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Stochastic analysis on infinite dimensional spaces and its related differential operators

Research Project

Project/Area Number 26400134
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionOkayama University

Principal Investigator

KAWABI HIROSHI  岡山大学, 自然科学研究科, 教授 (80432904)

Co-Investigator(Renkei-kenkyūsha) HIROKAWA MASAO  広島大学, 大学院工学研究院, 教授 (70282788)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords確率解析 / 微分作用素 / 中心極限定理 / 確率偏微分方程式 / rough path理論 / 結晶格子 / ベキ零被覆グラフ / ランダムウォーク / 確率論 / 一意性定理 / 離散幾何解析 / ラフパス理論 / マリアヴァン解析
Outline of Final Research Achievements

I mainly studied uniqueness problems of differential operators and the corresponding stochastic dynamics on infinite dimensional spaces via stochastic analysis. In particular, I considered Dirichlet forms given by space-time quantum fields with interactions of exponential type, called exp(\phi)_{2}-measure, in Euclidean QFT, and proved strong uniqueness of Dirichlet operators of the corresponding Dirichlet operator. I also discussed the related singular stochastic partial differential equation. Besides, I studied central limit theorems for non-symmetric random walks on both crystal lattices and nilpotent covering graphs from a viewpoint of discrete geometric analysis, and discussed its application to rough path theory. Finally, I proved that the space of Gaussian measures on an abstract Wiener space becomes a Hilbert-Riemannian manifold with non-positive sectional curvatures.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (36 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (2 results) Journal Article (6 results) (of which Peer Reviewed: 2 results,  Acknowledgement Compliant: 2 results,  Open Access: 2 results) Presentation (28 results) (of which Int'l Joint Research: 9 results,  Invited: 18 results)

  • [Int'l Joint Research] University of Bonn/University of Bielefeld(ドイツ)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] University of Bonn/University of Bielefeld(Germany)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Long time asymptotics of non-symmetric random walks on crystal lattices2017

    • Author(s)
      Satoshi Ishiwata, Hiroshi Kawabi, Motoko Kotani
    • Journal Title

      Journal of Functional Analysis

      Volume: 272 Issue: 4 Pages: 1553-1624

    • DOI

      10.1016/j.jfa.2016.11.011

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Central limit theorems for non-symmetric random walks on nilpotent covering graphs2017

    • Author(s)
      石渡 聡, 河備 浩司, 難波 隆弥
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2030 Pages: 9-16

    • Related Report
      2016 Annual Research Report
    • Open Access
  • [Journal Article] From non-symmetric random walks on nilpotent covering graphs to rough paths via discrete geometric analysis2016

    • Author(s)
      Satoshi Ishiwata, Hiroshi Kawabi, Ryuya Namba
    • Journal Title

      Mathematisches Forschungsinstitut Oberwolfach Report

      Volume: 24 Pages: 1359-1362

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Uniqueness of Dirichlet forms related to stochastic quantization under exponential interaction in two-dimensional finite volume2015

    • Author(s)
      Hiroshi Kawabi
    • Journal Title

      Oberwolfach Reports

      Volume: 11 Pages: 2736-2739

    • Related Report
      2015 Research-status Report
  • [Journal Article] 結晶格子上の非対称ランダムウォークの中心極限定理2015

    • Author(s)
      石渡 聡、河備 浩司、小谷 元子
    • Journal Title

      数理解析研究所講究録

      Volume: 1952 Pages: 30-37

    • Related Report
      2015 Research-status Report
  • [Journal Article] An explicit effect of non-symmetric random walks on the triangular lattice2015

    • Author(s)
      Satoshi Ishiwata, Hiroshi Kawabi, Tsubasa Teruya
    • Journal Title

      Math. J. Okayama Univ.

      Volume: 57 Pages: 129-148

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space2017

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Workshop on Markov processes and its geometric aspects
    • Place of Presentation
      Fukuoka University
    • Year and Date
      2017-03-08
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] From non-symmetric random walks on nilpotent covering graphs to distorted Brownian rough paths via discrete geometric analysis2017

    • Author(s)
      河備 浩司
    • Organizer
      首都大学東京数理解析・幾何学合同セミナー
    • Place of Presentation
      首都大学東京大学院理工学研究科
    • Year and Date
      2017-02-01
    • Related Report
      2016 Annual Research Report
  • [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space2017

