Advertisement

Journal of Statistical Physics

, Volume 159, Issue 6, pp 1495–1503 | Cite as

Equivalence of Subordinated Processes with Tempered \(\alpha \)-Stable Waiting Times and Fractional Fokker–Planck Equations in Space and Time Dependent Fields

  • Yun-Xiu Zhang
  • Hui Gu
  • Jin-Rong Liang
Article
  • 175 Downloads

Abstract

In this paper we introduce a subordinated stochastic process controlled by tempered \(\alpha \)-stable waiting times and prove the equivalence of this process and the fractional Fokker–Planck equation with space and time dependent diffusion coefficients in the influence of an external space and time dependent force.

Keywords

Subordinated processes Tempered \(\alpha \)-stable processes  Fractional Fokker–Planck equation 

Notes

Acknowledgments

This work is supported by the Science and Technology Commission of Shanghai Municipality (No. 11ZR1410300) and by Shanghai Leading Academic Discipline Project (No.B407).

References

  1. 1.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208 (2004)CrossRefADSMATHMathSciNetGoogle Scholar
  3. 3.
    Kou, S.C., Xie, X.S.: Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93, 180603 (2004)CrossRefADSGoogle Scholar
  4. 4.
    Kou, S.C.: Stochastic modeling in nanoscale Biophysics: subdiffusion within proteins. Ann. Appl. Stat. 2, 501–535 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Liang, J.-R., Wang, J., Lv, L.-J., Gu, H., Qiu, W.-Y., Ren, F.-Y.: Fractional Fokker-Planck equation and Black-Scholes formula in composite-diffusive regime. J. Stat. Phys. 146, 205–216 (2012)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Magdziarz, M., Gajda, J.: Anomalous dynamics of Black-Scholes model time-changed by inverse subordinators. Acta Phys. Pol. B 43, 1093–1110 (2012)CrossRefGoogle Scholar
  7. 7.
    Fogedby, H.C.: Lévy flights in quenched random force fields. Phys. Rev. E 58, 1690 (1998)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Metzler, R., Barkai, E., Klafter, J.: The derivation of fractional Fokker-Planck equations from a generalized Master equation. Europhys. Lett. 46, 431–436 (1999)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Sokolov, I.M., Klafter, J.: Field-induced dispersionn in subdiffusion. Phys. Rev. Lett. 97, 140602 (2006)CrossRefADSGoogle Scholar
  11. 11.
    Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Process. Appl. 119, 3238–3252 (2009)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75, 016708 (2007)CrossRefADSGoogle Scholar
  13. 13.
    Magdziarz, M., Weron, A., Klafter, J.: Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: the case of a time-dependent force. Phys. Rev. Lett. 101, 210601 (2008)CrossRefADSGoogle Scholar
  14. 14.
    Sato, K.-I.: Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  15. 15.
    Magdziarz, M., Gajda, J., Zorawik, T.: Comment on fractional Fokker-Planck equation with space and time dependent drift and diffusion. J. Stat. Phys. 154, 1241–1250 (2014)CrossRefADSMATHMathSciNetGoogle Scholar
  16. 16.
    Bronstein, I., Israel, Y., Kepten, E., Mai, S., Shav-Tal, Y., Barkai, E., Garini, Y.: Transient anomalous diffusion of telomeres in the nucleus of mammalian cells. Phys. Rev. Lett. 103, 018102 (2009)CrossRefADSGoogle Scholar
  17. 17.
    Jeon, J.-H., Monne, H.M.-S., Javanainen, M., Metzler, R.: Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109, 188103 (2012)CrossRefADSGoogle Scholar
  18. 18.
    Jeon, J.-H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C.: Berg-Sørensen, K., Oddershede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)Google Scholar
  19. 19.
    Gajda, J., Magdziarz, M.: Fractional Fokker-Planck equation with tempered \(\alpha -\)stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Rosiński, J.: Tempering stable processes. Stoc. Proc. Appl. 117, 677–707 (2007)CrossRefMATHGoogle Scholar
  21. 21.
    Magdziarz, M., Orzeł, S., Weron, A.: Option pricing in subdiffusive Bachelier model. J. Stat. Phys. 145, 187–203 (2011)CrossRefADSMATHMathSciNetGoogle Scholar
  22. 22.
    Henry, B.I., Langlands, T.A.M., Straka, P.: Fractional Fokker-Planck equations for subdiffusion with space- and time- dependent forces. Phys. Rev. Lett. 105, 170602 (2010)CrossRefADSGoogle Scholar
  23. 23.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. A Series of Comprehensive Studies in Mathematics, vol. 293. Springer, Berlin (1999)Google Scholar
  24. 24.
    Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763–772 (2009)CrossRefADSMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiChina
  2. 2.School of FinanceNanjing Audit UniversityNanjingChina
  3. 3.Department of MathematicsNanjing Forestry UniversityNanjingChina

Personalised recommendations