*Maximal Coupling*

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## April 19, 2016

### Rigidity for Markovian Maximal Couplings of Elliptic Diffusions

Writing about web page http://www.springer.com/-/1/AVQcC6GMAgfPWjhrECri

Banerjee, S., & Kendall, W. S. (2016). Rigidity for Markovian Maximal Couplings of Elliptic Diffusions. *Probab. Theory Related Fields*, **42**(to appear), 58pp. http://doi.org/10.1007/s00440-016-0706-4

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm (2013) proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups).

### Coupling the Kolmogorov Diffusion: maximality and efficiency considerations

Writing about web page https://arxiv.org/abs/1506.04804

Banerjee, S., & Kendall, W. S. (2016). Coupling the Kolmogorov Diffusion: maximality and efficiency considerations. *Advances in Applied Probability*, **48A**, to appear. See https://arxiv.org/abs/1506.04804 p, li { white-space: pre-wra

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral, or, slightly more generally, together with a finite number of iterated time integrals). In this case there can be no Markovian maximal coupling. Indeed, Markovian couplings cannot even be efficient (extending the terminology of Burdzy & Kendall, Efficient Markovian couplings: examples and counterexamples; *Annals of Applied Probability*, 2000, 10.2, 362–409). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling.