  Two models for quantum Brownian motion - the Oscillator Bath (OB) model and
the Random-Band-Matrix (RBM) model - are compared and a relation between the
spectral density function I(w) and the variance (Vab)^2 is established. The
extension to a fractal environment is then considered and the microscopic
origin of anomalous diffusion is discussed. In particular, it is shown that the
asymptotic behavior of the normalized velocity auto-correlation function (VACF)
is entirely determined by the band form factor. This allows for a microscopic
derivation of the "whip-back" effect.
