Reflection coupling and Wasserstein contractivity without convexity

We note that even if convexity of the potential U fails locally, overdamped Langevin diffusions in R(d) 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 is an element of [0, infinity) and K is an element of (0, infinity) if (x - y) center dot (del U(x) - del U(y)) is bounded from below by -L vertical bar x - y vertical bar(2) for vertical bar x - y vertical bar < R and by K vertical bar x - y vertical bar(2) for vertical bar x - y vertical bar >= R. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.