On the inverse problem of statistical physics: from irreversible semigroups to chaotic dynamics

We show that all measure preserving stationary Markov processes arise as projections of Kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos. This result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the Misra–Prigogine–Courbage theory of irreversibility in the more general context of the inverse problem of statistical physics. 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.