Probability Theory
Reflection coupling and Wasserstein contractivity without convexity
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1101-1104.

We note that even if convexity of the potential U fails locally, overdamped Langevin diffusions in Rd are contractions w.r.t. the Kantorovich–Rubinstein-Wasserstein distance based on an appropriately chosen concave distance function equivalent to the Euclidean distance. The choice of the distance function is then optimized to obtain a large exponential decay rate. The results yield dimension-independent bounds of optimal order in R,L[0,) and K(0,) if (xy)(U(x)U(y)) is bounded from below by L|xy|2 for |xy|<R and by K|xy|2 for |xy|R.

On considére diffusions de Langevin sur Rd dans un potentiel U non convex dans un ensemble borné. A lʼaide du couplage de réflection, on observe que ces diffusions sont des contractions pour la distance de Kantorovich–Rubinstein–Wasserstein basée sur une distance concave appropriée, équivalente à la distance Euclidienne. Le choix de la distance est optimisé pour obtenir un grand taux de décroissance exponentielle. Les résultats impliquent bornes optimales pour R,L[0,) et K(0,), indépendamment de la dimension, sous la condition que (xy)(U(x)U(y)) est borné inférieurement par L|xy|2 pour |xy|<R et par K|xy|2 pour |xy|R.

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DOI: 10.1016/j.crma.2011.09.003
Eberle, Andreas 1

1 University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany
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Eberle, Andreas. Reflection coupling and Wasserstein contractivity without convexity. Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1101-1104. doi : 10.1016/j.crma.2011.09.003. http://www.numdam.org/articles/10.1016/j.crma.2011.09.003/

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