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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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