Intrinsic randomness of unstable dynamics and Sz.-Nagy-Foias dilation theory

Abstract

Misra, Prigogine and Courbage (MPC) demonstrated the possibility of obtaining stochastic Markov processes from deterministic dynamics simply through a "change of representation" which involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. From a mathematical point of view, MPC theory is a theory of positivity preserving quasi-affine transformations that intertwine the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. In this work, dropping the positivity condition, a characterization of the contraction semigroups induced by quasi-affine transformations, the structure of the unitary groups admitting such intertwining relations and a prototype for the quasi-affinities are given on the basis of the Sz.-Nagy-Foia\c{s} dilation theory. The results are applied to MPC theory in the context of statistical mechanics.