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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.

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Authors and Affiliations

  1. Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo, 112-8610, Japan

    Kazumasa Kuwada

  2. Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Karl-Theodor Sturm

Authors
  1. Kazumasa Kuwada
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  2. Karl-Theodor Sturm
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Correspondence to Kazumasa Kuwada.

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Kuwada, K., Sturm, KT. Monotonicity of Time-Dependent Transportation Costs and Coupling by Reflection. Potential Anal 39, 231–263 (2013). https://doi.org/10.1007/s11118-012-9327-4

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