Fractional Poisson Fields

  • Nikolai Leonenko
  • Ely Merzbach


Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space \(\mathbb{R}_+^2\). Some properties are given and, in particular, we prove a long-range dependence property.


Poisson fields Long-range dependence Subordinator Inverse subordinator 

AMS 2000 Subject Classifications

60G22 60G60 60G55 


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  1. Baddeley A (2007) Spatial point processes and their applications. In: Stochastic geometry. Lecture notes in math, 1892. Springer, Berlin, pp 1–75CrossRefGoogle Scholar
  2. Barkai E (2001) Fractional Fokker-Planck equation, solution, and application. Phys Rev E 63:046118CrossRefGoogle Scholar
  3. Beghin L, Orsingher E (2009) Fractional Poisson processes and related planar random motions. Electron J Probab 14(61):1790–1827MATHMathSciNetGoogle Scholar
  4. Beghin L, Orsingher E (2010) Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron J Probab 15(22):684–709MATHMathSciNetGoogle Scholar
  5. Bingham NH (1971) Limit theorems for occupation times of Markov processes. Z Wahrscheinlichkeitstheorie und Verw Gebiete 17:1–22CrossRefMATHMathSciNetGoogle Scholar
  6. Cahoy DO, Uchaikin VV, Woyczynski WA (2010) Parameter estimation for fractional Poisson processes. J Statist Plann Inference 140(11):3106–3120CrossRefMATHMathSciNetGoogle Scholar
  7. Daley DJ (1999) The Hurst index for a long-range dependent renewal processes. Ann Probab 27(4):2035–2041CrossRefMATHMathSciNetGoogle Scholar
  8. Djrbashian MM (1993) Harmonic analysis and boundary value problems in the complex domain. Birkhauser Verlag, BaselCrossRefMATHGoogle Scholar
  9. Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math, Art ID 298628, 51 pGoogle Scholar
  10. Herbin E, Merzbach E (2006) A set-indexed fractional Brownian motion. J Theor Probab 19(2):337–364CrossRefMATHMathSciNetGoogle Scholar
  11. Ivanoff G, Merzbach E (2000) Set-indexed Martingales. Chapman & Hall, London, UKMATHGoogle Scholar
  12. Ivanoff BG, Merzbach E (2006) What is a multi-parameter renewal process? Stochastics 78(6):411–441MATHMathSciNetGoogle Scholar
  13. Janczura J, Wylomanska A (2009) Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys Polon B 40:1341–1351Google Scholar
  14. Laskin N (2003) Fractional Poisson process. Chaotic transport and complexity in classical and quantum dynamics. Commun Nonlinear Sci Numer Simul 8(3–4):201–213CrossRefMATHMathSciNetGoogle Scholar
  15. Leonenko NN, Ruiz-Medina MD, Taqqu MS (2011) Fractional elliptic, hyperbolic and parabolic random fields. Electron J Probab 16:1134–1172CrossRefMATHMathSciNetGoogle Scholar
  16. Leonenko NN, Meerschaert MM, Sikorskii A (2013a) Fractional Pearson diffusions. J Math Anal Appl 403:532–246CrossRefMATHMathSciNetGoogle Scholar
  17. Leonenko NN, Meerschaert MM, Sikorskii A (2013b) Correlation structure of fractional Pearson diffusions. Comput Math Appl. doi: 10.1016/j.camwa.2013.01.009
  18. Leonenko NN, Meerschaert MM, Sikorskii A (2013c) Correlation structure of time changed Levy processes (preprint)Google Scholar
  19. Mainardi F, Gorenflo R, Scalas E (2004) A fractional generalization of the Poisson processes. Vietnam J Math 32:53–64 (special issue)MATHMathSciNetGoogle Scholar
  20. Mainardi F, Gorenflo R, Vivoli A (2005) Renewal processes of Mittag-Leffler and Wright type. Fract Calc Appl Anal 8(1):7–38MATHMathSciNetGoogle Scholar
  21. Mainardi F, Gorenflo R, Vivoli A (2007) Beyond the Poisson renewal process: a tutorial survey. J Comput Appl Math 205(2):725–735CrossRefMATHMathSciNetGoogle Scholar
  22. Meerschaert MM, Sikorskii A (2012) Stochastic models for fractional calculus. De Gruyter, Berlin/BostonMATHGoogle Scholar
  23. Meerschaert MM, Nane E, Vellaisamy P (2011) The fractional Poisson process and the inverse stable subordinator. Electron J Probab 16(59):1600–1620MATHMathSciNetGoogle Scholar
  24. Merzbach E, Nualart D (1986) A characterization of the spatial Poisson process and changing time. Ann Probab 14(4):1380–1390CrossRefMATHMathSciNetGoogle Scholar
  25. Merzbach E, Shaki YY (2008) Characterizations of multiparameter Cox and Poisson processes by the renewal property. Stat Probab Lett 78:637–642CrossRefMATHMathSciNetGoogle Scholar
  26. Orsingher E, Polito F (2012) The space-fractional Poisson process. Stat Probab Lett 82:852–858CrossRefMATHMathSciNetGoogle Scholar
  27. Podlubny I (1999) Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering, vol 198. Academic Press, Inc., San Diego, CAMATHGoogle Scholar
  28. Repin ON, Saichev AI (2000) Fractional Poisson law. Radiophys Quantum Electron 43(9):738–741CrossRefMathSciNetGoogle Scholar
  29. Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley, New YorkMATHGoogle Scholar
  30. Veillette M, Taqqu MS (2010a) Numerical computation of first passage times of increasing Lévy processes. Methodol Comput Appl Probab 12(4):695–729CrossRefMATHMathSciNetGoogle Scholar
  31. Veillette M, Taqqu MS (2010b) Using differential equations to obtain joint moments of first-passage times of increasing Lé vy processes. Stat Probab Lett 80(7–8):697–705CrossRefMATHMathSciNetGoogle Scholar
  32. Wang X-T, Wen Z-X (2003) Poisson fractional processes. Chaos, Solitons Fractals 18(1):169–177CrossRefMATHMathSciNetGoogle Scholar
  33. Wang X-T, Wen Z-X, Zhang S-Y (2006) Fractional Poisson process. II, Chaos, Solitons Fractals 28(1):143–147CrossRefMATHMathSciNetGoogle Scholar
  34. Wang X-T, Zhang S-Y, Fan S (2007) Nonhomogeneous fractional Poisson processes. Chaos, Solitons Fractals 31(1):236–241CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

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