Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations

By Martin Hairer, Jonathan C. Mattingly, M. Scheutzow. 2009

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There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris’ theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such “asymptotic couplings” were central to (Mattingly and Sinai in Comm Math Phys 219(3):523–565, 2001; Mattingly in Comm Math Phys 230(3):461–462, 2002; Hairer in Prob Theory Relat Field 124:345–380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553–582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris’ celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are “small” (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a “small set” by the much weaker notion of a “d-small set,” which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris’ theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4424ca5a1a3ea536ecb6a49ed9d1e2f920b8cfc4

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Citations in the corpus

Quantitative contraction rates for Markov chains on general state spaces (2018, background): “in [23, 24, 3, 19] to study properties of Markov chains and diffusion processes.”
A simple framework to justify linear response theory (2009, background): “out control on W. Our aim is therefore to present conditions under which a spectral gap in a space of thetypeC1 V,W canbeveriﬁeddirectly,withoutrequiringaspectralgapresultina space of the type CV. In [HMS09], a general framework was laid out that gives conditions underwhicha“spectralgap”isobtainedin”
Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, methodology): “From a methodological perspective, we have demonstrated a particular application of the theory developed in [22], demonstrating its versatility for the analysis of rates of convergence in Markov chains.”
Spectral gap for the stochastic quantization equation on the 2-dimensional torus (2016, methodology): “Their method is based on the asymptotic coupling technique from [HMS11] and relies on the bounds from [MWe15].”
Exponential ergodicity for Markov processes with random switching (2013, methodology): “. The problem here is that bounded sets may not be small sets. We will therefore replace the notion of small set by the following notion of “closedness” between transition probabilities introduced in [24], which takes into account the topology of the underlying space X. Deﬁnition 3.5 (d-small set). ”
Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “The approach taken in [16], which is partially based on ideas from [23, 24, 9], is to build a multiplicative semimetric ρ out of a concave function of the underlying distance and a Lyapunov function that ensures contractivity at large distances, see Section 2.”
Subgeometric rates of convergence of Markov processes in the Wasserstein metric (2012, background): “To study such Markov processes Hairer, Mattingly and Scheutzow [11] introduce the following concept.”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, methodology): “This will allow us to derive quantitative bounds for asymptotic couplings in the sense of [25,26], i.”