Diffusion and memory effects for stochastic processes and fractional Langevin equations

We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker–Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the non-linear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker–Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative (defined according to Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969). In the generic linear or non-linear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution.