Coupling the Kolmogorov Diffusion: maximality and efficiency considerations

TL;DR

This paper investigates the limitations of Markovian couplings for Kolmogorov diffusions and introduces a simple, efficient non-Markovian coupling as a solution to these limitations.

Contribution

It demonstrates the impossibility of Markovian maximal and efficient couplings for Kolmogorov diffusions and provides a novel non-Markovian coupling that is efficient.

Findings

Markovian couplings cannot be maximal for Kolmogorov diffusions.

Markovian couplings cannot be efficient in this context.

A simple non-Markovian coupling is constructed that is efficient.

Abstract

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral, or, slightly more generally, together with a finite number of iterated time integrals). In this case there can be no Markovian maximal coupling. Indeed, Markovian couplings cannot even be efficient (extending the terminology of Burdzy and Kendall, Efficient Markovian couplings: examples and counterexamples; Annals of Applied Probability, 2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling.