  We consider the energy-driven stochastic state vector reduction equation for
the density matrix, which for pure state density matrices can be written in two
equivalent forms. We use these forms to discuss the decoupling of the noise
terms for independent subsystems, and to construct ``environmental'' stochastic
density matrices whose time-independent expectations are the usual quantum
statistical distributions. We then consider a measurement apparatus weakly
coupled to an external environment, and show that in mean field (Hartree)
approximation the stochastic equation separates into independent equations for
the apparatus and environment, with the Hamiltonian for the apparatus augmented
by the environmental expectation of the interaction Hamiltonian. We use the
Hartree approximated equation to study a simple accretion model for the
interaction of the apparatus with its environment, as part of a more general
discussion of when the stochastic dynamics predicts state vector reduction, and
when it predicts the maintenance of coherence. We also discuss the magnitude of
decoherence effects acting during the reduction process. Our analysis supports
the suggestion that a measurement takes place when the different outcomes are
characterized by sufficiently distinct environmental interactions for the
reduction process to be rapidly driven to completion.
