On the derivation of the Fokker-Plank equation by using of Fractional calculus

In this paper, fractional calculus has been used to find the spectral counterpart of the Fokker-Planck equations for non-linear systems driven by Lévy white noise processes. In particular it is shown thatone can obtain the equation ruling the characteristic function of the response to a non-linear system, withoutusing the Itô formula. Indeed, it is possible to reproduce the well-known results, already known in literature,by means of the characteristic function representation in terms of complex moments, recently proposed by thefirst two authors. The case of a-stable Lévy driven stochastic differential equation is also treated introducingan associated process constructed from the stable one.