Fractional representation of Fokker–Planck equation

From the definition of the characteristic function and Kramers–Moyal forward expansion, one can obtain the fractional Fokker–Planck equation (FFPE) in the domain of fractal time evolution with a critical exponent β (0<β 1) [El-Wakil SA, Zahran MA. Chaos, Solitons & Fractals 11 (2000) 791–98]. The solutions of Fokker–Planck equation will establish in three different cases of mean-square displacement as follows: (i) (x(t+τ)−x(t))2 τ, (ii) (x(t+τ)−x(t))2 τβ, 0<β 1, (iii) (x(t+τ)−x(t))2 x−θτβ, θ=dw−2.The distribution function of each case can be obtained in a closed form of Fox-function.