Space-time Bernoullicity of the lower and upper stationary processes for attractive spin systems. (English) Zbl 0727.60119

Steif, Jeffrey E.

Ann. Probab. 19, No. 2, 609-635 (1991).

Summary: We study spin systems, probabilistic cellular automata and interacting particle systems, which are Markov processes with state space \((0,1)^{{\mathbb{Z}}^ n}\). 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 \((0,1)^{{\mathbb{Z}}^ n}\) and index set \({\mathbb{Z}}\) are isomorphic (in the sense of ergodic theory) to an independent process indexed by \({\mathbb{Z}}\). In the translation invariant case, we prove the stronger fact that these stationary processes, viewed as (0,1)-valued processes with index set \({\mathbb{Z}}^ n\times {\mathbb{Z}}\) (space-time), are isomorphic to an independent process also indexed by \({\mathbb{Z}}^ n\times {\mathbb{Z}}\). Such processes are called Bernoulli shifts. Finally, we extend all of these results to continuous time.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G10	Stationary stochastic processes
68Q80	Cellular automata (computational aspects)
28D15	General groups of measure-preserving transformations

Keywords:

spin systems; cellular automata; interacting particle systems; stationary distributions; Bernoulli shifts