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Annales Henri Poincaré

, Volume 13, Issue 6, pp 1501–1510 | Cite as

From Diffusions on Graphs to Markov Chains via Asymptotic State Lumping

  • Adam BobrowskiEmail author
Open Access
Article

Abstract

We show that fast diffusions on finite graphs with semi permeable membranes on vertices may be approximated by finite-state Markov chains provided the related permeability coefficients are appropriately small. The convergence theorem involves a singular perturbation with singularity in both operator and boundary/transmission conditions, and the related semigroups of operators converge in an irregular manner. The result is motivated by recent models of synaptic depression.

Keywords

Markov Chain Line Graph Fast Diffusion Large Deviation Principle Semi Permeable Membrane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Banasiak J., Bobrowski A.: Interplay between degenerate convergence of semigroups and asymptotic analysis: a study of a singularly perturbed abstract telegraph system. J. Evol. Equ. 9(2), 293–314 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Banasiak J., Goswami A., Shindin S.: Aggregation in age and space structured population models: an asymptotic analysis approach. J. Evol. Equ. 11, 121–154 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bobrowski A.: Degenerate convergence of semigroups. Semigr. Forum 49(3), 303–327 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bobrowski A.: A note on convergence of semigroups. Ann. Polon. Math. 69(2), 107–127 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bobrowski A.: Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge (2005)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bobrowski A.: Degenerate convergence of semigroups related to a model of stochastic gene expression. Semigr. Forum 73(3), 345–366 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bobrowski A.: On limitations and insufficiency of the Trotter–Kato theorem. Semigr. Forum 75(2), 317–336 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bobrowski A., Bogucki R.: Semigroups generated by convex combinations of several Feller generators in models of mathematical biology. Studia Mathematica 189, 287–300 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bobrowski, A., Morawska, K.: From a PDE model to an ODE model of dynamics of synaptic depression. (2011) (submitted)Google Scholar
  10. 10.
    Deo N.: Graph Theory with Applications to Engineering and Computer Science. Prentice-Hall, Inc., Englewood Cliffs (1974)zbMATHGoogle Scholar
  11. 11.
    Engel K.-J., Nagel R.: One-parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  12. 12.
    Engel K.-J., Nagel R.: A Short Course on Operator Semigroups. Springer, New York (2006)zbMATHGoogle Scholar
  13. 13.
    Ethier S.N., Kurtz T.G.: Markov Processes. Characterization and Convergence. Wiley, New York (1986)zbMATHGoogle Scholar
  14. 14.
    Ewens W.J.: Mathematical Population Genetics. Springer, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Feller W.: Diffusion processes in genetics. Proceedings of 2nd Symposium on Probability and Statistics, Berkeley (1950)Google Scholar
  16. 16.
    Feller W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Feller W.: Diffusion processes in one dimension. Trans. Am. Math. Soc. 77(1), 1–31 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Feller W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1954)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fijavž M.K., Mugnolo D., Sikolya E.: Variational and semigroup methods for waves and diffusion in networks. Appl. Math. Optim. 55, 219– (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Freidlin M.I., Sheu S.-J.: Diffusion processes on graphs: stochastic differential equations, large deviation principle. Probab. Theory Relat. Fields 116, 181–220 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Freidlin M.I., Wentzel A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  22. 22.
    Freidlin M.I., Wentzel A.D.: Diffusion processes on graphs and the averaging principle. Ann. Prob. 21, 2215–2245 (1993)zbMATHCrossRefGoogle Scholar
  23. 23.
    Freidlin, M.I., Wentzel, A.D.: Random Perturbations of Hamiltonian Systems. AMS Mem. Am. Math. Soc. 109(523) (1994)Google Scholar
  24. 24.
    Ito K., Mc Kean H.P. Jr: Diffusion Processes and their Sample Paths. Springer, Berlin (1996) Reprint of the 1974 Edition. Classics in Mathematics SerieszbMATHGoogle Scholar
  25. 25.
    Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  26. 26.
    Kaźmierczak B., Lipniacki T.: Regulation of kinase activity by diffusion and feedback. J. Theor. Biol. 259, 291–296 (2009)CrossRefGoogle Scholar
  27. 27.
    Kostrykin, V., Potthoff, J., Shrader, R.: Contraction semigroups on metric graphs. In: Exner, P., Keating, J., Kuchment, P., Sunada, T., Teplyaev, A. (eds.). Analysis on Graphs and its Applications, vol. 77 of Proceedings of Symposium on Pure Mathematics, pp. 423–458. American Mathematical Society, Providence, R.I. (2008)Google Scholar
  28. 28.
    Kostrykin, V., Potthoff, J., Shrader, R.: Brownian motions on metric graphs. ArXiv:1102.4937v1 (2011)Google Scholar
  29. 29.
    Mandl P.: Analytical Treatment of One-Dimensional Markov Processes. Springer, New York (1968)zbMATHGoogle Scholar
  30. 30.
    Mugnolo D.: Gaussian estimates for a heat equation on a network. Netw. Het. Media 2, 55–79 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Mugnolo, D.: Semigroup methods for evolution equations on networks, pp. 1–75. (2007) (preprint)Google Scholar
  32. 32.
    Nittka R.: Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup. J. Funct. Anal. 257(5), 1429–1444 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)zbMATHGoogle Scholar
  34. 34.
    Taylor H.M., Karlin S.: Second Course in Stochastic Processes. Academic Press, London (1981)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Electrical Engineering and Computer ScienceLublin University of TechnologyLublinPoland

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