  We consider the non-Markovian Langevin evolution of a dissipative dynamical
system in quantum mechanics in the path integral formalism. After discussing
the role of the frequency cutoff for the interaction of the system with the
heat bath and the kernel and noise correlator that follow from the most common
choices, we derive an analytic expansion for the exact non-Markovian
dissipation kernel and the corresponding colored noise in the general case that
is consistent with the fluctuation-dissipation theorem and incorporates
systematically non-local corrections. We illustrate the modifications to
results obtained using the traditional (Markovian) Langevin approach in the
case of the exponential kernel and analyze the case of the non-Markovian
Brownian motion.
