  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).
