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

Abstract

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.

35 pages(v1). 39 pages. The presentation of the proof of Proposition 3.6 is improved. The proof of Lemma 4.5 is corrected (Proposition 4.4 is added for this). The proof of Lemma 4.8 is modified (v2)

References in corpus (2)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Dimensional contraction via Markov transportation distance

Cited by in corpus (1)

• $(K,N)$-convexity and the curvature-dimension condition for negative $N$