Geodesic
Probability Interpretation of the Integral of Fractional Order
Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 3, pp. 491-507

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We establish a relation between stable distributions in probability theory and the fractional integral. Moreover, it turns out that the parameter of the stable distribution coincides with the exponent of the fractional integral. It follows from an analysis of the obtained results that equations with the fractional time derivative describe the evolution of some physical system whose time degree of freedom becomes stochastic, i.e. presents a sum of random time intervals subject to a stable probability distribution. We discuss relations between the fractal Cantor set (Cantor strips) and the fractional integral. We show that the possibility to use this relation as an approximation of the fractional integral is rather limited.
Keywords: integral of fractional order, stable probability distributions
Mots-clés : Fokker–Planck. 
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     author = {A. A. Stanislavskii},
     title = {Probability {Interpretation} of the {Integral} of {Fractional} {Order}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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A. A. Stanislavskii. Probability Interpretation of the Integral of Fractional Order. Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 3, pp. 491-507. http://geodesic.mathdoc.fr/item/TMF_2004_138_3_a10/