An introduction to Coupling (2511.14489v1)

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).

• The paper presents explicit constructions of maximal couplings to quantify total variation distance and optimize convergence in probabilistic systems.
• It establishes exponential convergence to equilibrium in Markov chains by using a 'couple-and-stick' method with precise quantitative bounds.
• The analysis extends to dynamical systems via the Ruelle operator, using Wasserstein metrics to provide actionable mixing and contraction estimates.

"An Introduction to Coupling" – Technical Analysis and Summary

Overview and Scope

The paper "An introduction to Coupling" (2511.14489) offers a comprehensive review of coupling techniques in probability and ergodic theory, with explicit focus on their application to Markov processes, convergence to equilibrium, and dynamical systems via operators such as the Ruelle transfer operator. The discussion navigates through several core notions—total variation distance, maximal coupling, $\bar{d}$-distance, Wasserstein metrics—demonstrating their interplay with convergence rates and contraction properties. Throughout, the exposition is grounded in measure-theoretic formalism and leverages explicit constructions of couplings for rigorous quantitative bounds.

Fundamentals of Coupling: Definitions and Distances

The paper systematically introduces coupling as a probabilistic tool for relating two measures $\mu$ and $\nu$ on a common measurable space $\Omega$ through a "plan" $\Gamma$ on $\Omega \times \Omega$. Such $\Gamma$ is required to have $\mu$ and $\nu$ as marginals. The central objects of paper include:

• Total Variation Distance: Defined as $|\mu - \nu|_{tv}$, with duality formulation via bounded measurable functions, and operationalized through maximal couplings. The explicit measure-theoretic construction of the maximal coupling plan—which achieves equality in $|\mu_1-\mu_2|_{tv} = 2\pi\{(x,y): x \neq y\}$—is presented using Radon–Nikodym derivatives and infima.
• Wasserstein Distance: The metric $W_1(\mu, \nu)$ is formulated via cost minimization over couplings, and the Kantorovich duality provides a functional characterization relevant for establishing contraction properties.
• $\bar{d}$-Distance: For shift-invariant measures on symbolic spaces, the $\bar{d}$ metric is constructed via joinings ($\sigma$-invariant couplings), and its direct computation in the Bernoulli case yields the identity $\bar{d}(\mu,\nu)=|p_1-q_1|$ under $p_1 \leq q_1$. This establishes a bridge between dynamical and probabilistic structures.

Maximal Coupling: Constructions and Properties

Strong attention is paid to maximal coupling constructions. Through explicit measure-theoretic calculations, the plan achieving the total variation distance is shown to be an amalgam of the largest possible agreement (the diagonal part) and an independent coupling elsewhere. Quantitative relationships between the total variation and coupling probability of mismatch follow tightly from these constructions.

Key technical statements include:

• For any coupling $\Gamma$, $|\mu_1-\mu_2|_{tv} \leq 2\Gamma\{(x,y):x \neq y\}$, with equality for the maximal coupling.
• The shift map (dynamical operator) is contractive in total variation, i.e., pushing forward $\mu_1$ and $\mu_2$ by the shift cannot increase their total variation distance.

Coupling Times and Convergence Rates for Markov Processes

The probabilistic estimation of convergence rates via coupling times is a central theme. For Markov chains with strictly positive transition matrices:

• The main technical result asserts exponential convergence to equilibrium measured in total variation: for $\lambda$ the stationary distribution and $\nu$ any initial distribution,

$$|\lambda - P_\nu(Y_n \in \cdot)|_{tv} \leq 2(1-\rho)^n,$$

where $\rho$ is the minimal transition probability.

• The explicit construction of a coupling (meeting time) and its probabilistic analysis is carefully detailed, relying on the fact that, once coupled, paths remain together. The rationale is extended from product spaces to more general (non-independent) couplings.

The proof structure avails itself of stopping times and Markovian properties. The explicit "couple-and-stick" process—where two copies evolve independently up to their first coincidence and then coalesce—serves as the backbone for deriving exponential decay rates.

Coupling in the Dynamical Systems Context: Ruelle Operator

The Ruelle (transfer) operator and its dual are treated within the metric-measure framework, utilizing Wasserstein-1 distance for probability measures on symbolic spaces endowed with a cylinder-metric. The nontrivial assertion is that the dual Ruelle operator is a contraction in an appropriately chosen equivalent metric. The arguments employ bounded distortion estimates for the normalized potential and leverage the Lasota-Yorke inequality for Lipschitz regularity.

Fundamental steps include:

• Explicit control on the distortion of preimages under the dynamics facilitates comparison of pushed-forward Dirac measures.
• Construction of plans with high mass on small neighborhoods, giving lower bounds on the probability of coincidence in the coupling, and thus allowing to bound Wasserstein distances.
• The contraction constant $\alpha$ is given by

$$\alpha = \max\left\{1-\frac{a}{2}, \frac{1 + \alpha_1}{2}\right\},$$

where $a$ is the uniform lower bound on the coupling probability of "closeness" after $t$ steps, and $\alpha_1$ arises from the Lasota-Yorke inequality.

This establishes not merely uniqueness of equilibrium ($g$-measures) for normalized Lipschitz potentials but quantitative rates of convergence for iterated Ruelle conjugate actions on measures.

Implications and Future Directions

The methodology detailed in the paper has broad applicability for quantitative mixing estimates in Markov and dynamical systems, especially for symbolic systems, expanding maps, and processes admitting good distortion control on cylinders. The coupling-based proofs yield not only existence of equilibrium and mixing times but precise rates in normed spaces of measures.

Implications include:

• Practically, the explicit convergence rates can underwrite sharp mixing time estimates for Markov chains in high dimensions and justify approximate sampling strategies.
• Theoretically, the adaptation of coupling constructions to contractive metrics like Wasserstein facilitates cross-fertilization between information theory, optimal transport, and thermodynamic formalism.
• The analysis of the $\bar{d}$-distance in the context of joinings provides a dynamical metric of pertinence in classification and stability of stochastic processes and their invariant measures.

Natural extensions and open challenges include:

• Generalization to state spaces beyond finite alphabets, or to stochastic matrices lacking uniform positivity (i.e., non-primitive cases).
• Coupling-based estimates for systems with singularities or non-uniform structure, where decay of correlations may be subexponential.
• Further exploration of the interplay between different metrics (e.g., relative entropy, Wasserstein-$p$, and $\bar{d}$) and their contractivity under transfer operators in higher regularity space.

Conclusion

The paper furnishes detailed, mathematically rigorous constructions and estimates for coupling, emphasizing its centrality in both classical probability and ergodic-theoretic contexts. By formulating and solving concrete coupling problems for Markov chains, symbolic dynamical systems, and transfer operators, it elucidates exponential mixing properties and establishes contraction of the Ruelle operator in Wasserstein distance. The results underscore the multifaceted nature of coupling, bridging optimal transport, stochastic process theory, and dynamical systems, and providing a robust arsenal for both pure and applied probabilistic analysis.