Exponential contraction in Wasserstein distance on static and evolving manifolds

Abstract

Dedicated to the memory of Nicu BobocABSTRACT. In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures.

Approximation of Riemannian measures by Stein's method

“…For example, the uniform boundedness assumptions could be relaxed to allow a certain amount of growth, while in many cases the positive lower bound on Ric ψ is simply not necessary. The positive lower bound will be used to verify exponential convergence to equilibrium, but as shown for example by Cheng, Thalmaier and Zhang in the recent article [6], weaker conditions exist. In Section 7, we show how all assumptions on curvature can be dispensed with for the case ψ = 0 with M compact, using instead the existence of a spectral gap, yielding Theorems 7.6 and 7.6.…”