


[Contents]  [Index] 
Cooperation with: P. Dai Pra (Università degli Studi di Padova, Italy), P.Y. Louis (Berliner Graduiertenkolleg ``Stochastische Prozesse und Probabilistische Analysis'' (Graduate College ``Stochastic Processes and Probabilistic Analysis''), Université Lille 1, France)
Description:
Description: Stochastic processes called Probabilistic Cellular Automata, and denoted by PCA, are discretetime Markov chains with parallel updating (see, e.g., [1], [3], [6]). PCA models are useful in a large number of scientific areas, for example for parallel computing.
In [2], the authors present some links between the sets of reversible and stationary and Gibbs measures, respectively for general PCAs. They illustrate these results on the following particular class of reversible PCAs with transition probability P:
where and the interaction is of finite range.
In [4], [5], the author studies ergodic properties of PCAs
of the class . In particular, he shows that in the
attractive case (i.e. when ), if the set of Gibbs
measures associated to the potential
is reduced to one element which is moreover
weak mixing, then the PCA is ergodic, its equilibrium measure is
, and the convergence holds exponentially fast.
The proof is based on a specific monotone coupling of
PCAs. The domain of validity for this exponential ergodicity
of attractive PCAs seems to be optimal, since it is known that, in the
particular
case if k=1 and otherwise,
for all temperature larger than the critical one
the unique Gibbs measure is weak mixing.
References:



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