  We derive sufficient conditions for the memory kernel which guarantee
legitimate (completely positive and trace-preserving) dynamical map. It turns
out that these conditions provide a natural parameterizations of the dynamical
map being a generalization of Markovian semigroup. It is shown that this class
of maps cover almost all known examples -- from Markovian semigroup,
semi-Markov evolution up to collision models and their generalization.

Quantum Physics