Coupling and Applications

TL;DR

This paper provides a comprehensive overview of coupling methods in Markov processes, illustrating their use in analyzing ergodicity, convergence, and inequalities, with a focus on transport problems and probability distances.

Contribution

It introduces a self-contained framework for coupling arguments and demonstrates their applications across various fundamental properties of Markov processes.

Findings

Coupling describes the transport problem and introduces optimal coupling.

Applications include ergodicity, convergence rates, and inequalities.

Provides a unified approach to coupling in Markov process analysis.

Abstract

This paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and probability distance (or transportation-cost), then introduce applications of coupling to the study of ergodicity, Liouville theorem, convergence rate, gradient estimate, and Harnack inequality for Markov processes.