Share This Article:

Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

Abstract Full-Text HTML Download Download as PDF (Size:295KB) PP. 24-28
DOI: 10.4236/am.2010.11004    4,160 Downloads   7,346 Views   Citations

ABSTRACT

In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v1 and v2. We consider a special type of boundary which can trap the particle for a random time. We found closed-form expressions for the stationary distribution of the position of the particle not only for the alternating Markov process but also for a broad class of semi-Markov processes.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

A. Pogorui and R. Rodríguez-Dagnino, "Stationary Distribution of Random Motion with Delay in Reflecting Boundaries," Applied Mathematics, Vol. 1 No. 1, 2010, pp. 24-28. doi: 10.4236/am.2010.11004.

References

[1] S. Goldstein, “On Diffusion by Discontinuous Move-ments and on the Telegraph Equation,” The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 4, No. 2, 1951, pp. 129-156.
[2] M. Kac, “A Stochastic Model Related to the Telegrapher’s Equation,” Rocky Mountain Journal of Mathematics, Vol. 4, No. 3, 1974, pp. 497-509.
[3] E. Orsingher, “Hyperbolic Equations Arising in Random Models,” Stochastic Processes and their Applications, Vol. 21, No. 1, 1985, pp. 93-106.
[4] A. F. Turbin, “Mathematical Model of Einstein, Wiener, Levy,” in Russian, Fractal Analysis and Related Fields, Vol. 2, 1998, pp. 47-60.
[5] J. Masoliver, J. M. Porrà, and G. H. Weiss, “Solution to the Telegrapher’s Equation in the Presence of Reflecting and Partly Reflecting Boundaries,” Physical Review E, Vol. 48, No. 2, 1993, pp. 939-944.
[6] V. S. Korolyuk and A. V. Swishchuk, A. V. Semi- Mar-kov, “Random Evolutions,” Kluwer Academic Publishers, 1995.
[7] V. S. Korolyuk and V. V. Korolyuk, “Stochastic Models of Systems,” Kluwer Academic Publishers, 1999.
[8] V. S. Korolyuk and A. F. Turbin, “Mathematical Founda-tions of the State Lumping of Large Systems,” Kluwer Academic Publishers, 1994.
[9] I. I. Gikhman and A. V. Skorokhod, “Theory of Stochastic Processes, Vol. 2,” Springer-Verlag, New York, 1975.
[10] V. Balakrishnan, C. van den Broeck and P. Hangui, “First-Passage of Non-Markovian Processes: The Case of a Reflecting Boundary,” Physical Review A, Vol. 38, No. 8, 1988, pp. 4213-4222.

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.