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A class of non-commutative stochastic processes is defined. These processes are defined up to equivalence by their multi-time correlation kernels. A reconstruction theorem, generalizing the Kolmogorov theorem for classical processes, is proved. Markov processes and their associated semigroups are studied, and some non-quasi free examples are constructed on the Clifford algebra, with the use of a perturbation theory of Markov processes. The connection with the Hepp-Lieb models is discussed. § 0. IntroductionWe study a class of non-commutative stochastic processes which are determined up to equivalence by their multi-time correlations. They are analogues of classical processes in the sense of Doob [1], Meyer [2]; indeed, those processes are included as a special case.We define a stochastic process over a C*-algebra ^7, indexed by a set T, to consist of a C*-algebra j/, a family {j t : teT] of *-homomorphisms from 2$ into $0 and a state a> on 3$. This structure gives rise to a non-commutative stochastic process in the sense of Accardi [3], with local algebras defined by jtfj= v (j t (b): tel, b e &} for any subset / of T; observables which are "localized at different times" are not assumed to commute. We show (Proposition 1.1) that such a process is determined up to equivalence by its family of correlation kernels o)

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We study the class of stationary states and the domain of attraction of each of them, for a dynamical semigroup possessing a faithful normal stationary state. We give applications to the approach to stationarity of an open quantum system, and to models of the quantum measurement process.