Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Let Pt be the (Neumann) diffusion semigroup Pt 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 Wp(μ1Pt,μ2Pt)≤e−≪tWp(μ1,μ2),t≥0,p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$\end{document} holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction Wp(μ1Pt,μ2Pt)≤ce−≪tWp(μ1,μ2),p≥1,t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$\end{document} 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.