Recurrence and transience of quantum Markov semigroups constructed form quantum Bernoulli noises

Quantum exclusion semigroup constructed form quantum Bernoulli noises can be written on diagonal and off-diagonal operators space, respectively. The diagonal parts describe a classical Markov process. In this paper, we first produce a decomposition into the sum of irreducible components determined by the class states of the associated classical Markov process and prove that each irreducible component is recurrent or transient if and only if its diagonal restriction is recurrent or transient. We then give conditions for existence of invariant states and study properties of invariant states such as detailed balance, uniqueness and convergence towards as time approaches infinity.