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.
Electron. J. Probab.
14:
633-662
(2009).
DOI: 10.1214/EJP.v14-634