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We study contractions of Markov chains on general metric spaces with respectto some carefully designed distance-like functions, which are comparable to thetotal variation and the standard $L^p$-Wasserstein distances for $p \ge 1$. Wepresent explicit lower bounds of the corresponding contraction rates. Byemploying the refined basic coupling and the coupling by reflection, theresults are applied to Markov chains whose transitions include additivestochastic noises that are not necessarily isotropic. This can be useful in thestudy 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 ChainMonte Carlo, we show that chains driven by the $\alpha$-stable noise can havebetter contraction rates than corresponding chains driven by the Gaussiannoise, due to the heavy tails of the $\alpha$-stable distribution.

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