Excessive maps, “arrival times” and perturbation of dynamical semigroups

The notion of excessive map for dynamical semigroup is introduced, and it is shown that an excessive map defines an operation-valued measure describing the measurement of an “arrival time” related to the irreversible dynamics described by the semigroup. Any such arrival time determines a positive perturbation of the dynamical semigroup describing the dynamics after “arrivals”. Generators of the relevant perturbations are characterized, and several examples, both commutative and a non-commutative, are discussed, elucidating the problem of standard representation.