Journal of Statistical Physics

, Volume 141, Issue 1, pp 68–93 | Cite as

Fractional Non-Linear, Linear and Sublinear Death Processes

Article

Abstract

This paper is devoted to the study of a fractional version of non-linear Open image in new window , t>0, linear M ν(t), t>0 and sublinear \(\mathfrak{M}^{\nu}(t)\), t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T (t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

Keywords

Fractional diffusion Dzhrbashyan–Caputo fractional derivative Mittag-Leffler functions Linear death process Non-linear death process Sublinear death process Subordinated processes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bailey, N.: The Elements of Stochastic Processes with Applications to the Natural Sciences. Wiley, New York (1964) MATHGoogle Scholar
  2. 2.
    Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986) CrossRefMathSciNetADSMATHGoogle Scholar
  3. 3.
    Schneider, W.R., Wyss, W.: Fractional diffusion and wave equation. J. Math. Phys. 30(1), 134–144 (1988) CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi 133(1), 425–430 (1986) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Orsingher, E., Beghin, L.: Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Relat. Fields 128(1), 141–160 (2004) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Saxena, R.K., Mathai, A.M., Haubold, H.J.: Fractional Reaction-Diffusion Equations. Astrophys. Space Sci. 305, 289–296 (2006) CrossRefADSMATHGoogle Scholar
  8. 8.
    Bening, V.E., Korolev, V.Yu., Koksharov, S., Kolokoltsov, V.N.: Limit theorems for continuous-time random walks in the double-array limit scheme. J. Math. Sci. 146(4), 5959–5976 (2007) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Meerschaert, M.M., Benson, D.A., Scheffler, H.-P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65(4), 041103 (2002) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Zaslavsky, G.M.: Fractional kinetic equation for Hamiltonian chaos. Physica D 76, 110–122 (1994) CrossRefMathSciNetADSMATHGoogle Scholar
  11. 11.
    Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7(4), 753–764 (1997) CrossRefMathSciNetADSMATHGoogle Scholar
  12. 12.
    Saxena, R.K., Mathai, A.M., Haubold, H.J.: On Fractional Kinetic Equations. Astrophys. Space Sci. 282, 281–287 (2002) CrossRefADSGoogle Scholar
  13. 13.
    Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344, 657–664 (2004) CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 209, 299–310 (2004) CrossRefADSGoogle Scholar
  15. 15.
    Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002) CrossRefMathSciNetADSMATHGoogle Scholar
  16. 16.
    Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2006) Google Scholar
  17. 17.
    Repin, O.N., Saichev, A.I.: Fractional Poisson law. Radiophys. Quantum Electron. 43(9), 738–741 (2000) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Laskin, N.: Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003) CrossRefMathSciNetADSMATHGoogle Scholar
  19. 19.
    Mainardi, F., Gorenflo, R.: A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64 (2004) MathSciNetMATHGoogle Scholar
  20. 20.
    Cahoy, D.O.: Fractional Poisson processes in terms of alpha-stable densities. Ph.D. Thesis (2007) Google Scholar
  21. 21.
    Uchaikin, V.V., Sibatov, R.T.: A fractional Poisson process on a model of dispersive charge transport in semiconductors. Russ. J. Numer. Anal. Math. Model. 23(3), 283–297 (2008) CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Beghin, L., Orsingher, E.: Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14(61), 1790–1826 (2009) MathSciNetMATHGoogle Scholar
  23. 23.
    Uchaikin, V.V., Cahoy, D.O., Sibatov, R.T.: Fractional processes: from poisson to branching one. Int. J. Bifurc. Chaos 18(9), 2717–2725 (2008) CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Orsingher, E., Polito, F.: Fractional pure birth processes. Bernoulli 16(3), 858–881 (2010) CrossRefGoogle Scholar
  25. 25.
    Orsingher, E., Polito, F.: On a fractional linear birth-death process. Bernoulli; online since 3rd February 2010 (2010, to appear) Google Scholar
  26. 26.
    Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Physica A 370(1), 114–118 (2006) CrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000) CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Chudnovsky, A., Kunin, B.: A probabilistic model of brittle crack formation. J. Appl. Phys. 62(10), 4124–4129 (1987) CrossRefADSGoogle Scholar
  29. 29.
    Orsingher, E., Beghin, L.: Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37(21), 206–249 (2009) CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    DeBlassie, R.D.: Iterated Brownian motion in an open set. Ann. Appl. Probab. 14(3), 1529–1558 (2004) CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Donnelly, P., Kurtz, T., Marjoram, P.: Correlation and variability in birth processes. J. Appl. Probab. 30(2), 275–284 (1993) CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Beghin, L., Orsingher, E.: Iterated elastic Brownian motions and fractional diffusion equations. Stoch. Process. Appl. 119(6), 1975–2003 (2009) CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Kolokoltsov, V.N.: Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory Probab. Appl. 53(4), 594–609 (2009) CrossRefMATHGoogle Scholar
  34. 34.
    Kirschenhofer, P.: A note on alternating sums. Electron. J. Comb. 3(2), 1–10 (1996) MathSciNetGoogle Scholar
  35. 35.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison–Wesley, Boston (1994) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Enzo Orsingher
    • 1
  • Federico Polito
    • 1
  • Ludmila Sakhno
    • 2
  1. 1.Dipartimento di Statistica, Probabilità e Statistiche Applicate“Sapienza” Università di RomaRomeItaly
  2. 2.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

Personalised recommendations