  A dynamics between Newton and Langevin formalisms is elucidated within the
framework of the generalized Langevin equation. For thermal noise yielding a
vanishing zero-frequency friction the corresponding non-Markovian Brownian
dynamics exhibits anomalous behavior which is characterized by ballistic
diffusion and accelerated transport. We also investigate the role of a possible
initial correlation between the system degrees of freedom and the heat-bath
degrees of freedom for the asymptotic long-time behavior of the system
dynamics. As two test beds we investigate (i) the anomalous energy relaxation
of free non-Markovian Brownian motion that is driven by a harmonic velocity
noise and (ii) the phenomenon of a net directed acceleration in noise-induced
transport of an inertial rocking Brownian motor.
