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

作者：Lopes Artur^*; Neumann Adriana; Thieullen Philippe

来源：Journal of Statistical Physics, 2013, 152(5): 894-933.

DOI：10.1007/s10955-013-0796-7

• Continuous time Markov chain
• Perron Theorem
• Gibbs state
• Ruelle Operator
• Equilibrium state
• Entropy
• Pressure
• Large deviations
• Deviation function

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.

• 出版日期2013-9