Skip to main content
SpringerLink
Log in

Fractional Poisson Fields

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Baddeley A (2007) Spatial point processes and their applications. In: Stochastic geometry. Lecture notes in math, 1892. Springer, Berlin, pp 1–75

    Chapter  Google Scholar 

  • Barkai E (2001) Fractional Fokker-Planck equation, solution, and application. Phys Rev E 63:046118

    Article  Google Scholar 

  • Beghin L, Orsingher E (2009) Fractional Poisson processes and related planar random motions. Electron J Probab 14(61):1790–1827

    MATH  MathSciNet  Google Scholar 

  • Beghin L, Orsingher E (2010) Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron J Probab 15(22):684–709

    MATH  MathSciNet  Google Scholar 

  • Bingham NH (1971) Limit theorems for occupation times of Markov processes. Z Wahrscheinlichkeitstheorie und Verw Gebiete 17:1–22

    Article  MATH  MathSciNet  Google Scholar 

  • Cahoy DO, Uchaikin VV, Woyczynski WA (2010) Parameter estimation for fractional Poisson processes. J Statist Plann Inference 140(11):3106–3120

    Article  MATH  MathSciNet  Google Scholar 

  • Daley DJ (1999) The Hurst index for a long-range dependent renewal processes. Ann Probab 27(4):2035–2041

    Article  MATH  MathSciNet  Google Scholar 

  • Djrbashian MM (1993) Harmonic analysis and boundary value problems in the complex domain. Birkhauser Verlag, Basel

    Book  MATH  Google Scholar 

  • Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math, Art ID 298628, 51 p

  • Herbin E, Merzbach E (2006) A set-indexed fractional Brownian motion. J Theor Probab 19(2):337–364

    Article  MATH  MathSciNet  Google Scholar 

  • Ivanoff G, Merzbach E (2000) Set-indexed Martingales. Chapman & Hall, London, UK

    MATH  Google Scholar 

  • Ivanoff BG, Merzbach E (2006) What is a multi-parameter renewal process? Stochastics 78(6):411–441

    MATH  MathSciNet  Google Scholar 

  • Janczura J, Wylomanska A (2009) Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys Polon B 40:1341–1351

    Google Scholar 

  • Laskin N (2003) Fractional Poisson process. Chaotic transport and complexity in classical and quantum dynamics. Commun Nonlinear Sci Numer Simul 8(3–4):201–213

    Article  MATH  MathSciNet  Google Scholar 

  • Leonenko NN, Ruiz-Medina MD, Taqqu MS (2011) Fractional elliptic, hyperbolic and parabolic random fields. Electron J Probab 16:1134–1172

    Article  MATH  MathSciNet  Google Scholar 

  • Leonenko NN, Meerschaert MM, Sikorskii A (2013a) Fractional Pearson diffusions. J Math Anal Appl 403:532–246

    Article  MATH  MathSciNet  Google Scholar 

  • Leonenko NN, Meerschaert MM, Sikorskii A (2013b) Correlation structure of fractional Pearson diffusions. Comput Math Appl. doi:10.1016/j.camwa.2013.01.009

  • Leonenko NN, Meerschaert MM, Sikorskii A (2013c) Correlation structure of time changed Levy processes (preprint)

  • Mainardi F, Gorenflo R, Scalas E (2004) A fractional generalization of the Poisson processes. Vietnam J Math 32:53–64 (special issue)

    MATH  MathSciNet  Google Scholar 

  • Mainardi F, Gorenflo R, Vivoli A (2005) Renewal processes of Mittag-Leffler and Wright type. Fract Calc Appl Anal 8(1):7–38

    MATH  MathSciNet  Google Scholar 

  • Mainardi F, Gorenflo R, Vivoli A (2007) Beyond the Poisson renewal process: a tutorial survey. J Comput Appl Math 205(2):725–735

    Article  MATH  MathSciNet  Google Scholar 

  • Meerschaert MM, Sikorskii A (2012) Stochastic models for fractional calculus. De Gruyter, Berlin/Boston

    MATH  Google Scholar 

  • Meerschaert MM, Nane E, Vellaisamy P (2011) The fractional Poisson process and the inverse stable subordinator. Electron J Probab 16(59):1600–1620

    MATH  MathSciNet  Google Scholar 

  • Merzbach E, Nualart D (1986) A characterization of the spatial Poisson process and changing time. Ann Probab 14(4):1380–1390

    Article  MATH  MathSciNet  Google Scholar 

  • Merzbach E, Shaki YY (2008) Characterizations of multiparameter Cox and Poisson processes by the renewal property. Stat Probab Lett 78:637–642

    Article  MATH  MathSciNet  Google Scholar 

  • Orsingher E, Polito F (2012) The space-fractional Poisson process. Stat Probab Lett 82:852–858

    Article  MATH  MathSciNet  Google Scholar 

  • 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, CA

    MATH  Google Scholar 

  • Repin ON, Saichev AI (2000) Fractional Poisson law. Radiophys Quantum Electron 43(9):738–741

    Article  MathSciNet  Google Scholar 

  • Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley, New York

    MATH  Google Scholar 

  • Veillette M, Taqqu MS (2010a) Numerical computation of first passage times of increasing Lévy processes. Methodol Comput Appl Probab 12(4):695–729

    Article  MATH  MathSciNet  Google Scholar 

  • 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):697–705

    Article  MATH  MathSciNet  Google Scholar 

  • Wang X-T, Wen Z-X (2003) Poisson fractional processes. Chaos, Solitons Fractals 18(1):169–177

    Article  MATH  MathSciNet  Google Scholar 

  • Wang X-T, Wen Z-X, Zhang S-Y (2006) Fractional Poisson process. II, Chaos, Solitons Fractals 28(1):143–147

    Article  MATH  MathSciNet  Google Scholar 

  • Wang X-T, Zhang S-Y, Fan S (2007) Nonhomogeneous fractional Poisson processes. Chaos, Solitons Fractals 31(1):236–241

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

    Nikolai Leonenko

  2. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Ely Merzbach

Authors
  1. Nikolai Leonenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ely Merzbach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ely Merzbach.

Additional information

Nikolai Leonenko and Ely Merzbach were partially supported by a grant of the Commission of the European Communities PIRSES-GA-2008-230804 (Marie Curie) “Multi-parameter Multi-fractional Brownian Motion”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leonenko, N., Merzbach, E. Fractional Poisson Fields. Methodol Comput Appl Probab 17, 155–168 (2015). https://doi.org/10.1007/s11009-013-9354-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9354-7

Keywords

AMS 2000 Subject Classifications

Search

Navigation