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The duality in Bakry–Émery’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.

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