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熱分布の確率・幾何解析と最適輸送理論

Research Project

Project/Area Number 26707004
Research Category

Grant-in-Aid for Young Scientists (A)

Allocation TypePartial Multi-year Fund
Research Field Basic analysis
Research InstitutionTohoku University (2017)
Tokyo Institute of Technology (2014-2016)

Principal Investigator

桑田 和正  東北大学, 理学研究科, 教授 (30432032)

Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Granted (Fiscal Year 2017)
Budget Amount *help
¥7,540,000 (Direct Cost: ¥5,800,000、Indirect Cost: ¥1,740,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2016: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2014: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords熱分布 / 曲率次元条件 / 最適輸送 / Bakry-Emery理論 / Brown運動 / 動径過程 / 確率論 / 幾何学 / 解析学 / 関数方程式論 / 結合法 / Ricci流
Outline of Annual Research Achievements

F. Baudoin氏,A. Thalmaier氏らと劣Riemann多様体上の幾何解析に関する共同研究に着手し論文を執筆した.退化計量を近似する計量の1パラメータ族を利用するアイデアを用いて,Riemann幾何の諸定理を拡張した.既存の劣Riemann幾何での結果をより広い空間族で拡張したのみならず,既存の結果の議論を大幅に単純化した.その結果を基盤に,劣Riemann多様体上のブラウン運動の結合法に関する共同研究を進めている.無限次元曲率次元条件下でのポアンカレ不等式の等号成立条件,および,曲率次元条件をみたす測度距離空間上の動径過程の解析に関して,前年度に得た成果を整備し詳細を詰め共同研究論文を執筆した.前者は,剛性定理の理解を一段深めるものであり,また測度距離空間上での近年の解析技法の有効性を明らかにした.後者は今後測度距離空間上の確率解析を展開する基礎を与えた.W-エントロピーの剛性定理では前年度からの共同研究を継続し論文執筆の最終段階に入った.特異空間上で問題を扱うことで,剛性定理の新しい側面を明らかにした.これらの諸結果は,曲率次元条件に基づく特異空間上の幾何解析および確率解析の新展開を示唆するものと言える.
また5月のBaudoin氏招聘に合わせて研究集会”one day workshop on stochastic differential geometry”を開催し,関連分野の研究者との研究打ち合わせを行った.特にBaudoin氏とは,前述の共同研究に関する詳しい研究打ち合わせを行った.

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

測度距離空間上での鏡映カップリングの構築について,その着想の根底にあった関数不等式を用いる方法のみでは限界があり,最適輸送を組み合わせた新しい理論が必要であろうことが明らかになった.そのため,研究の推進には新たなアイデアを更に投入する必要が生じている.
今年度の成果の多くは,前年度の成果を整備し論文を執筆した,という形となっている.その中にあって,劣Riemann多様体上の幾何解析および確率解析については大きな進展があったと言える.ただし当初の計画を完全に達成するまでの結果には至っていない.特に測度と距離関数を用いた曲率次元条の定式化の劣Riemann幾何の枠組みへの拡張および鏡映カップリングの構築については,なお新しい着想を要する.
非対称性生成作用素,空間非一様曲率次元条件,時間依存計量下の幾何解析については,当初の計画の重要な部分を他の研究者に先行されている.それらの結果の拡張には,より新しいアイデアを必要としている状況にある.

Strategy for Future Research Activity

測度距離空間上の結合法については,Riemann多様体上での既存の方法の再解釈を含め,今一度多角的な視点からのアプローチを再検討する.
また現在ボン大学に滞在中であることを利用して,本研究課題全般についてK.-Th. Sturm氏およびその研究グループの構成員との連絡,研究打ち合わせを密に取る.
劣Riemann多様体上の幾何解析および確率解析については,Baudoin氏およびThalmaier氏と連携を取りながら研究を進める.

