Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Asymptotic expansion for transport processes in semi-Markov media

Authors: A. A. Pogorui and Ramón M. Rodríguez-Dagnino
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 83 (2010).
Journal: Theor. Probability and Math. Statist. 83 (2011), 127-134
MSC (2010): Primary 60J25; Secondary 35C20
Published electronically: February 2, 2012
MathSciNet review: 2768853
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study asymptotic expansions for a solution of the singularly perturbed equation for a functional of a semi-Markov random evolution on the line. By using the method for solutions of singularly perturbed equations, we obtain the solution in the form of a series of regular and singular terms. The first regular term satisfies a diffusion-type equation, and the first singular term is a semi-group with the infinitesimal operator of the respective related bivariate process. Each regular and singular term can be calculated recursively.

References [Enhancements On Off] (What's this?)

  • 1. V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Kluwer Academic Publishers, 1998. MR 1753470 (2002b:60169)
  • 2. V. S. Korolyuk and A. V. Swishchuk, Evolution of Systems in Random Media, CRC Press, Boca Raton, FL, U.S.A., 1995. MR 1413300 (98g:60116)
  • 3. I. I. Gikhman and A. V. Skorokhod, The theory of Stochastic Processes, vol. 2, Springer-Verlag, New York, 1975. MR 0375463 (51:11656)
  • 4. V. S. Korolyuk and A. F. Turbin, Markov Renewal Processes in System Reliability Problems, Naukova Dumka, Kiev, 1982. (Russian) MR 695006 (85e:60095)
  • 5. V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of the State Lumping of Large Systems, Kluwer Academic Publishers, 1993. MR 1281385 (95e:60071)
  • 6. A. B. Vasil'eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations, Vysshaya Shkola, Moscow, 1990. (Russian) MR 1108181 (92i:34072)
  • 7. S. Albeverio, V. S. Korolyuk, and I. V. Samoilenko, Asymptotic expansion of semi-Markov random evolutions, Stochastics: An International Journal of Probability and Stochastics Processes, 81 (October 2009), no. 5, 477-502. MR 2569263 (2010k:60311)
  • 8. I. V. Samoilenko, Asymptotic expansion of Markov random evolution, Ukrainian Mathematical Bulletin 3 (2006), no. 3, 394-407. MR 2330681 (2008d:60098)
  • 9. A. Pogorui, Asymptotic expansion for the distribution of a Markovian random motion, Random Operators & Stochastic Equations 17 (2009), 189-196. MR 2560865 (2010i:60217)
  • 10. M. A. Pinsky, Lectures on Random Evolutions, World Scientific Publishing, 1991. MR 1143780 (93b:60160)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60J25, 35C20

Retrieve articles in all journals with MSC (2010): 60J25, 35C20

Additional Information

A. A. Pogorui
Affiliation: Zhytomyr State Ivan Franko University, Velyka Berdychivs’ka St. 40, Zhytomyr 10008, Ukraine

Ramón M. Rodríguez-Dagnino
Affiliation: Centro de Electrónica y Telecomunicaciones, ITESM, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, N.L., México

Keywords: Asymptotic expansion, semi-Markov, random evolution, singular perturbed equation
Received by editor(s): October 20, 2009
Published electronically: February 2, 2012
Additional Notes: We thank ITESM, Campus Monterrey, through the Research Chair in Telecommunications, for the support provided in the development of this work
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society