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Probability distribution function for systems driven by superheavy-tailed noise

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Abstract.

We develop a general approach for studying the cumulative probability distribution function of localized objects (particles) whose dynamics is governed by the first-order Langevin equation driven by superheavy-tailed noise. Solving the corresponding Fokker-Planck equation, we show that due to this noise the distribution function can be divided into two different parts describing the surviving and absorbing states of particles. These states and the role of superheavy-tailed noise are studied in detail using the theory of slowly varying functions.

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References

  1. W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, 1984)

  2. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, 2nd edn. (World Scientific, Singapore, 2004)

  3. H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer-Verlag, Berlin, 1989)

  4. C.W. Gardiner, Handbook of Stochastic Methods, 2nd edn. (Springer-Verlag, Berlin, 1990)

  5. L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)

    Article  ADS  Google Scholar 

  6. F. Jülicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997)

    Article  ADS  Google Scholar 

  7. P. Reimann, Phys. Rep. 361, 57 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. P. Hänggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)

    Article  ADS  Google Scholar 

  9. N.V. Agudov, B. Spagnolo, Phys. Rev. E 64, R035102 (2001)

    Article  ADS  Google Scholar 

  10. A.A. Dubkov, N.V. Agudov, B. Spagnolo, Phys. Rev. E 69, 061103 (2004)

    Article  ADS  Google Scholar 

  11. S.I. Denisov, A.N. Vitrenko, W. Horsthemke, Phys. Rev. E 68, 046132 (2003)

    Article  ADS  Google Scholar 

  12. S.I. Denisov, A.N. Vitrenko, W. Horsthemke, P. Hänggi, Phys. Rev. E 73, 036120 (2006)

    Article  ADS  Google Scholar 

  13. S. Jespersen, R. Metzler, H.C. Fogedby, Phys. Rev. E 59, 2736 (1999)

    Article  ADS  Google Scholar 

  14. P.D. Ditlevsen, Phys. Rev. E 60, 172 (1999)

    Article  ADS  Google Scholar 

  15. D. Schertzer, M. Larchevêque, J. Duan, V.V. Yanovsky, S. Lovejoy, J. Math. Phys. 42, 200 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. D. Brockmann, I.M. Sokolov, Chem. Phys. 284, 409 (2002)

    Article  ADS  Google Scholar 

  17. A. Dubkov, B. Spagnolo, Fluct. Noise Lett. 5, L267 (2005)

    Article  MathSciNet  Google Scholar 

  18. S.I. Denisov, W. Horsthemke, P. Hänggi, Phys. Rev. E 77, 061112 (2008)

    Article  ADS  Google Scholar 

  19. Lévy Flights and Related Topics in Physics, edited by M.F. Shlesinger, G.M. Zaslavsky, U. Frisch (Springer-Verlag, Berlin, 1995)

  20. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. A.V. Chechkin, V.Y. Gonchar, J. Klafter, R. Metzler, Adv. Chem. Phys. 133, 439 (2006)

    Article  Google Scholar 

  22. A. Dubkov, B. Spagnolo, V.V. Uchaikin, Int. J. Bifurc. Chaos 18, 2649 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. S.I. Denisov, P. Hänggi, H. Kantz, Europhys. Lett. 85, 40007 (2009)

    Article  ADS  Google Scholar 

  24. S. Havlin, H. Weissman, Phys. Rev. B 37, 487 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  25. S. Havlin, G.H. Weiss, J. Stat. Phys. 58, 1267 (1990)

    Article  ADS  Google Scholar 

  26. J. Dräger, J. Klafter, Phys. Rev. Lett. 84, 5998 (2000)

    Article  ADS  Google Scholar 

  27. S.I. Denisov, H. Kantz, Europhys. Lett. 92, 30001 (2010)

    Article  ADS  Google Scholar 

  28. S.I. Denisov, H. Kantz, P. Hänggi, J. Phys. A: Math. Theor. 43, 285004 (2010)

    Article  Google Scholar 

  29. M. Chaichian, A. Demichev, Path Integrals in Physics (Institute of Physics Publishing, Bristol, 2001), Vol. 1

  30. P. Del Moral, A. Doucet, Stoch. Anal. Appl. 22, 1175 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. S.I. Denisov, W. Horsthemke, P. Hänggi, Eur. Phys. J. B 68, 567 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987)

  33. W. Feller, An Introduction to Probability Theory and its Applications, 2nd edn. (Wiley, New York, 1971), Vol. 2

  34. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon & Breach, New York, 1993)

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Authors and Affiliations

  1. Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187, Dresden, Germany

    S. I. Denisov & H. Kantz

  2. Sumy State University, Rimsky-Korsakov Street 2, 40007, Sumy, Ukraine

    S. I. Denisov

Authors
  1. S. I. Denisov
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  2. H. Kantz
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Correspondence to S. I. Denisov.

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Denisov, S., Kantz, H. Probability distribution function for systems driven by superheavy-tailed noise. Eur. Phys. J. B 80, 167–175 (2011). https://doi.org/10.1140/epjb/e2011-10758-1

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  • DOI: https://doi.org/10.1140/epjb/e2011-10758-1

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