Description of a class of Markov processes “equivalent” toK-shifts

Abstract

It results from recent works of Prigogine and collaborators that one can construct a nonunitary operator which realizes an “equivalence” between the positive actions of a reversible dynamical system and an irreversible Markov process going to equilibrium. We consider here this construction and we prove that (a) forK-shifts the transition probability of the associated Markov process is concentrated in the stable manifold of the transformed point by the shift with a point mass concentrated on the deterministic trajectory; and (b) for Bernoulli shifts the measures which go to equilibrium are the same for the deterministic system and the Markov process.