Duality on gradient estimates and Wasserstein controls

TL;DR

This paper establishes a duality between Wasserstein control and gradient estimates, extending known results to more general settings and enabling direct derivation of Wasserstein bounds from gradient estimates without curvature bounds.

Contribution

It introduces a general duality framework linking Wasserstein control and gradient estimates, applicable to subelliptic heat flows on Lie groups.

Findings

Derived Wasserstein control from gradient estimates without curvature bounds

Extended duality results to subelliptic heat flows on Lie groups

Provided a coupling method for heat distributions with controlled relative distance

Abstract

We establish a duality between L^p-Wasserstein control and L^q-gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.