1239/aap/1005091360Google Scholar
Di Crescenzo, A. (2002). Exact transient analysis of a planar random motion with three directions. Stoch. Stoch. Rep. 72, 175–189.10.1080/10451120290019186Google Scholar
Dynkin, E. B. (1965). Markov Processes, Vols 1, 2. Academic Press, New York.10.1007/978-3-662-00031-1Google Scholar
Foong, S. K. (1992). First passage time, maximum displacement and Kac's solution to the telegrapher equation. Phys. Rev. A 46, 707–710.10.1103/PhysRevA.46.R707Google Scholar
Foong, S. K., and Kanno, S. (1994). Properties of the telegrapher's random process with or without a trap. Stoch. Process. Appl. 53, 147–173.10.1016/0304-4149(94)90061-2Google Scholar
Gautam, N., Kulkarni, V. G., Palmowski, Z., and Rolski, T. (1999). Bounds for fluid models driven by semi-Markov inputs. Prob. Eng. Inf. Sci. 13, 429–475.10.1017/S026996489913403XGoogle Scholar
Kolesnik, A. D., and Turbin, A. F. (1998). The equation of symmetric Markovian random evolution in a plane. Stoch. Process. Appl. 75, 67–87.10.1016/S0304-4149(98)00003-9Google Scholar
Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queueing: Models and Applications in Science and Engineering, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 321–338.Google Scholar
Lovely, P. S., and Dahlquist, F. W. (1975). Statistical measures of bacterial motility and chemotaxis. J. Theoret. Biol. 50, 477–496.10.1016/0022-5193(75)90094-6Google Scholar
Masoliver, J., and Weiss, G. H. (1992). First passage times for a generalized telegrapher's equation. Physica A 183, 537–548.10.1016/0378-4371(92)90299-6Google Scholar
Masoliver, J., and Weiss, G. H. (1993). On the maximum displacement of a one-dimensional diffusion process described by the telegrapher's equation. Physica A 195, 93–100.10.1016/0378-4371(93)90255-3Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchhoff's law. Stoch. Process. Appl. 34, 49–66.10.1016/0304-4149(90)90056-XGoogle Scholar
Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators Stoch. Equat. 3, 9–21.Google Scholar
Orsingher, E. (2000). Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69, 1–10.10.1080/17442500008834229Google Scholar
Perry, D., Stadje, W., and Zacks, S. (1999). Contributions to the theory of first-exit times for some compound processes in queueing theory. Queueing Systems 33, 369–379.10.1023/A:1019140616021Google Scholar
Siegert, A. F. J. (1951). On the first passage time probability problem. Physical Rev. 2 81, 617–623.10.1103/PhysRev.81.617Google Scholar
Stadje, W. (1987). The exact probability distribution of a two-dimensional random walk. J. Statist. Phys. 46, 207–216.10.1007/BF01010341Google Scholar
Stadje, W. (1988). Asymptotic normality in a two-dimensional random walk model for cell motility. J. Statist. Phys. 51, 615–635.10.1007/BF01028475Google Scholar
Stadje, W. (1989). Exact probability distributions for non-correlated random walk models. J. Statist. Phys. 56, 415–435.10.1007/BF01044444Google Scholar
Stroock, D. W. (1974). Some stochastic processes which arise from a model of the motion of a bacterium. Z. Wahrscheinlichkeitsth. 28, 305–315.10.1007/BF00532948Google Scholar
Weiss, G. H. (1983). Random walks and their applications. Amer. Sci. 71, 65–71.Google Scholar
Zacks, S. (2003). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.10.1239/jap/1082999081Google Scholar