Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution. (10th June 2016)

• Record Type: Journal Article
• Authors: Casse, Jérôme
• Is Part Of: Advances in applied probability. Volume 48:Number 2(2016)
• Page Start: 369
• Page End: 391
• Publication Date: 2016-06-10
• Subjects: Probabilistic cellular automata, -- invariant measure, -- Markov chain; Primary 37B15, -- 60G10, -- 60J05, -- Secondary 60K35; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2016.5
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: This paper is devoted to probabilistic cellular automata (PCAs) onN, Z orZ / n Z, 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.