Journal of Statistical Physics

, Volume 170, Issue 4, pp 700–730 | Cite as

Fractional Poisson Fields and Martingales

  • Giacomo Aletti
  • Nikolai Leonenko
  • Ely Merzbach


We present new properties for the Fractional Poisson process (FPP) and the Fractional Poisson field on the plane. A martingale characterization for FPPs is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.


Fractional Poisson fields Inverse subordinator Martingale characterization Second order statistics Fractional differential equations 

Mathematics Subject Classification

Primary: 60G55 60G60 Secondary: 60G44 60G57 62E10 60E07 



N. Leonenko and E. Merzbach wish to thank G. Aletti for two visits to University of Milan.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Giacomo Aletti
    • 1
  • Nikolai Leonenko
    • 2
  • Ely Merzbach
    • 3
  1. 1.ADAMSS CenterUniversità degli Studi di MilanoMilanItaly
  2. 2.Cardiff UniversityCardiffUK
  3. 3.Bar-Ilan UniversityRamat GanIsrael

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