Development of geometric analysis based on coupling methods for diffusion processes
| Project/Area Number |
22740083
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Basic analysis
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| Research Institution | Ochanomizu University |
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Principal Investigator |
KUWADA Kazumasa お茶の水女子大学, 大学院人間文化創成科学研究科, 准教授 (30432032)
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| Project Period (FY) |
2010-04-01 – 2014-03-31
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| Project Status |
Completed (Fiscal Year 2013)
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| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | 拡散過程 / カップリング / Ricci曲率 / Wasserstein距離 / Ricci流 / 曲率次元条件 / 熱半群の微分評価 / Hopf-Lax半群 / リッチ曲率 / 熱分布の微分評価 / ヴァッサーシュタイン距離 / リッチ流 / ホップ・ラックス半群 / アレクサンドロフ空間 |
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| Research Abstract |
On singular spaces admitting no differentiable structure, we identify two different formulations of heat flows as gradient flows. As a result, we prove an implication from a formulation of the presence of a lower Ricci curvature bound via optimal transport to another one via the gradient estimate of heat semigroups.
By using stochastic analytic techniques, we extend an estimate of the Wasserstein distance between heat distributions which appears as a kind of curvature condition in the above argument to the one on Ricci flows. In addition, we extend it to the one associated with the Wasserstein-like distance defined by using Perelman's L-distance.
We obtain a formulation of a characteristic property of the mirror coupling in terms of optimal transport. By using this formulation, we proved that the same property holds on a class of singular spaces given by limit of Riemannian manifolds.
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Report
(5 results)
Research Products
(79 results)
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[Presentation] Heat flow on Alexandrov spaces2011
Author(s)
K. Kuwada
Organizer
35th conference on Stochastic Processes and their Applications
Place of Presentation
Hotel Mision de Los Angeles (オアハカ, メキシコ)
Year and Date
2011-06-21
Related Report
Invited
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