1,624 research outputs found
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 -Wasserstein control for heat distributions at
different times. The -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
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