Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-27T05:17:18.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes Markov processes

Published online by Cambridge University Press:  22 February 2016

Alexander D. Kolesnik
Alexander D. Kolesnik*
Affiliation:
Academy of Sciences of Moldova
*
∗ Postal address: Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academy Street 5, Kishinev 2028, Moldova. Email address: kolesnik@math.md
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.

Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line ℝ. The processes Xk(t), k = 1, 2, describe stochastic motions at finite constant velocities c1 > 0 and c2 > 0 that start at the initial time instant t = 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1 > 0 and λ2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) - X2(t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

Keywords

Telegraph processtelegraph equationpersistent random walkprobability distribution functionEuclidean distance

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J60: Diffusion processes 60J65: Brownian motion
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. 82C70: Transport processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1172 - 1193
Copyright
© Applied Probability Trust 

References

Bartlett, M. S. (1957). Some problems associated with random velocity. Publ. Inst. Statist. Univ. Paris 6, 261–270.Google Scholar
Bartlett, M. S. (1978). A note on random walks at constant speed. Adv. Appl. Prob. 10, 704–707.Google Scholar
Bogachev, L. and Ratanov, N. (2011). Occupation time distributions for the telegraph process. Stoch. Process. Appl. 121, 1816–1844.Google Scholar
Cane, V. R. (1967). Random walks and physical processes. Bull. Internat. Statist. Inst. 42, 622–640.Google Scholar
Cane, V. R. (1975). Diffusion models with relativity effects. In Perspectives in Probability and Statistics, Applied Probability Trust, Sheffield, pp. 263–273.Google Scholar
Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.Google Scholar
Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 84–96.Google Scholar
Foong, S. K. (1992). First-passage time, maximum displacement, and Kac's solution of the telegrapher equation. Phys. Rev. A 46, 707–710.Google 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.CrossRefGoogle Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.Google Scholar
Kabanov, Yu. M. (1993). Probabilistic representation of a solution of the telegraph equation. Theory Prob. Appl. 37, 379–380.CrossRefGoogle Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.Google Scholar
Kaplan, S. (1964). Differential equations in which the Poisson process plays a role. Bull. Amer. Math. Soc. 70, 264–267.CrossRefGoogle Scholar
Kisyński, J. (1974). On M. Kac's probabilistic formula for the solution of the telegraphist's equation. Ann. Polon. Math. 29, 259–272.Google Scholar
Kolesnik, A. (1998). The equations of Markovian random evolution on the line. J. Appl. Prob. 35, 27–35.Google Scholar
Kolesnik, A. D. (2012). Moment analysis of the telegraph random process. Bull. Acad. Ştiinţe Ripub. Moldova Math. 1 (68), 90–107.Google Scholar
Kolesnik, A. D. and Ratanov, N. (2013). Telegraph Processes and Option Pricing. Springer, Heidelberg.Google Scholar
Masoliver, J. and Weiss, G. H. (1992). First passage times for a generalized telegrapher's equation. Physica A 183, 537–548.CrossRefGoogle 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.Google Scholar
Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.Google Scholar
Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986). Integrals and Series. Supplementary Chapters. Nauka, Moscow (in Russian).Google Scholar
Ratanov, N. E. (1997). Random walks in an inhomogeneous one-dimensional medium with reflecting and absorbing barriers. Theoret. Math. Phys. 112, 857–865.Google Scholar
Ratanov, N. E. (1999). Telegraph evolutions in inhomogeneous media. Markov Process. Relat. Fields 5, 53–68.Google Scholar
Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.Google Scholar
Turbin, A. F. and Samoīlenko, I. V. (2000). A probabilistic method for solving the telegraph equation with real-analytic initial conditions. Ukrainian Math. J. 52, 1292–1299.CrossRefGoogle Scholar