  We consider the decoherence of phase space histories in a class of quantum
Brownian motion models, consisting of a particle moving in a potential $V(x)$
in interaction with a heat bath at temperature $T$ and dissipation gamma, in
the Markovian regime. The evolution of the density operator for this open
system is thus described by a non-unitary master equation. The phase space
histories of the system are described by a class of quasiprojectors.
Generalizing earlier results of Hagedorn and Omn\`es, we show that a phase
space projector onto a phase space cell $\Gamma$ is approximately evolved under
the master equation into another phase space projector onto the classical
dissipative evolution of $\Gamma$, and with a certain amount of degradation due
to the noise produced by the environment. We thus show that histories of phase
space samplings approximately decohere, and that the probabilities for these
histories are peaked about classical dissipative evolution, with a width of
peaking depending on the size of the noise.
