A PROBABILISTIC INTERPRETATION OF THE DZHRBASHYAN FRACTIONAL INTEGRAL

Dazhi Zhao, Guozhu Yu, Tao Yu () and Lu Zhang ()
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Dazhi Zhao: School of Sciences, Southwest Petroleum University, Chengdu 610500, P. R. China†Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu 610500, P. R. China
Guozhu Yu: ��School of Mathematics, Southwest Jiaotong University, Chengdu 610031, P. R. China
Tao Yu: �School of Mathematics, Sichuan University, Chengdu 610065, P. R. China
Lu Zhang: �School of Mathematics, Sichuan University, Chengdu 610065, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 08, 1-8

Abstract: Physical and probabilistic interpretations of the fractional derivatives and integrals are basic problems to their applications. In this paper, we establish a relation between the Dzhrbashyan fractional integral and the expectation of a corresponding random variable by constructing the cumulative distribution function. As examples, interpretations of the Riemann–Liouville fractional integral and Kober integral operator are given. Furthermore, probabilistic interpretations of the Caputo fractional derivative and the fractional integral of a function with respect to another function are discussed too. With the help of probabilistic interpretations proposed in this paper, models described by fractional derivatives and integrals can be endowed with corresponding statistical meanings, while some statistical physics models can be rewritten in fractional calculus too.

Keywords: Probabilistic Interpretation; Dzhrbashyan Fractional Integral; Distribution Function; Physical Interpretation; Statistical Physics (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21502698

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