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A counterexample for the optimality of Kendall-Cranston coupling. (English) Zbl 1134.60011
Consider a Riemannian manifold $$M$$. If we are given two points $$x$$ and $$y$$ of $$M$$, we are interested in the construction of two (correlated) Brownian motions $$X$$ and $$Y$$, starting, respectively, from $$x$$ and $$y$$, such that $$X$$ and $$Y$$ meet at some time $$T$$ and are equal after that time. Such a construction is called a coupling. Here, the authors look for an optimal coupling, for which the probability of $$\{T\leq t\}$$ would be greater than for other couplings (for a fixed $$t>0$$). Actually, they consider the classical Kendall-Cranston coupling which is optimal in some cases (if $$M$$ has good symmetry), but prove with a counterexample that this coupling is not always optimal.

##### MSC:
 60D05 Geometric probability and stochastic geometry 58J65 Diffusion processes and stochastic analysis on manifolds 60J65 Brownian motion
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