  We study the consequences of adopting the memory dependent, non-Markovian,
physics with the memory-less over-damped approximation usually employed to
investigate Brownian particles. Due to the finite correlation time scale
associated with the noise, the stationary behavior of the system is not
described by the Boltzmann-Gibbs statistics. However, the presence of a very
weak external white noise can be used to regularize the equilibrium properties.
Surprisingly, the coupling to another bath effectively restores the dynamical
aspects missed by the over-damped treatment.
