  In the case of a quantum-classical hybrid system with a finite number of
degrees of freedom, the problem of characterizing the most general dynamical
semigroup is solved, under the restriction of being quasi-free. This is a
generalization of a Gaussian dynamics, and it is defined by the property of
sending (hybrid) Weyl operators into Weyl operators in the Heisenberg
description. The result is a quantum generalization of the L\'evy-Khintchine
formula; Gaussian and jump contributions are included. As a byproduct, the most
general quasi-free quantum-dynamical semigroup is obtained; on the classical
side the Liouville equation and the Kolmogorov-Fokker-Planck equation are
included. As a classical subsystem can be observed, in principle, without
perturbing it, information can be extracted from the quantum system, even in
continuous time; indeed, the whole construction is related to the theory of
quantum measurements in continuous time. While the dynamics is formulated to
give the hybrid state at a generic time $t$, we show how to extract multi-time
probabilities and how to connect them to the quantum notions of positive
operator valued measure and instrument. The structure of the generator of the
dynamical semigroup is analized, in order to understand how to go on to non
quasi-free cases and to understand the possible classical-quantum interactions;
in particular, all the interaction terms which allow to extract information
from the quantum system necessarily vanish if no dissipation is present in the
dynamics of the quantum component. A concrete example is given, showing how a
classical component can input noise into a quantum one and how the classical
system can extract information on the behaviour of the quantum one.
