Invariant Measures and Decay of Correlations of a Class of Ergodic Probabilistic Cellular Automata

Cristian Coletti (1) , Pierre Tisseur (1)

1 CMCC - Centro de Matematica, Computação e Cognição

Using an extended version of the duality concept between two stochastic processes, we give new ergodicity conditions for two states probabilistic cellular automata (PCA) of any dimensions and any radius. Under these assumptions, in the one dimensional case, we study some properties of the unique invariant measure and show that it is shift mixing. Also, the decay of correlation is studied in detail. In this sense, the extended concept of duality gives exponential decay of correlation. When the extended concept of duality can not be applied we are able to get, once again, exponential decay of correlation using well known results from the theory of branching processes.

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Cristian Coletti, Pierre Tisseur. Invariant Measures and Decay of Correlations of a Class of Ergodic Probabilistic Cellular Automata. 2009. ⟨hal-00407940v1⟩