Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

• Published in: Potential analysis Vol. 53; no. 3; pp. 1123 - 1144
• Main Author: Wang, Feng-Yu
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2020; Springer Nature B.V
• Subjects: Curvature; Diffusion; Functional Analysis; Geometry; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; Riemann manifold; 60J75; 47G20; 60G52; Diffusion semigroup; Wasserstein distance; SDEs with multiplicative noise; Riemannian manifold; Curvature condition
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-019-09800-z

Let P t be the (Neumann) diffusion semigroup P t generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant ≪> 0 if and only if W p ( μ 1 P t , μ 2 P t ) ≤ e − ≪ t W p ( μ 1 , μ 2 ) , t ≥ 0 , p ≥ 1 holds for all probability measures μ 1 and μ 2 on M , where W p is the L p Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction W p ( μ 1 P t , μ 2 P t ) ≤ c e − ≪ t W p ( μ 1 , μ 2 ) , p ≥ 1 , t ≥ 0 for some constants c ,≪> 0 for a class of diffusion semigroups with negative curvature where the constant c is essentially larger than 1. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.