熱分布の確率・幾何解析と最適輸送理論

• Project/Area Number: 26707004
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Tohoku 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: ¥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

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。

Research Products

• [Int'l Joint Research] コネチカット大学(米国)
• [Int'l Joint Research] ルクセンブルク大学(ルクセンブルク)
• [Int'l Joint Research] ベルゲン大学(ノルウェー)
• [Int'l Joint Research] SISSA(イタリア)
• [Int'l Joint Research] トロント大学(カナダ)
• [Int'l Joint Research]
• [Int'l Joint Research] 中国科学院(中国)
• [Int'l Joint Research] パデュー大学(米国)
• [Int'l Joint Research] SISSA(イタリア)
• [Int'l Joint Research] フライブルグ大学(ドイツ)
• [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)
• [Int'l Joint Research] ボン大学(ドイツ)
• [Journal Article] Zero noise limit of a stochastic differential equation involving a local time (2018)
  • Author(s): Kazumasa Kuwada and Taro Matsumura
  • Journal Title: Osaka Journal of Mathematics Volume: 印刷中
  • NAID: 120006542324
  • Peer Reviewed
• [Journal Article] Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition (2017)
  • 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
  • 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 spaces (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates (2015)
  • 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
  • Peer Reviewed
• [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: Dirichlet forms and their geometry
  • Place of Presentation: 東北大学
  • Year and Date: 2017-03-18
  • Int'l Joint Research / Invited
• [Presentation] Monotonicity and Rigidity of the W-entropy on RCD (0,N) spaces (2017)
  • 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
  • Invited
• [Presentation] Monotonicity and rigidity of the W-entropy on RCD^* (0, N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: Metric Measure Spaces and Ricci Curvature
  • Int'l Joint Research / Invited
• [Presentation] Rigidity for the spectral gap on RCD (K,\infty) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: 39th conference on stochastic processes and their applications (invited special session)
  • Int'l Joint Research / Invited
• [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: 2017 IMS-China international conference on statistics and probability
  • Int'l Joint Research / Invited
• [Presentation] Radial processes on RCD^* (K,N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: Geometric analysis on smooth and nonsmooth spaces
  • Int'l Joint Research / Invited
• [Presentation] Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: 23rd Nevanlinna Colloquium
  • Int'l Joint Research / Invited
• [Presentation] Some remarks on the W-entropy on RCD (0,N) spaces (2017)
  • Author(s): Kazumasa Kuwada
  • Organizer: Dirichlet forms and its geometric aspects
  • Place of Presentation: 福岡大学
  • Invited
• [Presentation] Monotonicity and Rigidity of the W-entropy on RCD^* (0,N) spaces (2016)
  • Author(s): Kazumasa Kuwada
  • Organizer: Heat Kernels, Stochastic Processes and Functional Inequalities
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach
  • Year and Date: 2016-11-28
  • Int'l Joint Research / Invited
• [Presentation] Monotonicity and Rigidity of the W-entropy on RCD (0,N) spaces (2016)
  • Author(s): Kazumasa Kuwada
  • Organizer: Optimal transport and applications
  • Place of Presentation: Scuola Normale Superiore, Pisa, Italy
  • Year and Date: 2016-11-07
  • 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
  • Invited
• [Presentation] Monotonicity and Rigidity of the W-entropy on RCD^* (0,N) spaces (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Estimates of transportation costs between heat distributions (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Analysis of heat distributions on metric measure spaces with a lower Ricci curvature bound (2016)
  • Author(s): Kazumasa Kuwada
  • Organizer: PDE workshop at Sendai
  • Place of Presentation: 東北大学
  • Year and Date: 2016-05-26
  • Invited
• [Presentation] Analysis of heat distributions on Riemannian metric measure spaces with a lower Ricci curvature bound (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] A dimensional Wasserstein contraction characterizing the curvature-dimension condition (2016)
  • 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
  • Invited
• [Presentation] Optimal transport, heat flow and coupling of Brownian motions (2015)
  • 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
  • Invited
• [Presentation] A self-improvement of the Bakry-Ledoux gradient estimate (2015)
  • Author(s): Kazumasa Kuwada
  • Organizer: Geometry and Probability
  • Place of Presentation: 東京工業大学
  • Year and Date: 2015-11-09
  • Int'l Joint Research / Invited
• [Presentation] Coupling by reflection of Brownian motions on metric measure spaces with a lower Ricci curvature bounds (2015)
  • Author(s): Kazumasa Kuwada
  • Organizer: Stochastic Analysis
  • Place of Presentation: 京都大学
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] A dimensional Wasserstein contraction characterizing the curvature-dimension condition (2015)
  • 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
  • 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
  • Invited
• [Presentation] Optimal transport, heat flow and coupling of Brownian motions (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] On estimates of transportation costs for heat distributions (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Coupling by reflection of Brownian motions on RCD (K, ∞) spaces (2015)
  • 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
  • Invited
• [Presentation] On the speed in transportation costs of heat distributions (2014)
  • 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
  • Invited
• [Presentation] On the speed in transportation costs of heat distributions (2014)
  • 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
  • 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
  • Invited
• [Presentation] On the speed in transportation costs of heat distributions (2014)
  • 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
• [Presentation] Wasserstein controls for heat distributions (2014)
  • 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
  • Invited
• [Presentation] On the speed in transportation costs of heat distributions (2014)
  • 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
  • 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
• [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
• [Funded Workshop] One day workshop on stochastic differential geometry (2017)
• [Funded Workshop] New Trends in Stochastic Analysis (2015)
  • Place of Presentation: 国際高等研究所
  • Year and Date: 2015-07-06