2014/10/12 by Jérôme Casse, Casse, Jérôme
Computer Science · Mathematics · #37B15 #60G10 #60J05 #60K35 #Cellular Automata and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:37B15 #msc:60G10 #msc:60J05 #msc:60K35
paper · pdf · doi:10.48550/arxiv.1410.3159
arxiv created 2014/10/12 · openalex publication_date 2014/10/12 · arxiv updated 2014/10/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper is devoted to probabilistic cellular automata (PCA) on ℕ, ℤ or ℤ/nℤ, depending of 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 invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCA) when the alphabet E is finite. Here we obtain conditions on the transition kernel of 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 to pass from a finite alphabet to a general alphabet comes from some problems of measurability, and a large part of this work is devoted to clarify these issues.