Couplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting Article Swipe

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Ergodicity Hypoelliptic operator Mathematics Markov chain Uniqueness Applied mathematics Exponential function Markov process Invariant (physics) Exponential growth Pure mathematics Order (exchange) Mathematical analysis Partial differential equation Statistics Mathematical physics Economics Linear differential equation Finance

Oleg Butkovsky , Michael Scheutzow ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.1007/s00220-020-03834-w · OA: W2954992998

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 and Mattingly (Electron J Probab 16:658–738, 2011).