An introduction to Coupling

TL;DR

This paper reviews the use of couplings in various mathematical problems, providing detailed proofs and convergence results, especially for finite-state Markov chains, and discusses recent advances in the field.

Contribution

It offers a comprehensive overview of coupling techniques, including new proofs and convergence results, enhancing understanding of their applications in probability and dynamical systems.

Findings

Dual of Ruelle operator is a contraction in 1-Wasserstein distance

Exponential convergence to equilibrium for finite-state Markov chains

Detailed exposition of coupling methods in mathematical problems

Abstract

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).