Convergence towards equilibrium of Probabilistic Cellular Automata

We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, and under some assumption ( 풜) we establish ergodicity for general attractive probabilistic cellular automata on Sℤd, where S is finite: this means the convergence towards equilibrium of these Markovian parallel dynamics, in the uniform norm, exponentially fast. For a class of reversible PCA dynamics on {-1,+1}ℤd, with a naturally associated Gibbsian potential 휑, we prove that a Weak Mixing condition for 휑 implies the validity of the assumption (풜), thus the 'exponential ergodicity' of the dynamics towards the unique Gibbs measure associated to 휑 holds. On some particular examples of this PCA class, we verify that our assumption (풜) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some precise PCA, the 'exponential ergodicity' holds as soon as there is no phase transition