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.Comment: 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

Coupling of Brownian motions and Perelman's L-functional

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.Comment: 20 page

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.Comment: Typos corrected, some improved bound