We investigate the diffusion motion of a Brownian particle which is ac
ted upon by both a friction force with memory effect and a noise. The
noise is expressed as f(X, t) is similar to X-sigma F(t), sigma > 0, w
here X and t are the displacement and time, respectively, and F(t) has
the long-time correlation effect [F(0) F(t)] is similar to t(-beta),
0 < beta < 1, beta = 1, 1 < beta < 2. The generalized Langevin equatio
n, the corresponding Fokker-Planck equation and its solution at large
time are established. A variety of anomalous diffusion patterns are pr
oposed. The correlation effects of noise may bring about that the effe
ctive diffusion coefficient is dependent on both the displacement and
time, and the probability density for finding the Brownian particle is
a non-Gaussian distribution. O'Shaughnessy and Procaccia's results on
the diffusion on fractals can be derived from our model.