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We develop a general framework for studying ergodicity of order-preservingMarkov semigroups. We establish natural and in a certain sense optimalconditions for existence and uniqueness of the invariant measure andexponential convergence of transition probabilities of an order-preservingMarkov process. As an application, we show exponential ergodicity andexponentially fast synchronization-by-noise of the stochasticreaction-diffusion equation in the hypoelliptic setting. This refines andcomplements corresponding results of Hairer, Mattingly (2011).

Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting