exponential ergodicity
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2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joanna Kubieniec

Abstract In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a state-dependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.


2020 ◽  
Vol 379 (3) ◽  
pp. 1001-1034
Author(s):  
Oleg Butkovsky ◽  
Michael Scheutzow

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


2020 ◽  
Vol 20 (06) ◽  
pp. 2040006 ◽  
Author(s):  
Benjamin Gess ◽  
Pavlos Tsatsoulis

We prove uniform synchronization by noise with rates for the stochastic quantization equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [O. Butkovsky and M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, preprint (2019), arXiv:1907.03725]. In particular, it is shown that this framework can be applied to the case of state spaces given in terms of Hölder spaces of negative exponent.


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