  Motivated by the search for a quantum analogue of the macroscopic fluctuation
theory, we study quantum spin chains dissipatively coupled to quantum noise.
The dynamical processes are encoded in quantum stochastic differential
equations. They induce dissipative friction on the spin chain currents. We show
that, as the friction becomes stronger, the noise induced dissipative effects
localize the spin chain states on a slow mode manifold, and we determine the
effective stochastic quantum dynamics of these slow modes. We illustrate this
approach by studying the quantum stochastic Heisenberg spin chain.
