Markov Chains and Dynamical Systems: The Open System Point of View

TL;DR

This paper explores the deep connections between Markov chains and dynamical systems within the context of open systems, revealing how stochastic processes emerge from deterministic dynamics through information loss.

Contribution

It establishes a unified framework linking Markov chains and dynamical systems, highlighting the role of information loss and reversibility in the transition from deterministic to stochastic models.

Findings

Markov chains can be derived from dynamical systems via information loss.

Loss of algebra morphism property characterizes the transition to stochasticity.

Solutions of stochastic differential equations are deterministic dynamical systems on product spaces.

Abstract

This article presents several results establishing connections be- tween 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 character- ized 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 solu- tions of stochastic dierential 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.