“Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice \(\{1,\dots,d\}^N\) (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator \(L=\mathfrak{L}_A-I\), where \(\mathfrak{L}_A\) is a discrete time Ruelle operator (transfer operator), and \(A:\{1,\dots,d\}^N\to\mathbb{R}\) is a given fixed Lipschitz function. The associated continuous time stationary Markov chain defines the a priori probability.
Given a Lipschitz interaction \(V:\{1,\dots,d\}^N\to\mathbb{R}\), we are interested in the Gibbs (equilibrium) state for such \(V\). This is another continuous time stationary Markov chain. In order to analyze this problem we use a continuous time Ruelle operator (transfer operator) naturally associated to \(V\). Among other things we 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 (see also [A. O. Lopes et al., “Entropy and variational principle for one-dimensional lattice systems with a general a-priori probability: positive and zero temperature”, arXiv:1210.3391]), 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 [Trans. Am. Math. Soc. 321, No. 2, 505–524 (1990; Zbl 0714.60019)]. In the last appendix of the paper we explain why the techniques we develop here have the capability to be applied to the analysis of convergence of a certain version of the Metropolis algorithm.”