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

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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 that one can obtain the equation ruling the characteristic function of the response to a non-linear system, without using 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 the first two authors. The case of a-stable Lévy driven stochastic differential equation is also treated introducing an associated process constructed from the stable one.