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 Levy stable stochastic forcing. A precise determination of
this equation is obtained by substituting a Levy 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 Levy 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 Levy stable distributions. Furthermore, with the help of
important examples, we show the applicability of the Fractional Fokker-Planck
equation in physics.Comment: 22 pages; To Appear in Physica