Time evolution, correlations, and linear response of non-Markov processes

We investigate the time evolution of stochastic non-Markov processes as they occur in the coarse-grained description of open and closed systems. We show that semigroups of propagators exist for all multivariate probability distributions, the generators of which yield a set of time-convolutionless master equations. We discuss the calculation of averages and time-correlation functions. Further, linear response theory is developed for such a system. We find that the response function cannot be expressed as an ordinary time-correlation function. Some aspects of the theory are illustrated for the two-state process and the Gauss process.