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In this paper we develop a general framework for constructing and analysingcoupled Markov chain Monte Carlo samplers, allowing for both (possiblydegenerate) diffusion and piecewise deterministic Markov processes. For manyperformance criteria of interest, including the asymptotic variance, the taskof finding efficient couplings can be phrased in terms of problems related tooptimal transport theory. We investigate general structural properties, provinga singularity theorem that has both geometric and probabilisticinterpretations. Moreover, we show that those problems can often be solvedapproximately and support our findings with numerical experiments. For theparticular objective of estimating the variance of a Bayesian posterior, ouranalysis suggests using novel techniques in the spirit of antithetic variates.Addressing the convergence to equilibrium of coupled processes we furthermorederive a modified Poincar\'e inequality.

Constructing sampling schemes via coupling: Markov semigroups and optimal transport