Probabilistic cellular automata with general alphabets letting a Markov chain invariant

TL;DR

This paper investigates probabilistic cellular automata with general alphabets, establishing conditions for the existence of invariant Markov chains and linking this to solutions of a cubic integral equation, while addressing measurability challenges.

Contribution

It extends previous finite-alphabet results to general alphabets, providing new conditions based on transition kernels and solving a key integral equation.

Findings

Invariant Markov chains correspond to solutions of a cubic integral equation.

Conditions for invariance are characterized for general alphabet PCA.

Measurability issues are addressed to extend the theory beyond finite alphabets.

Abstract

This paper is devoted to probabilistic cellular automata (PCA) on $\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$, 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.