Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

By Feng-Yu Wang. Potential Analysis, 2016

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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 $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$ 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 $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$ 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/45920a6bd70e0a5e541ea69221e4b4f05d763efe