Markovian Maximal Coupling of Markov Processes

TL;DR

This paper characterizes Markovian maximal couplings of Markov processes, including jump processes, and constructs a unique coupling for subordinated Brownian motion with an explicit generator.

Contribution

It extends the theory of Markovian maximal couplings to jump processes and provides explicit constructions and generator calculations for these couplings.

Findings

Markovian maximal couplings are characterized by total variation and Wasserstein distance equality.

Constructed the unique Markovian maximal coupling for subordinated Brownian motion.

Derived the generator of the coupling via state-space dependent mirror coupling.

Abstract

Markovian maximal couplings of Markov processes are characterized by an equality of total variation and a distance of Wasserstein type. If a Markovian maximal coupling is a Feller process, the generator can be calculated, e.g. for reflection coupled Brownian motion. Apart from processes with continuous paths also jump processes are treated for the first time. For subordinated Brownian motion a Markovian maximal coupling is constructed by subordinating reflection coupled Brownian motion. This coupling is the unique Markovian maximal coupling and its generator is determined by state-space dependent mirror coupling of the corresponding L\'evy measures.