On Multifractality and Fractional Derivatives

It is shown phenomenologically that the fractional derivative = Du of order of a multifractal function has a power-law tail in its cumulative probability, for a suitable range of 's. The exponent is determined by the condition , where _p is the exponent of the structure function of order p. A detailed study is made for the case of random multiplicative processes (Benzi et al., Physica D65:352 (1993)) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the _p graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g., because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data.