FROM PROBABILISTIC DESCRIPTIONS TO DETERMINISTIC DYNAMICS

We answer the question of dilating a Markov semigroup to a deterministic evolution in a larger space. We show that certain Markov processes may arise as projections of Kolmogorov dynamical systems. This result shows in particular the physical significance of the Misra-Prigogine-Courbage theory of irreversibility. Because we want positivity preserving transformations in both case, our procedure, although analogous to the Sz-Nagy-Foias dilation theory, has a different viewpoint, that of the natural extension of exact systems.