One dimensional Markov random fields, Markov chains and Topological Markov fields

TL;DR

This paper proves that one-dimensional finite-valued stationary Markov Random Fields are equivalent to Markov chains, introducing the new concept of topological Markov fields and extending results to continuous-time cases.

Contribution

The paper establishes that all one-dimensional stationary finite-valued MRFs are Markov chains and introduces the topological Markov field property, expanding understanding in symbolic dynamics.

Findings

Any one-dimensional stationary finite-valued MRF is a Markov chain.

Support of MRFs is non-wandering and a topological Markov field.

Continuous-time finite-valued stationary MRFs are also Markov chains.

Abstract

In this paper we show that any one-dimensional stationary, finite-valued Markov Random Field (MRF) is a Markov chain, without any mixing condition or condition on the support. Our proof makes use of two properties of the support $X$ of a finite-valued stationary MRF: 1) $X$ is non-wandering (this is a property of the support of any finite-valued stationary process) and 2) $X$ is a topological Markov field (TMF). The latter is a new property that sits in between the classes of shifts of finite type and sofic shifts, which are well-known objects of study in symbolic dynamics. Here, we develop the TMF property in one dimension, and we will develop this property in higher dimensions in a future paper. While we are mainly interested in discrete-time finite-valued stationary MRF's, we also consider continuous-time, finite-valued stationary MRF's, and show that these are (continuous-time) Markov chains as well.