A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations

TL;DR

This paper develops a thermodynamic formalism for continuous time Markov chains on Bernoulli space, introducing entropy, pressure, and large deviations analysis, extending classical discrete time results to continuous time settings.

Contribution

It introduces a continuous time Ruelle operator, proves a Perron-Frobenius theorem, and establishes a variational principle for pressure in this context.

Findings

Established a continuous time Perron-Frobenius theorem for Lipschitz potentials.

Defined a negative entropy and formulated a variational principle for pressure.

Analyzed large deviations for empirical measures in continuous time Markov chains.

Abstract

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice 1,...,d}^N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator $L={\mc L}_A -I$, where \mc L_A is a discrete time Ruelle operator (transfer operator), and A:{1,...,d}^N \to R is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the\emph{a priori}probability. Given a Lipschitz interaction V:\{1,...,d\}^{\bb N}\to \mathbb{R}, we are interested in Gibbs (equilibrium) state for such $V$. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative, and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer.