## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 2 (1991), 609-635.

### Space-Time Bernoullicity of the Lower and Upper Stationary Processes for Attractive Spin Systems

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 2 (1991), 609-635.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990444

**Digital Object Identifier**

doi:10.1214/aop/1176990444

**Mathematical Reviews number (MathSciNet)**

MR1106279

**Zentralblatt MATH identifier**

0727.60119

**JSTOR**

links.jstor.org

**Subjects**

Primary: 28D15: General groups of measure-preserving transformations

Secondary: 60G10: Stationary processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Bernoullicity attractive spin systems couplings $\overline{d}$-metric

#### Citation

Steif, Jeffrey E. Space-Time Bernoullicity of the Lower and Upper Stationary Processes for Attractive Spin Systems. Ann. Probab. 19 (1991), no. 2, 609--635. doi:10.1214/aop/1176990444. https://projecteuclid.org/euclid.aop/1176990444