Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufficient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we find many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we find a Markov chain which admits no maximal Markovian coupling for specified starting points.
"Characterization of maximal Markovian couplings for diffusion processes." Electron. J. Probab. 14 633 - 662, 2009. https://doi.org/10.1214/EJP.v14-634