  Stochastic convergence of discrete time Markov processes has been analysed
based on a dual Lyapunov approach. Using some existing results on ergodic
theory of Markov processes, it has been shown that existence of a properly
subinvariant function (counterpart of the Lyapunov density in deterministic
systems) implies sweeping of a Markov process out of the sets where this
function is integrable. Such a function can be used as a certificate of
convergence in probability of a stochastic system. We apply this technique to
Markov processes induced by a quantum system with non-demolition measurement
and propose dual Lyapunov certificates to certify sweeping.
