Rigidity for Markovian Maximal Couplings of Elliptic Diffusions

TL;DR

This paper explores the geometric and algebraic conditions under which Markovian maximal couplings exist for elliptic diffusions on manifolds, extending prior results from Brownian motion to more general diffusions.

Contribution

It characterizes the geometric structures and symmetries necessary for the existence of Markovian maximal couplings of elliptic diffusions on manifolds.

Findings

Existence of MMC relates to the dimension of the isometry group.

Diffusions with MMC are characterized by Killing and dilation vector fields.

Classification of spaces supporting MMC depends on diffusion geometry.

Abstract

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm (2013) proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups).