  We develop a general framework to study quantum trajectories resulting from
repeated random measurements subject to stationary noise, and generalize
results of K\"ummerer and Maassen to this setting. The resulting trajectory of
quantum states is a time-inhomogeneous Markov chain in a random environment.
K\"ummerer and Maassen introduced the concept of dark subspaces for noise-free
processes, establishing that their absence is equivalent to asymptotic
purification of the system state. We clarify the notion of dark subspaces in
the disordered setting by defining a measurable correspondence consisting of a
collection of random subspaces satisfying a darkness condition. We further
prove that asymptotic purification occurs if and only if this collection of
random dark subspaces is empty. Several examples of these phenomena are
provided.
