A Pathwise Ergodic Theorem for Quantum Trajectories
  If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states. 
Burkhard Kuemmerer, Hans Maassen