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June 2016 Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution
Jérôme Casse
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Adv. in Appl. Probab. 48(2): 369-391 (June 2016).


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.


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Jérôme Casse. "Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution." Adv. in Appl. Probab. 48 (2) 369 - 391, June 2016.


Published: June 2016
First available in Project Euclid: 9 June 2016

zbMATH: 1366.37024
MathSciNet: MR3511766

Primary: 37B15 , 60G10 , 60J05
Secondary: 60K35

Keywords: invariant measure , Markov chain , Probabilistic cellular automata

Rights: Copyright © 2016 Applied Probability Trust


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Vol.48 • No. 2 • June 2016
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