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).

中文翻译：

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在亚椭圆环境中通过 SPDE 的比较原理和指数遍历性进行耦合

我们开发了一个通用框架来研究保序马尔可夫半群的遍历性。我们为保序马尔可夫过程的不变测度和转移概率的指数收敛的存在性和唯一性建立了自然的和在某种意义上的最优条件。作为一个应用，我们展示了亚椭圆环境中随机反应扩散方程的指数遍历性和指数快速噪声同步。这完善和补充了 Hairer, Mattingly (2011) 的相应结果。