No abstract

We study a class of dynamical semigroups [Formula: see text] that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system [Formula: see text] driven by a Markov chain [Formula: see text]. We show that the almost sure large time behavior of the system can be extracted from the large [Formula: see text] asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator [Formula: see text]. As a physical application, we consider the case where the [Formula: see text]’s are the reduced dynamical maps describing the repeated interactions of a system [Formula: see text] with thermal probes [Formula: see text]. 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.