Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

2016/03/18 by Feng‐Yu Wang, Wang, Feng-Yu · 8 citations
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1603.05749

openalex publication_date 2016/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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(μ₁P_t, μ₂P_t)≤ \e^{-≪ t} W_p (μ₁,μ₂), t≥ 0, p≥ 1 holds for all probability measures μ₁ and μ₂ 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(μ₁P_t, μ₂P_t)≤ c\e^{-≪ t} W_p (μ₁,μ₂), 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.

Citations

• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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