Yet Another Look at Harris’ Ergodic Theorem for Markov Chains

By Martin Hairer, Jonathan C. Mattingly. 2008

418 citations recorded by the source corpus.

The aim of this note is to present an elementary proof of a variation of Harris’ ergodic theorem of Markov chains.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/69224c3b6393e0fb0a4c9d64085b69c806d2e369

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Accelerated Linear Convergence of Stochastic Momentum Methods in Wasserstein Distances (2019, background): “It is noted in [HM11] that dψ has the following alternative expression.”
Subgeometric rates of convergence of Markov processes in the Wasserstein metric (2012, background): “Let us also mention the recent paper of Hairer and Mattingly [10], which contains a new simple proof of the exponential ergodicity of a Markov process under LDC and the local Dobrushin condition.”
Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “Inspired by [23], a similar semi-metric has been considered in [16].”
A non‐conservative Harris ergodic theorem (2019, methodology): “ments 42 References 43 1. Introduction We are interested in the asymptotic behavior of positive semigroups acting on a weighted space of measures. For conservative semigroups, Harris’ ergodic theorem [42, 54, 40] provides a Date: March 12, 2019. 2010 Mathematics Subject Classiﬁcation. Primary 47A35;”
An Introduction to Stochastic PDEs (2009, background): “onstant Cdepending on xand y. Furthermore, they require more background in advanced probability theory than what is assumed for the scope of these notes. The elementary proof given here is taken from [HM08b] and is based on the arguments ﬁrst exposed in [HM08a]. It has the disadvantage of being less”
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.”
Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder” (2014, methodology): “Here, we rely on the following version given (an proved) in [18].”
Geometric ergodicity of the Bouncy Particle Sampler (2018, neutral): “Denoting, for t (cid:62) t 0 and k ∈ { 0 , . . . , n ( t ) } , ν k = δ ( x,y ) P 0 ,t − n ( t ) s 1 From (66), for any t (cid:62) t 0 , k ∈ { 1 , . . . , n ( t ) } where we deﬁned Lemma 22.”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, background): “A breakthrough toward the applicability to high- and infinite-dimensional models has been made by Hairer and Mattingly in [29], and in the subsequent publications [25, 30].”
A simple framework to justify linear response theory (2009, methodology): “We therefore only give a sketch of the proof and refer to [Nor86, HM09a , Nua06] for details.”
Posterior Computation with the Gibbs Zig-Zag Sampler (2020, methodology): “peciﬁed in Lemma 7. By Lemma 8 and a simple Gronwall inequality,¨ it follows that Psatisﬁes a Lyapunov inequality of the form 8t 0;etL GZZV rV+ h1 C with suitable r2(0;1) and h2R. By Theorem 3.4 of Hairer and Mattingly (2011), it follows that the embedded Markov chain associated with Pis geometrical”
Propagating Lyapunov functions to prove noise-induced stabilization (2011, background): “3 in [18], there exist constants ᾱ ∈ (0, 1) and β > 0 such that ρβ(μ1PT , μ2PT ) ≤ ᾱρβ(μ1, μ2).”