Fractional Differential Equations and Multifractality

There has been a mushrooming interest in the linear Fokker-Planck Equation (FPPE) which corresponds to the generating equation of Lévy's anomalous diffusion. We already pointed out some theoretical and empirical limitations of the linear FPPE for various geophysical problems: the medium is in fact considered as homogeneous and the exponent of the power law of the pdf tails should be smaller than 2. We showed that a nonlinear extension based on a nonlinear Langevin equation forced by a Lévy stable motion overcomes these limitations. We show that in order to generate multifractal diffusion, and more generally multifractal fields, we need to furthermore consider fractional time derivatives in the Langevin equation and in FPPE. We compare our approach with the Continuous-Time Random Walk (CTWR) approach.