Lévy anomalous diffusion and fractional Fokker–Planck equation

We demonstrate that the Fokker–Planck equation can be generalized into a ‘fractional Fokker–Planck’ equation, i.e., an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Lévy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Lévy stable source to the classical Gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non-trivial fractional operator which corresponds to the possible asymmetry of the Lévy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the fractional Fokker–Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Lévy stable distributions. Furthermore, with the help of important examples, we show the applicability of the fractional Fokker–Planck equation in physics.