Geometric analysis for non-symmetric generators on Riemannian manifolds

• Project/Area Number: 17K05215
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Yamagata University
• Principal Investigator: Ishiwata Satoshi 山形大学, 理学部, 准教授 (70375393)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 熱核 / ポアンカレ定数 / 連結和 / 非対称ランダム・ウォーク / べき零被覆グラフ / 中心極限定理 / 非対称ランダムウォーク / ポアンカレ不等式

Outline of Final Research Achievements

In this research project, we investigate geoemtric analysis on a connected sum of non-compact Riemannian manifolds. In particular, we obtain optimal estimates of the long time behavior of the heat kernel and the Poincare constant on manifold with ends. Almostly, the project was done as a joint work with Professor Alexander Grigor'yan from Bielefeld University in Germany and Professor Laurent Saloff-Coste from Cornell University in the US.

Academic Significance and Societal Importance of the Research Achievements

従来、トポロジーなどで使われていた空間の連結和という操作は幾何解析学とは相性が悪く、研究が進んでいなかった。本研究ではこの連結和という解析的には扱いにくい対象上の解析にチャレンジし、適切な仮定のもと、最良の結果を得た。この結果により、例えば結節点のあるような物体の中をランダムに動く粒子や信号がどちらに動きやすいか？という問題についての論理的な根拠を与え、従来よりも効率のよいシステムの開発に貢献することができる。

Research Products

• [Int'l Joint Research] Universitat Bielefeld(ドイツ)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] University of Bielefeld(ドイツ)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] Bielefeld University(ドイツ)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] Bielefeld University(ドイツ)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] Universitat Bielefeld(ドイツ)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] Universitat Bielefeld(Germany)
• [Int'l Joint Research] Cornell University(米国)
• [Journal Article] Poincar? constant on manifolds with ends (2023)
  • Author(s): Grigor'yan Alexander、Ishiwata Satoshi、Saloff‐Coste Laurent
  • Journal Title: Proceedings of the London Mathematical Society Volume: - Issue: 6 Pages: 1961-2012
  • DOI: 10.1112/plms.12522
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Geometric analysis on manifolds with ends (2021)
  • Author(s): Alexander Grigor’yan,Satoshi Ishiwata, and Laurent Saloff-Coste
  • Journal Title: Advances in Analysis and Geometry Volume: 3 Pages: 325-344
  • DOI: 10.1515/9783110700763-011
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I (2020)
  • Author(s): Ishiwata Satoshi、Kawabi Hiroshi、Namba Ryuya
  • Journal Title: Electronic Journal of Probability Volume: 25 Issue: none Pages: 1-46
  • DOI: 10.1214/20-ejp486
  • Peer Reviewed / Open Access
• [Journal Article] Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II (2020)
  • Author(s): Ishiwata Satoshi、Kawabi Hiroshi、Namba Ryuya
  • Journal Title: Potential Analysis Volume: online Issue: 1 Pages: 1-40
  • DOI: 10.1007/s11118-020-09851-7
  • Peer Reviewed
• [Journal Article] 非コンパクトリーマン多様体上の熱核評価 - 最近の発展 - (2018)
  • Author(s): 石渡聡
  • Journal Title: 数学 Volume: 71 Pages: 77-92
  • NAID: 130007975509
  • Peer Reviewed
• [Journal Article] Heat kernel estimates on connected sums of parabolic manifolds (2018)
  • Author(s): Alexander Grigor'yan, Satoshi Ishiwata, L. Saloff-Coste
  • Journal Title: Journal de Mathematiques Pures et Appliquees Volume: 113 Pages: 155-194
  • DOI: 10.1016/j.matpur.2018.03.002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Can one observe the bottleneckness of a space by the heat distribution? (2017)
  • Author(s): Satoshi Ishiwata
  • Journal Title: Mathematical Physics and Computer Simulation, Volgograd State University Volume: 20 Pages: 77-88
  • Peer Reviewed
• [Presentation] Poincar´e constant on manifolds with ends (2023)
  • Author(s): Satoshi Ishiwata
  • Organizer: Open Japanese-German conference on stochastic analysis and applications
  • Int'l Joint Research / Invited
• [Presentation] 多様体の離散近似 (2022)
  • Author(s): 石渡聡
  • Organizer: オンライン勉強会：機械学習の数理
  • Invited
• [Presentation] Heat kernel estimates on connected sums of parabolic manifolds (2022)
  • Author(s): 石渡聡
  • Organizer: Hokkaido University online seminar
  • Invited
• [Presentation] Geometric analysis on manifolds with ends (2021)
  • Author(s): Satoshi Ishiwata
  • Organizer: Himeji Conference on Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Poincare constant on manifold with ends (2019)
  • Author(s): S. Ishiwata
  • Organizer: Analysis and PDEs on Manifolds and Fractals
  • Int'l Joint Research / Invited
• [Presentation] Poincare constant on manifolds with ends (2018)
  • Author(s): Satoshi Ishiwata
  • Organizer: 2018 Spring Probability Workshop, Institute of Mathematics, Academia Sinica
  • Int'l Joint Research / Invited
• [Presentation] Poincare Inequality on manifolds with ends (2017)
  • Author(s): Satoshi Ishiwata
  • Organizer: Analysis and PDEs on Manifolds
  • Int'l Joint Research / Invited
• [Presentation] Poincare Inequality on manifolds with ends (2017)
  • Author(s): Satoshi Ishiwata
  • Organizer: Global properties in potential theory of continuous and discrete spaces
  • Invited
• [Presentation] Poincare Inequality on manifolds with ends (2017)
  • Author(s): Satoshi Ishiwata
  • Organizer: Cornell University Probability seminar
  • Invited