We consider a Poisson process with random intensity for which the distribution of intervals between jumps is described by an equation with fractional derivatives. The distribution of random intensity of this process and the generating function of the jump number are obtained in explicit form. It is noted that the studied fractional Poisson law can be used for statistical description of chaotic processes of different physical nature demonstrating the phenomenon of anomalous diffusion.
KeywordsGenerate Function Explicit Form Poisson Process Fractional Derivative Physical Nature
Unable to display preview. Download preview PDF.
- 1.M. F. Shlesinger, G. M. Zaslavsky, and U. Frish, eds., Lecture Notes in Physics. Levy Flights and Related Topic in Physics, Springer, Berlin (1994).Google Scholar
- 2.J. Klafter, M. F. Shlesinger, and G. Zumofen, Phys. Today, 49, 33 (1996).Google Scholar
- 3.A. I. Saichev and G. M. Zaslavsky, Chaos, 7, No. 4, 753 (1997).Google Scholar
- 4.E. Barkai and J. Klafter, Phys. Rev. E, 57, 5237 (1998).Google Scholar
- 5.W. Feller, An Introduction to Probability Theory and Its Applications, 2 [Russian translation], Mir, Moscow (1984).Google Scholar
- 6.S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).Google Scholar
- 7.A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (Bate-man Manuscript Project), 3, McGraw-Hill, New York (1953).Google Scholar