Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

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 $\ll >0$ if and only if $$W_p(μ_1P_t, μ_2P_t)\le \e^{-\ll t} W_p (μ_1,μ_2),\ \ t\ge 0, p\ge 1 $$ holds for all probability measures ${\mu }_1$ and ${\mu }_2$ on $M$, where $W_p$ is the $L^p$ Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_p(μ_1P_t, μ_2P_t)\le c\e^{-\ll t} W_p (μ_1,μ_2),\ \ p\ge 1, t\ge 0$$ for some constants $c,\ll >0$ 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.

26 pages

References in corpus (1)

• Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications

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