Dynamical Gibbs Variational Principles for Irreversible Interacting Particle Systems with Applications to Attractor Properties

Interacting particle systems are countable systems of locally interacting Markov processes and are often used as toy models for stochastic phenomena with an underlying spatial structure. Even though the definition of an interacting particle system often looks very simple and the major technical issues of existence and uniqueness have long been settled, it is in general surprisingly difficult to say anything non-trivial about their behavior. Some of the main challenges deal with the long-time behavior of the systems. Because of the possible non-uniqueness of the time-stationary distribution the analysis is very delicate and various techniques have been developed to study limit theorems or attractor properties. One particular technique that will play a major role in this talk is due to Richard Holley and involves using the relative entropy functional with respect to some specification $\gamma$ as a Lyapunov function for the measure-valued differential equation that describes the time evolution of the system in the space of measures. We first explain the general idea behind the method on the simple example of a continuous time Markov chain on a finite state space and then discuss how one can extend it to irreversible interacting particle systems in infinite volumes.