  We consider a time-dependent small quantum system weakly coupled to an
environnement, whose effective dynamics we address by means of a Lindblad
equation. We assume the Hamiltonian part of the Lindbladian is slowly varying
in time and the dissipator part has small amplitude. We study the properties of
the evolved state of the small system as the adiabatic parameter and coupling
constant both go to zero, in various asymptotic regimes. In particular, we
analyse the deviations of the transition probabilities of the small system
between the instantaneous eigenspaces of the Hamiltonian with respect to their
values in the purely Hamiltonian adiabatic setup, as a function of both
parameters.
