  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.
