  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.
