We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian H, acting on h, interacting with a Gaussian, gauge invariant, reservoir. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible, we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
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