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April, 1991 Space-Time Bernoullicity of the Lower and Upper Stationary Processes for Attractive Spin Systems
Jeffrey E. Steif
Ann. Probab. 19(2): 609-635 (April, 1991). DOI: 10.1214/aop/1176990444


In this paper, we study spin systems, probabilistic cellular automata and interacting particle systems, which are Markov processes with state space $\{0, 1\}^{\mathbf{Z}^n}$. Restricting ourselves to attractive systems, we consider the stationary processes obtained when either of two distinguished stationary distributions is used, the smallest and largest stationary distributions with respect to a natural partial order on measures. In discrete time, we show that these stationary processes with state space $\{0, 1\}^{\mathbf{Z}^n}$ and index set $\mathbf{Z}$ are isomorphic (in the sense of ergodic theory) to an independent process indexed by $\mathbf{Z}$. In the translation invariant case, we prove the stronger fact that these stationary processes, viewed as $\{0, 1\}$-valued processes with index set $\mathbf{Z}^n \times \mathbf{Z}$ (space-time), are isomorphic to an independent process also indexed by $\mathbf{Z}^n \times \mathbf{Z}$. Such processes are called Bernoulli shifts. Finally, we extend all of these results to continuous time.


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Jeffrey E. Steif. "Space-Time Bernoullicity of the Lower and Upper Stationary Processes for Attractive Spin Systems." Ann. Probab. 19 (2) 609 - 635, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0727.60119
MathSciNet: MR1106279
Digital Object Identifier: 10.1214/aop/1176990444

Primary: 28D15
Secondary: 60G10 , 60K35

Keywords: $\overline{d}$-metric , attractive spin systems , Bernoullicity , Couplings

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • April, 1991
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