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We prove that every stationary state in the annihilator of all Krausoperators of a weak coupling limit type Markov generator consists of twopieces, one of them supported on the interaction-free subspace and the secondone on its orthogonal complement. In particular, we apply the previous resultto describe in detail the structure of a slightly modified quantum transportmodel due to Arefeva, Kozyrev and Volovich (modified AKV's model) studied firstin Ref.\cite{gggq}, in terms of generalized annihilation and creationoperators.

Stationary states of weak coupling limit type Markov generators and quantum transport models