CC BY 4.0 UnportedButkovsky, OlegScheutzow, Michael2021-10-212021-10-212020https://oa.tib.eu/renate/handle/123456789/7085https://doi.org/10.34657/6132We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction–diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658–738, 2011). © 2020, The Author(s).enghttps://creativecommons.org/licenses/by/4.0/530510Comparison PrincipleExponential ErgodicitySPDEsHypoelliptic SettingMarkov processArticle