This paper investigates the motion of a Brownian particle experiencing both a friction (biased) force and a randomly fluctuating force with a long-time-correlation function ${\mathit{C}}_{\mathit{f}}$(t)∼${\mathit{t}}^{\mathrm{-}\mathrm{\beta }}$, 0<β<1, 1<β<2, and β=1, instead of a Dirac δ function. The generalized Langevin equation and Fokker-Planck equation and corresponding solution are presented. It is shown that when 0<β<1 or 1<β<2, the diffusion motion of the Brownian particle is the anomalous diffusion that is related to fractal Brownian motion (FBM). But when β=1 the diffusion motion is anomalous diffusion with no connection to FBM. The effects of friction retardation result in a probability density function for finding the particle at displacement X at time t that depends on the initial value of velocity of the particle. The approach in this paper may provide a systematic method for the study of particles diffusing in fractal media.

• Received 22 July 1991

DOI:https://doi.org/10.1103/PhysRevA.45.833