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

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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.

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References

  1. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  4. Kou, S.C.: Stochastic modeling in nanoscale Biophysics: subdiffusion within proteins. Ann. Appl. Stat. 2, 501–535 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Magdziarz, M., Gajda, J.: Anomalous dynamics of Black-Scholes model time-changed by inverse subordinators. Acta Phys. Pol. B 43, 1093–1110 (2012)

    Article  Google Scholar 

  7. Fogedby, H.C.: Lévy flights in quenched random force fields. Phys. Rev. E 58, 1690 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  10. Sokolov, I.M., Klafter, J.: Field-induced dispersionn in subdiffusion. Phys. Rev. Lett. 97, 140602 (2006)

    Article  ADS  Google Scholar 

  11. Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Process. Appl. 119, 3238–3252 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75, 016708 (2007)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  14. Sato, K.-I.: Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

  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)

    Article  ADS  MathSciNet  Google Scholar 

  20. Rosiński, J.: Tempering stable processes. Stoc. Proc. Appl. 117, 677–707 (2007)

    Article  MATH  Google Scholar 

  21. Magdziarz, M., Orzeł, S., Weron, A.: Option pricing in subdiffusive Bachelier model. J. Stat. Phys. 145, 187–203 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  23. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. A Series of Comprehensive Studies in Mathematics, vol. 293. Springer, Berlin (1999)

  24. Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763–772 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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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).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

    Yun-Xiu Zhang & Jin-Rong Liang

  2. School of Finance, Nanjing Audit University, Nanjing, 211815, China

    Hui Gu

  3. Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, China

    Yun-Xiu Zhang

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  1. Yun-Xiu Zhang
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  2. Hui Gu
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  3. Jin-Rong Liang
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Correspondence to Jin-Rong Liang.

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Zhang, YX., Gu, H. & Liang, JR. Equivalence of Subordinated Processes with Tempered \(\alpha \)-Stable Waiting Times and Fractional Fokker–Planck Equations in Space and Time Dependent Fields. J Stat Phys 159, 1495–1503 (2015). https://doi.org/10.1007/s10955-014-1184-7

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  • DOI: https://doi.org/10.1007/s10955-014-1184-7

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