From irreversible Markov semigroups to chaotic dynamics

We answer qualitatively the inverse coarse-graining problem of statistical physics, namely, which microscopic dynamics give rise to a given physically observed Markov semigroup as a result of exact coarse graining? We prove the fact that all Markov chains arise as projections of dynamical systems in larger spaces and show that in particular the irreversible Markov chains arise as projections of chaotic systems of Kolmogorov type. This result generalizes our previous results on the Misra-Prigogine-Courbage semigroups. Because we want positivity-preserving transformations, our procedure although analogous to the Sz-Nagy-Foias dilation theory has a different viewpoint, that of positive dilations.