Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates

TL;DR

This paper extends the duality between Bakry-Émery gradient estimates and Wasserstein controls to refined estimates with high generality, establishing new equivalences involving dimension bounds and Wasserstein controls for heat distributions.

Contribution

It introduces a new equivalence between Bakry-Ledoux's refined gradient estimate and an $L^2$-Wasserstein control condition involving dimension bounds, extending the theory to high generality.

Findings

Established an equivalent condition to Bakry-Ledoux's gradient estimate involving Wasserstein control.

Studied $L^p$-versions of these estimates on Riemannian manifolds using coupling methods.

Extended the duality and control framework to refined estimates with high generality.

Abstract

The duality in Bakry-\'Emery's gradient estimates and Wasserstein controls for heat distributions is extended to that in refined estimates in a high generality. As a result, we find an equivalent condition to Bakry-Ledoux's refined gradient estimate involving an upper dimension bound. This new condition is described as a $L^2$-Wasserstein control for heat distributions at different times. The $L^p$-version of those estimates are studied on Riemannian manifolds via coupling method.