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Potential Analysis

, Volume 27, Issue 4, pp 353–380 | Cite as

Markov Chain Approximations for Symmetric Jump Processes

  • Ryad Husseini
  • Moritz KassmannEmail author
Article

Abstract

Markov chain approximations of reversible jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a reversible jump process with state space ℝ d , the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space H α/2(ℝ d ), α∈(0,2), that is equivalent to the standard one.

Keywords

Jump processes Markov chains Lévy measure Central-limit theorem 

Mathematics Subject Classifications (2000)

60J75 60F05 60B10 60J27 60G52 

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References

  1. 1.
    Aldous, D.: Stopping times and tightness. Ann. Probab. 6(2), 335–340 (1978)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)zbMATHGoogle Scholar
  3. 3.
    Barlow, M.T., Bass, R.F., Chen, Zh.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. (2007) (in press)Google Scholar
  4. 4.
    Bass, R.F., Kassmann, M.: Hölder continuity of harmonic functions with respect to operators of variable orders. Comm. Partial Differential Equation 30, 1249–1259 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bass, R.F., Kumagai, T.: Symmetric Markov chains on ℤd with unbounded range. Trans. Amer. Math. Soc. (2007) (in press)Google Scholar
  6. 6.
    Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354(7), 2933–2953 (electronic) (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen, Zh.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108(1), 27–62 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23(Suppl. 2), 245–287 (1987)MathSciNetGoogle Scholar
  9. 9.
    Donsker, M.D.: An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. 1951(6), 12 (1951)MathSciNetGoogle Scholar
  10. 10.
    Ethier, St. N., Kurtz, Th. G.: Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, Characterization and Convergence (1986)Google Scholar
  11. 11.
    Husseini, R., Kassmann, M.: Jump processes, \(\mathcal{L}\)-harmonic functions and continuity estimates. Preprint. http://www.iam.uni-bonn.de/~kassmann/husseini-kassmann-cont-preprint-2.pdf (2006)
  12. 12.
    Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4), 833–860 (1995)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Ma, Z.-M., Röckner, M., Zhang, T.-S. Approximation of arbitrary Dirichlet processes by Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 34(1), 1–22 (1998)zbMATHCrossRefGoogle Scholar
  14. 14.
    Ma, Z.-M., Röckner, M., Sun, W.: Approximation of Hunt processes by multivariate Poisson processes. Acta Appl. Math. 63, 233–243 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  16. 16.
    Stroock, D.W., Varadhan S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)zbMATHGoogle Scholar
  17. 17.
    Stroock, D.W., Zheng, W.: Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33, 619–649 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany

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