A quantum dynamical semigroup is the natural extensions of a Markovian process to quantum open systems. This maps guarantees von Neumann´s conditions in the evolution of the density matrix ρ(t): hermiticity, positivity and Tr[ρ]. Here we show the equivalence---in the weak coupling limit---between tracing-out the bath variables and a phenomenological stochastic approach to get the quantum semigroup. The usual approach comes from the total microscopic dynamics (system interacting with a bath). The second formalism assigns a thermal wave function to the open system that evolves with a Schröedinger-Langevin (Sch-L) equation. In this stochastic equation the interaction between the system and the bath is introduced by a dissipative operator and a stochastic term in the Schröedinger equation.
From the Sch-L eq., the generator can be written considering a positive operator erasing the initial conditions of ρ and a fluctuating superoperator coming from the stochastic term in the Sch-L eq. The equivalence between tracing-out and the stochastic approach, allows to characterize the quantum expressions for terms of the generator. We also provide an stochastic non Markovian evolution that represents, in mean value, the dynamic of the open quantum system.
