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2019 Quantitative contraction rates for Markov chains on general state spaces
Andreas Eberle, Mateusz B. Majka
Electron. J. Probab. 24: 1-36 (2019). DOI: 10.1214/19-EJP287

Abstract

We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb R^d$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.

Citation

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Andreas Eberle. Mateusz B. Majka. "Quantitative contraction rates for Markov chains on general state spaces." Electron. J. Probab. 24 1 - 36, 2019. https://doi.org/10.1214/19-EJP287

Information

Received: 21 August 2018; Accepted: 2 March 2019; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 07055664
MathSciNet: MR3933205
Digital Object Identifier: 10.1214/19-EJP287

Subjects:
Primary: 60J05, 60J22, 65C05, 65C30, 65C40

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