An introduction to Coupling

In this review paper, we describe the use of couplings in several different mathematical problems. We consider the total variation norm, maximal coupling, and the $\bar{d}$-distance. We present a detailed proof of a result recently proved: the dual of the Ruelle operator is a contraction with respect to $1$-Wasserstein distance. We also show exponential convergence to equilibrium in the state space for finite-state Markov chains when the transition matrix $\mathcal{P}$ has all entries positive.} In this new version, we describe in more detail the line of reasoning followed in the work previously published as a chapter in ``Modeling, Dynamics, Optimization and Bioeconomics II'', Springer Verlag (2017).