Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

Abstract

This paper is devoted to probabilistic cellular automata (PCAs) on N,Z or Z / nZ, depending on two neighbors with a general alphabet E (finite or infinite, discrete or not). We study the following question: under which conditions does a PCA possess a Markov chain as an invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCAs) when the alphabet E is finite. Here we obtain conditions on the transition kernel of a PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation. One of the difficulties in passing from a finite alphabet to a general alphabet comes from the problem of measurability, and a large part of this work is devoted to clarifying these issues.