Expenditure Plan for Carryover Budget

30年度が最終年度であるため、記入しない。

Report

(4 results)
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (51 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (12 results) Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (31 results) (of which Int'l Joint Research: 17 results,  Invited: 30 results) Book (1 results) Remarks (1 results) Funded Workshop (2 results)

  • [Int'l Joint Research] コネチカット大学(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] ルクセンブルク大学(ルクセンブルク)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] ベルゲン大学(ノルウェー)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] SISSA(イタリア)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research]

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] 中国科学院(中国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] パデュー大学(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] SISSA(イタリア)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] フライブルグ大学(ドイツ)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] リヨン第1大学/パリ第6大学/ブレーズ・パスカル大学(フランス)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] ボン大学(ドイツ)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Zero noise limit of a stochastic differential equation involving a local time2018

    • Author(s)
      Kazumasa Kuwada and Taro Matsumura
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 印刷中

    • NAID

      120006542324

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition2017

    • Author(s)
      Francois Bolley, Ivan Gentil, Arnaud Guillin and Kazumasa Kuwada
    • Journal Title

      Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

      Volume: 印刷中 Pages: 845-880

    • DOI

      10.2422/2036-2145.201611_009

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces2015

    • Author(s)
      M. Erbar, K. Kuwada and K.-T. Sturm
    • Journal Title

      Invent. Math.

      Volume: 201 Issue: 3 Pages: 993-1071

    • DOI

      10.1007/s00222-014-0563-7

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates2015

    • Author(s)
      K. Kuwada
    • Journal Title

      Calc. Var. Partial Differential Equations

      Volume: 54 Issue: 1 Pages: 127-161

    • DOI

      10.1007/s00526-014-0781-2

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Dirichlet forms and their geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2017-03-18
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Monotonicity and Rigidity of the W-entropy on RCD (0,N) spaces2017

    • Author(s)
      桒田 和正
    • Organizer
      岡山-広島解析・確率論セミナー2017
    • Place of Presentation
      岡山大学
    • Year and Date
      2017-02-21
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Monotonicity and rigidity of the W-entropy on RCD^* (0, N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Metric Measure Spaces and Ricci Curvature
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Rigidity for the spectral gap on RCD (K,\infty) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      39th conference on stochastic processes and their applications (invited special session)
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      2017 IMS-China international conference on statistics and probability
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Radial processes on RCD^* (K,N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Geometric analysis on smooth and nonsmooth spaces
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      23rd Nevanlinna Colloquium
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some remarks on the W-entropy on RCD (0,N) spaces2017

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Dirichlet forms and its geometric aspects
    • Place of Presentation
      福岡大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Monotonicity and Rigidity of the W-entropy on RCD^* (0,N) spaces2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Heat Kernels, Stochastic Processes and Functional Inequalities
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach
    • Year and Date
      2016-11-28
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Monotonicity and Rigidity of the W-entropy on RCD (0,N) spaces2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Optimal transport and applications
    • Place of Presentation
      Scuola Normale Superiore, Pisa, Italy
    • Year and Date
      2016-11-07
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 最適輸送理論,曲率次元条件と熱分布2016

    • Author(s)
      桒田 和正
    • Organizer
      確率論サマースクール
    • Place of Presentation
      信州大学
    • Year and Date
      2016-09-06
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Monotonicity and Rigidity of the W-entropy on RCD^* (0,N) spaces2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Geometric analysis on Riemannian and metric spaces
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-09-05
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Estimates of transportation costs between heat distributions2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      CMC conference: Analysis, Geometry and Optimal Transport
    • Place of Presentation
      Korea Institute for Advanced Study
    • Year and Date
      2016-06-20
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Analysis of heat distributions on metric measure spaces with a lower Ricci curvature bound2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      PDE workshop at Sendai
    • Place of Presentation
      東北大学
    • Year and Date
      2016-05-26
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Analysis of heat distributions on Riemannian metric measure spaces with a lower Ricci curvature bound2016

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      The 1st Hong Kong/Kyoto Workshop on Fractal Geometry and Related Areas
    • Place of Presentation
      香港科技大学(中国)
    • Year and Date
      2016-03-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition2016

    • Author(s)
      Francois Bolley, Ivan Gentil, Arnaud Guillin and Kazumasa Kuwada
    • Organizer
      Metric Geometry and its Applications
    • Place of Presentation
      復旦大学
    • Year and Date
      2016-02-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A dimensional Wasserstein contraction characterizing the curvature-dimension condition2016

    • Author(s)
      Francois Bolley, Ivan Gentil, Arnaud Guillin and Kazumasa Kuwada
    • Organizer
      マルコフ過程とその周辺
    • Place of Presentation
      ヴェルク横須賀
    • Year and Date
      2016-01-09
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Optimal transport, heat flow and coupling of Brownian motions2015

