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- ItemCouplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting(Berlin ; Heidelberg : Springer, 2020) Butkovsky, Oleg; Scheutzow, MichaelWe 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).