Ergodic Behavior of Markov Processes

Generalized couplings and ergodic rates for SPDEs and other Markov models

“…One of the main results of our paper, Theorem 2.3, provides a transparent and fairly simple algorithm for the construction of a distance-like function d with required properties. We strongly believe this to be a substantial complement to the approach initiated in [23] and developed in [6], [12], [25], which hopefully makes this method well applicable for a wide variety of Markov systems.…”

“…Their ergodic properties cannot be analyzed with classical methods from, e.g., [33] and require special treatment. One of the first papers in this direction was [23]; the results obtained there have been later generalized and extended in [6], [12], [25,Chapter 4]. These works provide sufficient conditions for exponential or subexponential convergence of transition probabilities of a Markov process towards its invariant measure in the Wasserstein metric.…”

“…Typically the transition probabilities of these processes with different initial points are mutually singular, and thus ergodicity in total variation clearly fails. For just one example of such a system, we refer to [39], where the intrinsic memory effect was observed for an SFDE: once the value of the segment process is known at some time moment t, the values at all the previous time moments are uniquely and measurably defined; see also the discussions in [23] and [25,Section 4.1]. Clearly, the intrinsic memory effect yields mutual singularity of transition probabilities for the Markov process; in addition, such an effect prevents the mixing property of the system, which is a prerequisite for the classical approach.…”

“…The strategy in a sense mimics the classical one; namely, it shows that if a Markov process visits a d-small set often enough, then, under some additional constraint on a Markov kernel (the so-called non-expansion property), there exists a unique IPM and the transition probabilities weakly converge to the IPM with a rate that is quantified by means of a properly chosen probability metric; again the convergence rate is determined by the recurrence properties of the chain. For further extensions and generalizations within this approach, we refer to [6], [12], [25,Chapter 4].…”

“…In this context, there is a clearly seen difference between the stabilization of transition probabilities (with non-specified rate) on the one hand, and explicit ergodic rates on the other hand. The first one typically guarantees the Law of Large Numbers (or more generally, averaging principle), while the second one gives rise to the Central Limit Theorem and diffusion approximation-type theorems; see [25,Section 5.4, comment 2 and Remark 6.3.9]. For further discussion, references, and systematic treatment of the limit theorems for systems with weakly ergodic Markov perturbations we refer to [25,Chapters 5,6].…”