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超準解析による極限定理の幾何学的解釈

Research Project

Project/Area Number 17654036
Research Category

Grant-in-Aid for Exploratory Research

Allocation TypeSingle-year Grants
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

小谷 元子  Tohoku University, 大学院・理学研究科, 教授 (50230024)

Co-Investigator(Kenkyū-buntansha) 竹田 雅好  東北大学, 大学院・理学研究科, 教授 (30179650)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2007: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsランダム・ウォーク / 結晶格子 / 大偏差 / グロモフ・ハウスドルフ収束 / 被覆グラフ / 距離空 / 超準解析 / ランダムウォーク / 大偏差原理 / 距離空間
Research Abstract

結晶格子上のランダムウォークの大偏差に関して,幾何学的考察を行った.結晶格子とは,自由アーベル群が自由に作用し,商空間が有限グラフとなる無限グラフである.この周期性から,結晶格子を無限遠から観察すると一様な図形に見える.より正確に述べる.結晶格子をグラフ距離によって距離空間と考え,距離をスケール変換した距離空間の1パラメーター族を得る.このスケールをゼロに近付けたときのグロモフ・ハウスドルフ位相による極限距離空間を,結晶格子の無限遠での接錐という.結晶格子のようなアーベル周期性をもつ距離空間の無限遠での接錐の存在は,グロモフによって知られているが,この極限距離空間を具体的に,また,グラフの幾何の言葉で特徴つけた.距離球はコンパクト凸多面体となり,その端点が,具体的に商空間の閉路の構造で決まることを調べた.更に,この極限空間の単位距離球が,ランダム・ウォークの大偏差源氏に現れるレート関数の本質的定義域と一致することを示し,Math. Z.に発表した.これを非アーベル被覆の場合に拡張することを目標に,ランダム・スネーク,R樹木,超準極限に関するグロモフ,シャピロ,ドルータの結果など,関連研究について情報収集し,検討を行った.その結果,超準解析による定式化を書き下すことができたが,論文として公表するには至らなかった.ランダム・ウォークの長時間挙動,及び磁場付き推移作用素のスペクトルと結晶格子の幾何に関する総説をまとめAmer .Math. Soc. Sugaku Expositoryに公表した.

Report

(3 results)
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (16 results)

All 2008 2007 2006 2005 Other

All Journal Article (9 results) (of which Peer Reviewed: 2 results) Presentation (6 results) Book (1 results)

  • [Journal Article] Lp-independence of spectral bounds of Schrodinger type semigroups.2007

    • Author(s)
      M. Takeda
    • Journal Title

      J. Funct. Anal 252

      Pages: 550-565

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Branching Brownian motions on Riemannian manifolds: expectation of the number of branches hitting closed sets2007

    • Author(s)
      M. Takeda
    • Journal Title

      Potential Anal. 27

      Pages: 61-72

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large deviation and the tangent cone at infinity of a crystal lattice2006

    • Author(s)
      M.Korani, T.Sunada
    • Journal Title

      Math. Z. 254

      Pages: 837-870

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Gaugeability for Feynman-Kac functionals with applications to symmetric alpha stable process2006

    • Author(s)
      M.Takeda
    • Journal Title

      Proc. Amer. Math. Soc. 134

      Pages: 2729-2738

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Large deviation and the tangent cone at infinity of a crystal lattice2006

    • Author(s)
      M.Kotani, T.Sunada
    • Journal Title

      Mathematische Zeitschrift (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 離散幾何解析学に見る対称性2006

    • Author(s)
      小谷 元子
    • Journal Title

      「応用数理」フォーラム 16巻1号(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetric $alpha$-stable processes2005

    • Author(s)
      Y.Shiozawa, M.Takeda
    • Journal Title

      Potential Analysis 23

      Pages: 135-151

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Gaugeability for Feynman-Kac functionals with applications to symmetric $alpha$-stable processeS

    • Author(s)
      M.Takeda
    • Journal Title

      Proc.Amer.Math.Soc (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Differentiability of spectral functions for symmetric $alpha$-stable processes

    • Author(s)
      M.Takeda, K.Tsuchida
    • Journal Title

      Trans.Amer.Math.Soc. (to appear)

    • Related Report
      2005 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2008

    • Author(s)
      M. Kotani
    • Organizer
      Shonan, International Conference on Nconcommutative Geometry and Physic
    • Place of Presentation
      Shonan, Japan
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2008

    • Author(s)
      M. Kotani
    • Organizer
      Expamders in Pure and Applied Mathematics
    • Place of Presentation
      IPAM, UCLA, USA
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2007

    • Author(s)
      M. Kotani
    • Organizer
      Geometry Seminar
    • Place of Presentation
      Fudan University, Shaghai, China
    • Year and Date
      2007-12-07
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2007

    • Author(s)
      M. Kotani
    • Organizer
      Stochastic calculus on manifolds, graphs, and random structure
    • Place of Presentation
      Hausdorff Institute, Bonn, Germany
    • Year and Date
      2007-10-10
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2007

    • Author(s)
      M. Kotani
    • Organizer
      Conference on Geometric Analysis
    • Place of Presentation
      KIAS, Seoul, Korea
    • Related Report
      2007 Annual Research Report
  • [Presentation] Geometric Aspects of Random Walks on a crystal lattice2007

    • Author(s)
      M. Kotani
    • Organizer
      Analysis on Graphs and Fractals,
    • Place of Presentation
      Cardiff(UK)
    • Related Report
      2007 Annual Research Report
  • [Book] 文庫で読む科学2007

    • Author(s)
      小谷元子
    • Publisher
      岩波書店
    • Related Report
      2007 Annual Research Report

URL: 

Published: 2005-04-01   Modified: 2016-04-21  

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