It is well known that the well known mirror coupling is an optimal coupling of euclidean Brownian motions. In general optimal couplings are not unique. Using a simple uniqueness result from mass transportation theory with a concave cost function, we show that the mirror coupling is the unique optimal coupling among Markov couplings. Whether this result also holds for Riemannian Brownian motion on a compact Riemannian manifold is an unsolved and highly interesting problem. We show that the problem can be reduced to the uniqueness problem for a mass transportation problem for a cost function defined by the heat kernel. More generally, we attempt to develop a theory of coupling for manifold-valued semimartingales. It can be formulated as a theory of mass transportation theory in the (infinite dimensional) path space over the manifold. Various forms of cost function have been defined in the literature but so far none of them is completely satisfactory. We propose a new definition of cost function which we hope will lead to a more satisfactory theory of mass transportation for the path space. This talk is based in part on the joint work with

T. Sturm and I. Popescu.