Stochastic comparisons for general probabilistic cellular automata

In this work we study stochastic comparisons for probabilistic cellular automata where the set of values each automaton can take is an arbitrary finite set endowed with a partial or total order. We give necessary and sufficient conditions on the parameters which define the processes for their stochastic ordering. The equivalence is proved establishing the existence of increasing Markovian couplings of these processes. When such couplings exist, we give the explicit expression of their operators. As a consequence of these results, we get necessary and sufficient conditions on the transition probabilities of probabilistic cellular automata for their monotonicity.