A Thermodynamic Formalism for Continuous Time Markov Chains with Values on the Bernoulli Space: Entropy, Pressure and Large Deviations

• Published in: Journal of statistical physics Vol. 152; no. 5; pp. 894 - 933
• Main Authors: Lopes, Artur, Neumann, Adriana, Thieullen, Philippe
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-09-2013; Springer; Springer Verlag
• Subjects: Algorithms; Analysis; Dynamical Systems; Markov processes; Mathematical and Computational Physics; Mathematics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Thermodynamics; Perron Theorem; Equilibrium state; Continuous time Markov chain; Gibbs state; Entropy; Large deviations; Pressure; Deviation function; Ruelle Operator
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-013-0796-7

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice
(also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator
, where
is a discrete time Ruelle operator (transfer operator), and
is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the
a priori
probability.
Given a Lipschitz interaction
, 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 (see also Lopes et al. in Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature. Arxiv, 2012), 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 (Tamsui Oxf. J. Manag. Sci. 321(2):505–524,
1990
). 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.