    • Author(s)
      桑田 和正
    • Organizer
      偏微分方程式に付随する確率論的問題
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2015-12-09
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] A self-improvement of the Bakry-Ledoux gradient estimate2015

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Geometry and Probability
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-11-09
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Coupling by reflection of Brownian motions on metric measure spaces with a lower Ricci curvature bounds2015

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Stochastic Analysis
    • Place of Presentation
      京都大学
    • Year and Date
      2015-09-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A dimensional Wasserstein contraction characterizing the curvature-dimension condition2015

    • Author(s)
      Francois Bolley, Ivan Gentil, Arnaud Guillin and Kazumasa Kuwada
    • Organizer
      Stochastic Analysis and Applications
    • Place of Presentation
      東北大学
    • Year and Date
      2015-08-31
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 最適輸送理論,リーマン的曲率次元条件と熱分布2015

    • Author(s)
      桑田 和正
    • Organizer
      第62回 幾何学シンポジウム
    • Place of Presentation
      東京理科大学
    • Year and Date
      2015-08-27
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Optimal transport, heat flow and coupling of Brownian motions2015

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Stochastic Processes and Applications
    • Place of Presentation
      National University of Mongolia (モンゴル)
    • Year and Date
      2015-07-27
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On estimates of transportation costs for heat distributions2015

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      New Trends in Stochastic Analysis
    • Place of Presentation
      国際高等研究所
    • Year and Date
      2015-07-06
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Coupling by reflection of Brownian motions on RCD (K, ∞) spaces2015

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      New Trends in Optimal Transport
    • Place of Presentation
      ボン大学(ドイツ)
    • Year and Date
      2015-03-02 – 2015-03-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the speed in transportation costs of heat distributions2014

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      ERC Workshop on Optimal Transportation and Applications
    • Place of Presentation
      ピサ国立大学(イタリア)
    • Year and Date
      2014-10-27 – 2014-10-31
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the speed in transportation costs of heat distributions2014

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Stability for the asymptotic behavior of PDEs, stochastic processes and their discretization
    • Place of Presentation
      リヨン第一大学(フランス)
    • Year and Date
      2014-09-29 – 2014-10-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 最適輸送,曲率次元条件と熱分布2014

    • Author(s)
      桑田和正
    • Organizer
      幾何学阿蘇研究集会
    • Place of Presentation
      休暇村南阿蘇(阿蘇郡高森町)
    • Year and Date
      2014-09-17 – 2014-09-20
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the speed in transportation costs of heat distributions2014

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      UK-Japan Stochastic Analysis School
    • Place of Presentation
      ウォーリック大学(イギリス)
    • Year and Date
      2014-09-01 – 2014-09-05
    • Related Report
      2014 Annual Research Report
  • [Presentation] Wasserstein controls for heat distributions2014

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      Stochastic Processes, Analysis and Mathematical Physics
    • Place of Presentation
      関西大学
    • Year and Date
      2014-08-25 – 2014-08-29
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the speed in transportation costs of heat distributions2014

    • Author(s)
      Kazumasa Kuwada
    • Organizer
      7th International Conference on Stochastic Analysis and its Applications
    • Place of Presentation
      ソウル国立大学(韓国)
    • Year and Date
      2014-08-06 – 2014-08-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Book] 最適輸送理論とリッチ曲率2017

    • Author(s)
      桑江一洋,塩谷隆,太田慎一,高津飛鳥,桑田和正
    • Total Pages
      141
    • Publisher
      日本数学会
    • Related Report
      2016 Annual Research Report
  • [Remarks] 挑戦的研究賞の受賞者紹介|関連資料|東京工業大学産学連携推進本部

    • URL

      http://www.sangaku.titech.ac.jp/document/researchers/201604.html

    • Related Report
      2015 Annual Research Report
  • [Funded Workshop] One day workshop on stochastic differential geometry2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] New Trends in Stochastic Analysis2015

    • Place of Presentation
      国際高等研究所
    • Year and Date
      2015-07-06
    • Related Report
      2015 Annual Research Report

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Published: 2014-04-04   Modified: 2022-02-28  

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