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

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.