  Using the path integral representation of the density matrix propagator of
quantum Brownian motion, we derive its asymptotic form for times greater than
the localization time, $ (\hbar / \gamma k T )^{\half}$, where $\gamma$ is the
dissipation and $T$ the temperature of the thermal environment. The
localization time is typically greater than the decoherence time, but much
shorter than the relaxation time, $\gamma^{-1}$. We use this result to show
that the reduced density operator rapidly evolves into a state which is
approximately diagonal in a set of generalized coherent states. We thus
reproduce, using a completely different method, a result we previously obtained
using the quantum state diffusion picture (Phys.Rev. D52, 7294 (1995)). We also
go beyond this earlier result, in that we derive an explicit expression for the
weighting of each phase space localized state in the approximately diagonal
density matrix, as a function of the initial state. For sufficiently long times
it is equal to the Wigner function, and we confirm that the Wigner function is
positive for times greater than the localization time (multiplied by a number
of order 1).
