Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds

TL;DR

This paper introduces a novel class of non-Markovian couplings for sub-Riemannian Brownian motions on certain manifolds, leveraging global isometries to achieve maximal couplings with a reflection principle, improving bounds on coupling times.

Contribution

It develops a unified, simple method for constructing maximal, non-Markovian couplings on various sub-Riemannian manifolds using global isometries, extending prior work and providing explicit coupling time bounds.

Findings

Couplings are maximal and based on global isometries.

Coupling time reduces to hitting time of a Brownian component.

Applications include bounds for heat semigroup inequalities.

Abstract

We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general Markovian or co-adapted couplings cannot give the sharp rate for the coupling time in this case. Non-Markovian couplings of this type on sub-Riemannian manifolds were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by B\'en\'efice to SL(2,R) and SU(2), using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. The coupled processes satisfy a reflection principle with respect to their coupling time, so that the coupling time reduces to the hitting time for one component of the Brownian motion, which is useful in explicitly bounding the tail probability of the coupling time. Further, it's natural to use this coupling as the second stage of a two-stage coupling when considering points on different vertical fibers. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup.