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2021 Fiscal Year Annual Research Report

Analysis and geometry of fractals and stochastic processes on them from field-transverse viewpoints

Research Project

Project/Area Number 18H01123
Research InstitutionKyoto University

Principal Investigator

梶野 直孝  京都大学, 数理解析研究所, 准教授 (90700352)

Co-Investigator(Kenkyū-buntansha) 白石 大典  京都大学, 情報学研究科, 准教授 (00647323)
中島 誠  名古屋大学, 多元数理科学研究科, 准教授 (60635902)
田中 亮吉  京都大学, 理学研究科, 准教授 (80629759)
Project Period (FY) 2018-04-01 – 2022-03-31
Keywordsフラクタル上の解析学 / エネルギー測度 / ウォーク次元 / 調和関数 / 乗法型確率熱方程式 / 曲面群 / 3次元ループ除去ランダムウォーク
Outline of Annual Research Achievements

2021年度(および繰越期間である2022年度)は研究代表者の梶野および研究分担者の中島・田中・白石がそれぞれ研究計画調書に記した担当課題(1)-(7)および関連する新たな研究課題について,国内外の研究集会やセミナーに参加し研究発表および情報収集を行った.
梶野は課題(3)「エネルギー測度」について,Mathav Murugan氏(UBC)との2019年度の共同研究の結果は適用できないがエネルギー測度は特異になっているような具体例の族の構築に関する論文,および一般化Sierpinski carpetのwalk次元が2より真に大きいことの初等的証明を与えた論文を出版した.また課題(2)「領域上の調和測度」とも関連して,自己相似フラクタル上の自己相似Dirichlet形式に関する調和関数であってそのエネルギー測度に関するL^2-内積の下での熱核がGauss型評価を満たすものの全体が一定のコンパクト性を有することを証明し,Murugan氏との先行する共同研究に組み入れる形で論文にまとめ出版した.
中島は乗法型確率熱方程式の2次元の場合の解から構成される関数の族の研究を行い,劣臨界領域と呼ばれるパラメータ領域において解の摂動を求めEdwards-Wilkinson型の確率偏微分方程式が現れることを示した.解析手法を改善することで3次元以上の研究では個別に計算していた評価をまとめて評価できるようになった.
田中は双曲群の古典的なクラスでそれ自体具体的な研究がかなりの程度可能である曲面群について研究を行った.具体的にはグラフの調和埋め込みについての一意化定理を証明し,成果を論文としてまとめ出版した.
白石はXinyi Li氏(北京大学)との共同研究において,3次元ループ除去ランダムウォークのスケール極限のヘルダー連続性に関する結果をループ除去ランダムウォークの成長指数を用いて与えた.

Research Progress Status

令和3年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和3年度が最終年度であるため、記入しない。

  • Research Products

    (37 results)

All 2023 2022 2021 Other

All Int'l Joint Research (5 results) Journal Article (8 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 8 results,  Open Access: 5 results) Presentation (18 results) (of which Int'l Joint Research: 11 results,  Invited: 13 results) Remarks (5 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Country Name
      CANADA
    • Counterpart Institution
      University of British Columbia
  • [Int'l Joint Research] Universite de Montpellier(フランス)

    • Country Name
      FRANCE
    • Counterpart Institution
      Universite de Montpellier
  • [Int'l Joint Research] University of California, San Diego(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      University of California, San Diego
  • [Int'l Joint Research] University of Chicago(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      University of Chicago
  • [Int'l Joint Research] Peking University(中国)

    • Country Name
      CHINA
    • Counterpart Institution
      Peking University
  • [Journal Article] On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions2023

    • Author(s)
      Naotaka Kajino and Mathav Murugan
    • Journal Title

      Inventiones mathematicae

      Volume: 231 Pages: 263-405

    • DOI

      10.1007/s00222-022-01148-3

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] An elementary proof that walk dimension is greater than two for Brownian motion on Sierpinski carpets2023

    • Author(s)
      Naotaka Kajino
    • Journal Title

      Bulletin of the London Mathematical Society

      Volume: 55 Pages: 508-521

    • DOI

      10.1112/blms.12742

    • Peer Reviewed
  • [Journal Article] Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions2023

    • Author(s)
      Shuta Nakajima and Makoto Nakashima
    • Journal Title

      Electronic Journal of Probability

      Volume: 28 Pages: 1-38

    • DOI

      10.1214/22-EJP885

    • Peer Reviewed / Open Access
  • [Journal Article] On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates II: Some borderline examples2022

    • Author(s)
      Naotaka Kajino
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 394 Pages: 223-251

    • DOI

      10.1007/978-981-19-4672-1_13

    • Peer Reviewed
  • [Journal Article] The Holder continuity of the scaling limit of three-dimensional loop-erase loop-erased random walk2022

    • Author(s)
      Xinyi Li and Daisuke Shiraishi
    • Journal Title

      Electronic Journal of Probability

      Volume: 27 Pages: 1-37

    • DOI

      10.1214/22-EJP869

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Topological flows for hyperbolic groups2021

    • Author(s)
      Ryokichi Tanaka
    • Journal Title

      Ergodic Theory and Dynamical Systems

      Volume: 41 Pages: 3474-3520

    • DOI

      10.1017/etds.2020.101

    • Peer Reviewed
  • [Journal Article] Uniformizing surfaces via discrete harmonic maps2021

