Fractional (space-time) Fokker–Planck equation

By using Kramers–Moyal forward expansion and the definition of characteristic function (CF) with some consideration related to derivatives of fractional order, one can obtain the fractional space-time Fokker–Planck equation (FFPE) The obtained equation could be related to a dynamical system subject to fractional Brownian motion. Therefore, the solution of FFPE will be established on three different cases that correspond to different physical situations related to the mean-square displacement, (x(t+τ)−x(t))2 σ(x,t)τβ.