Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces

TL;DR

This paper develops a thermodynamic formalism for topological Markov chains on Borel standard spaces, establishing existence and properties of equilibrium states, and applying these results to Markov chains and interacting random paths.

Contribution

It introduces a thermodynamic formalism for bounded continuous potentials on general Borel spaces, including existence, structural properties of equilibrium states, and applications to Markov chains and random paths.

Findings

Existence of equilibrium states for bounded continuous potentials.

Convexity and structural properties of equilibrium states.

Application to invariant measures and stability of Markov chains.

Abstract

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is a general Borel standard space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states do not have a purely additive part. We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite dimensional diffusion.