  In this paper we develop a general framework for constructing and analysing
coupled Markov chain Monte Carlo samplers, allowing for both (possibly
degenerate) diffusion and piecewise deterministic Markov processes. For many
performance criteria of interest, including the asymptotic variance, the task
of finding efficient couplings can be phrased in terms of problems related to
optimal transport theory. We investigate general structural properties, proving
a singularity theorem that has both geometric and probabilistic
interpretations. Moreover, we show that those problems can often be solved
approximately and support our findings with numerical experiments. For the
particular objective of estimating the variance of a Bayesian posterior, our
analysis suggests using novel techniques in the spirit of antithetic variates.
Addressing the convergence to equilibrium of coupled processes we furthermore
derive a modified Poincar\'e inequality.
