Existence conditions for invariant probability measures with applications to an SIS model

We investigate the existence of non-trivial invariant probability measures for a probabilistic cellular automaton modeling a susceptible-infected-susceptible process on Zd. To this end, we introduce a topological framework that ensures the existence of invariant measures for a general class of continuous operators. Applying this approach to the model, we establish the existence of at least two distinct, linearly independent translation-invariant measures in the non-ergodic regime. Moreover, we prove that any non-trivial invariant measure in this setting cannot be a Bernoulli product measure.