We study a class of dynamical semigroups that emerge, by a Feynman-Kac type formalism, from a random quantum dynamical system driven by a Markov chain. We show that the almost sure large time behavior of the system can be extracted from the large time asymptotics of the semigroup, which is in turn directly related to the spectral properties of its generator L. As a physical application, we consider the case where the quantum dynamical system describes the repeated interactions of a system S with thermal probes. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.