  We analyse the evolution of a quantum oscillator in a finite temperature
environment using the quantum state diffusion (QSD) picture. Following a
treatment similar to that of reference [7] we identify stationary solutions of
the corresponding It\^o equation. We prove their global stability and compute
typical time scales characterizing the localization process. The recovery of
the density matrix in approximately diagonal form enables us to verify the
approach to thermal equilibrium in the long time limit and we comment on the
connection between QSD and the decoherent histories approach.
