Thermodynamic Formalism Out of Equilibrium Part I: Pressure Out of Equilibrium, and Azuma Inequality and Gibbs Processes

We study the thermodynamic formalism of topological Markov shifts where the potential varies by a random walk on an exterior system. We define the {\em pressure out of equilibrium} which is associated to such a family of potentials, and prove a corresponding variational principle. We present an appl