A PROBABILISTIC INTERPRETATION OF THE DZHRBASHYAN FRACTIONAL INTEGRAL

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 theCaputo 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.