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Maximal couplings and geometry

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Maximal couplings are couplings of Markov processes where the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian couplings are coupling strategies where neither process is allowed to look into the future of the other before making the next transition. These are easier to describe and play a fundamental role in many branches of probability and analysis. Hsu and Sturm proved that the reflection coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two di fferent points. Later, Kuwada proved that to have a MMC for Brownian motions on a Riemannian manifold, the manifold should have a reflection structure, and thus proved the first result connecting a purely probabilistic phenomenon (MMC) to the geometry of the underlying space. In this work, we investigate general elliptic diffusions on Riemannian manifolds, and show how the geometry (dimension of the isometry group and flows of isometries) plays a fundamental role in classifying the space and the generator of the diffusion for which an MMC exists. We also describe these di ffusions in terms of Killing vector fi elds (generators of rigid motions on manifolds) and dilation vector elds around a point.

This is joint work with W.S. Kendall.

This talk is part of the Probability series.

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