Matthias Erbar, Kazumasa Kuwada and Karl-Theodor Sturm On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces Invent. Math., 201(3): 993-1071 2015 http://dx.doi.org/10.1007/s00222-014-0563-7
2013
Kazumasa Kuwada and Karl-Theodor Sturm Monotonicity of time-dependent transportation costs and coupling by reflection Potential Analysis, 39(3): 231-263 2013 http://dx.doi.org/10.1007/s11118-012-9327-4
Abstract: Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.