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

We 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, Mattingly (2011).