Diffusion term of the Fokker Planck equation for fractional differential equations enforced by white noise

Abstract

In many applications of engineering interest, like in the viscoelastic modelling of solids, the equation of motion can have one or more terms containing derivatives of non-integer order. It follows that finding the probability density function (PDF) of a fractional differential equation enforced by white noise is a serious problem because the It� calculus is not still valid. Indeed, the future state of the response depends upon its entire past history, i.e. the system is not Markovian.

In this paper the loss of Markovianity problem in case of fractional differential equations enforced by a Gaussian white is overcome by using the self-similarity property and the diffusion term of the fractional Fokker Planck (FFPK) equation is calculated showing that it is ruled by a fractional derivative of order 2H, being H the Hurst index.