Gaussian quantum Markov semigroup arising from weak coupling limits

We characterize Gaussian QMS arising in the weak coupling limit (WCL) of a system interacting with some reservoirs. It turns out that, under some natural assumptions on the system Hamiltonian, there exists a GKLS representation of the generator with Kraus operators that are either annihilation or creation operators and Hamiltonian H which is diagonalizable. We also discuss some properties of invariant states.