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 ∈ [ 0 , ∞ ) and K ∈ ( 0 , ∞ ) if ( x − y ) ⋅ ( ∇ U ( x ) − ∇ U ( y ) ) is bounded from below by − L | x − y | 2 for | x − y | < R and by K | x − y | 2 for | x − y | ⩾ R .