Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

Affiliation(s)

Tecnológico de Monterrey (ITESM), Electrical and Computer Engineering. Sucursal de correos, Monterrey, México.

Tecnológico de Monterrey (ITESM), Electrical and Computer Engineering. Sucursal de correos, Monterrey, México.

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.

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.

Cite this paper

nullA. 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.

nullA. Pogorui and R. Rodríguez-Dagnino, "Stationary Distribution of Random Motion with Delay in Reflecting Boundaries,"

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.

[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.