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

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).

References in corpus (1)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations

Cited by in corpus (3)

• Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$
• Ergodicity for singular-degenerate porous media equations
• Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$