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

## TL;DR

This paper introduces a general framework for analyzing ergodicity of order-preserving Markov semigroups, providing optimal conditions for invariant measures and demonstrating exponential ergodicity and synchronization in hypoelliptic stochastic reaction-diffusion equations.

## Contribution

It develops a broad theoretical framework for ergodicity of order-preserving Markov processes and applies it to hypoelliptic SPDEs, refining previous results.

## Key findings

- Established optimal conditions for invariant measure existence and uniqueness.
- Proved exponential convergence of transition probabilities.
- Demonstrated exponential synchronization-by-noise in hypoelliptic SPDEs.

## Abstract

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).