Potential Analysis

, Volume 39, Issue 3, pp 231–263 | Cite as

Monotonicity of Time-Dependent Transportation Costs and Coupling by Reflection

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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.

Keywords

Transportation cost Coupling by reflection Diffusion process Curvature-dimension condition Total variation 

Mathematics Subject Classifications (2010)

58J65 53C21 60H30 60J60 58J35 
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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Graduate School of Humanities and SciencesOchanomizu UniversityTokyoJapan
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany

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