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) partial derivative (beta)p(x, t)/partial derivativet(beta) = (-i)(7) D(x)(7)sigma (x, t)p(x, t), 0 < <beta> less than or equal to 1, 0 < <gamma> less than or equal to 2. The obtained equation could he 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 + tau) - x(t))(2)] similar to sigma (x, t)tau (beta). (C) 2001 Elsevier Science Ltd. All rights reserved.