    • Author(s)
      Toru Kajigaya and Ryokichi Tanaka
    • Journal Title

      Annales Henri Lebesgue

      Volume: 4 Pages: 1767-1807

    • DOI

      10.5802/ahl.116

    • Peer Reviewed / Open Access
  • [Journal Article] Random walks on the discrete affine group2021

    • Author(s)
      Jeremie Brieussel, Ryokichi Tanaka and Tianyi Zheng
    • Journal Title

      Groups, Geometry, and Dynamics

      Volume: 15 Pages: 935-963

    • DOI

      10.4171/GGD/616

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Sub-Gaussian heat kernel bounds and singularity of energy measures for symmetric diffusions2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      OIST Analysis on Metric Spaces Seminar
    • Invited
  • [Presentation] On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      7th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals
    • Int'l Joint Research / Invited
  • [Presentation] Quasisymmetric Gaussian uniformization is impossible for Brownian motion on the Sierpinski carpet2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      International Workshop on Dirichlet Forms and Related Topics
    • Int'l Joint Research / Invited
  • [Presentation] On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates II: Some borderline examples2022

    • Author(s)
      梶野 直孝
    • Organizer
      2022年度日本数学会秋季総合分科会・統計数学分科会
  • [Presentation] Quasisymmetric Gaussian uniformization is impossible for Brownian motion on the Sierpi\'{n}ski carpet2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      Open Japanese-German conference on stochastic analysis and applications
    • Int'l Joint Research / Invited
  • [Presentation] Quasisymmetric Gaussian uniformization is impossible for Brownian motion on the Sierpinski carpet2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      Heat Kernels, Stochastic Processes and Functional Inequalities
    • Int'l Joint Research
  • [Presentation] Conformal walk dimension: its universal value and the non-attainment for the Sierpinski carpet2022

    • Author(s)
      Naotaka Kajino
    • Organizer
      Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities
    • Int'l Joint Research
  • [Presentation] Gaussian fluctuations of stochastic heat equation and KPZ equation in higher dimension in $L^2$ regime2022

    • Author(s)
      Makoto Nakashima
    • Organizer
      Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields 2022
    • Int'l Joint Research / Invited
  • [Presentation] ランダムウォークの衝突回数の末尾確率に対する上下評価2021

    • Author(s)
      梶野 直孝
    • Organizer
      無限粒子系、確率場の諸問題XVI
  • [Presentation] Conformal walk dimension: its universal value and the non-attainment for the Sierpinski carpet2021

    • Author(s)
      Naotaka Kajino
    • Organizer
      Pacific Workshop on Probability and Statistical Physics
    • Int'l Joint Research / Invited
  • [Presentation] On the conformal walk dimension II: Non-attainment for some Sierpinski carpets2021

    • Author(s)
      梶野 直孝
    • Organizer
      2021年度確率論シンポジウム
  • [Presentation] Fluctuation in L^2-region for stochastic heat equation and KPZ equation in higher dimension2021

    • Author(s)
      中島 誠
    • Organizer
      関西確率論セミナー
    • Invited
  • [Presentation] Gaussian fluctuations of stochastic heat equation and KPZ equation in higher dimension in subcritical regime2021

    • Author(s)
      Makoto Nakashima
    • Organizer
      Rigorous Statistical Mechanics and Related Topics
    • Int'l Joint Research / Invited
  • [Presentation] 高次元確率熱方程式とKPZ方程式のL2-領域における解の摂動2021

    • Author(s)
      中島 誠
    • Organizer
      The 19th Symposium Stochastic Analysis on Large Scale Interacting Systems
    • Invited
  • [Presentation] The Manhattan curve and rough similarity rigidity2021

    • Author(s)
      Ryokichi Tanaka
    • Organizer
      Pacific Dynamics Seminar
    • Int'l Joint Research / Invited
  • [Presentation] The Manhattan curve and rough similarity rigidity2021

    • Author(s)
      Ryokichi Tanaka
    • Organizer
      Ohio State University, Ergodic Theory Seminar
    • Int'l Joint Research / Invited
  • [Presentation] Glauber-Exclusion dynamics: rapid mixing regime2021

    • Author(s)
      田中 亮吉
    • Organizer
      The 19th Symposium Stochastic Analysis on Large Scale Interacting Systems
    • Invited
  • [Presentation] Convergence of three-dimensional loop-erased random walk in the natural parameterization2021

    • Author(s)
      Daisuke Shiraishi
    • Organizer
      The 10th International Conference on Stochastic Analysis and its Applications
    • Int'l Joint Research / Invited
  • [Remarks] Publications and preprints

    • URL

      https://www.kurims.kyoto-u.ac.jp/~nkajino/publications.html

  • [Remarks] 梶野 直孝|京都大学数理解析研究所

    • URL

      https://www.kurims.kyoto-u.ac.jp/ja/list/kajino.html

  • [Remarks] Makoto Nakashima's web site

    • URL

      https://www.math.nagoya-u.ac.jp/~nakamako/

  • [Remarks] Ryokichi Tanaka

    • URL

      https://ryokichitanaka.github.io

  • [Remarks] 白石 大典(情報学研究科 情報学専攻応用解析学講座) | 京都大学 教育研究活動データベース

    • URL

      https://kdb.iimc.kyoto-u.ac.jp/profile/ja.f6a26c885ba820ce.html

  • [Funded Workshop] Analysis and geometry of fractals and metric spaces: Recent developments and future prospects2023

URL: 

Published: 2023-12-25  

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