ON SOME SIMILARITIES AND DIFFERENCES BETWEEN FRACTIONAL PROBABILITY DENSITY SIGNED MEASURE OF PROBABILITY AND QUANTUM PROBABILITY

摘要

A probability density of fractional (or fractal) order is defined by the probability incre_ment pr{x < X < x + dx} = pα(x)(dx), 0 < α < 1, and appears to be quite suitable to deal with random variables defined in a fractal space. Combining this definition with the fractional Taylor's series f (x + h) = E_α (D_x~α h~a) f (x) (E_α (.) denotes the Mittag_Leffler function) provided by the modified Riemann_Liouville derivative, one can expand a probability calculus parallel to the standard one. This approach could be considered as a framework for the derivation of some space fractional partial differential diffusion equations in coarse-grained spaces. It is shown firstly that there is some relation be_tween fractional probability and signed measure of probability, and secondly that when α = 1/2, there is some identity between this fractal probability and quantum probability. Shortly, a wavefunction could be thought of as a fractal probability density of order 1/2. One exhibits further relations with possibility theory and relative information. Lastly, one arrives at a new informational entropy based on the inverse of the Mittag_Leffler function.