Markov Chains and Dynamical Systems: the Open System Point of View *

This article presents several results establishing connections between Markov chains and dynamical systems, from the point of view of open systems in physics. We show how all Markov chains can be understood as the information on one component that we get from a dynamical system on a product system, when losing information on the other component. We show that passing from the deterministic dynamics to the random one is characterized by the loss of algebra morphism property; it is also characterized by the loss of reversibility. In the continuous time framework, we show that the solutions of stochastic differential equations are actually deterministic dynamical systems on a particular product space. When losing the information on one component, we recover the usual associated Markov semigroup.