Strict Kantorovich contractions for Markov chains and Euler schemes with general noise

TL;DR

This paper investigates contraction properties of Markov chains with general noise, providing explicit bounds and showing that heavy-tailed noises like alpha-stable distributions can lead to better convergence rates than Gaussian noise.

Contribution

The paper introduces explicit contraction bounds for Markov chains with general noise, including non-isotropic and heavy-tailed distributions, and applies these to Euler schemes for SDEs.

Findings

Heavy-tailed alpha-stable noise can improve contraction rates.

Explicit lower bounds for contraction rates are derived.

Refined coupling methods are employed for analysis.

Abstract

We study contractions of Markov chains on general metric spaces with respect to some carefully designed distance-like functions, which are comparable to the total variation and the standard $L^p$-Wasserstein distances for $p \ge 1$. We present explicit lower bounds of the corresponding contraction rates. By employing the refined basic coupling and the coupling by reflection, the results are applied to Markov chains whose transitions include additive stochastic noises that are not necessarily isotropic. This can be useful in the study of Euler schemes for SDEs driven by L\'evy noises. In particular, motivated by recent works on the use of heavy tailed processes in Markov Chain Monte Carlo, we show that chains driven by the $\alpha$-stable noise can have better contraction rates than corresponding chains driven by the Gaussian noise, due to the heavy tails of the $\alpha$-stable distribution.