Probability calculus of fractional order and fractional Taylor’s series application to Fokker–Planck equation and information of non-random functions

Abstract

A probability distribution of fractional (or fractal) order is defined by the measure μ{d x} = p( x)(d x) α , 0 < α < 1. Combining this definition with the fractional Taylor’s series f ( x + h ) = E α ( D x α h α ) f ( x ) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier’s transform of fractional order using the Mittag–Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker–Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.