    • Author(s)
      河備 浩司
    • Organizer
      マルコフ過程とその周辺
    • Place of Presentation
      とくぎんトモニプラザ
    • Year and Date
      2017-01-07
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space2016

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Stochastic Partial Differential Equations and Related Fields
    • Place of Presentation
      University of Bielefeld
    • Year and Date
      2016-10-10
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 結晶格子上の非対称ランダムウォークの長時間漸近挙動2016

    • Author(s)
      河備 浩司
    • Organizer
      九州確率論セミナー
    • Place of Presentation
      九州大学大学院数理学研究院
    • Year and Date
      2016-07-30
    • Related Report
      2016 Annual Research Report
  • [Presentation] From non-symmetric random walks on nilpotent covering graphs to rough paths via discrete geometric analysis2016

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Rough Paths, Regularity Structures and Related Topics
    • Place of Presentation
      Mathematisches Forschunsinstitut Oberwolfach
    • Year and Date
      2016-05-06
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stationary phase for rough differential equations2016

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Okayama Spring Seminar on Stochastic Analysis
    • Place of Presentation
      岡山大学
    • Year and Date
      2016-03-31
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] ベキ零被覆グラフ上の非対称ランダムウォークの中心極限定理(I)2016

    • Author(s)
      河備 浩司
    • Organizer
      広島・岡山解析確率論セミナー
    • Place of Presentation
      広島大学
    • Year and Date
      2016-02-23
    • Related Report
      2015 Research-status Report
  • [Presentation] 結晶格子上の非対称ランダムウォークの長時間漸近挙動2016

    • Author(s)
      河備 浩司
    • Organizer
      首都大学東京数理解析セミナー
    • Place of Presentation
      首都大学東京
    • Year and Date
      2016-02-08
    • Related Report
      2015 Research-status Report
  • [Presentation] 結晶格子上の非対称ランダムウォークの長時間挙動2016

    • Author(s)
      河備 浩司
    • Organizer
      Kunitachi One-Day Symposium on the Recent Developments in the Mathematical Science
    • Place of Presentation
      一橋大学
    • Year and Date
      2016-02-06
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Rough Path理論入門:オーバービュー2016

    • Author(s)
      河備 浩司
    • Organizer
      大阪大学理学部数学教室談話会
    • Place of Presentation
      大阪大学
    • Year and Date
      2016-01-04
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space2015

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Workshop ``Geometry and Probability"
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-11-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 抽象Wiener空間上のGauss測度族のWasserstein幾何2015

    • Author(s)
      高津 飛鳥、河備 浩司
    • Organizer
      日本数学会幾何学分科会
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-15
    • Related Report
      2015 Research-status Report
  • [Presentation] Functional CLTs for non-symmetric random walks on crystal lattices2015

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      国際研究集会 ``Stochastic Analysis and Applications"
    • Place of Presentation
      東北大学
    • Year and Date
      2015-09-03
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Precise long time asymptotics of non-symmetric random walks on crystal lattices2015

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Summer School on Dirichlet Forms and Stochastic Analysis
    • Place of Presentation
      関西大学
    • Year and Date
      2015-08-25
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Long time asymptotics of non-symmetric random walks on crystal lattices2015

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      RIMS camp-style seminar ``New Trends in Stochastic Analysis"
    • Place of Presentation
      国際高等研究所
    • Year and Date
      2015-07-07
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Long time asymptotics of non-symmetric random walks on crystal lattices2015

    • Author(s)
      河備 浩司
    • Organizer
      阪大確率論セミナー
    • Place of Presentation
      大阪大学
    • Year and Date
      2015-06-30
    • Related Report
      2015 Research-status Report
  • [Presentation] Stationary phase for rough differential equations2015

    • Author(s)
      河備 浩司
    • Organizer
      立命館大学数理ファイナンスセミナー
    • Place of Presentation
      立命館大学
    • Year and Date
      2015-03-05
    • Related Report
      2014 Research-status Report
  • [Presentation] Long time behavior of non-symmetric random walks on crystal lattices2015

    • Author(s)
      河備 浩司
    • Organizer
      東京確率論セミナー
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-01-26
    • Related Report
      2014 Research-status Report
  • [Presentation] 結晶格子上の非対称ランダムウォークの長時間漸近挙動2014

    • Author(s)
      河備 浩司
    • Organizer
      日本数学会北海道支部講演会・支部総会 特別講演
    • Place of Presentation
      北海道大学
    • Year and Date
      2014-12-03
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Long time behavior of non-symmetric random walks on crystal lattices2014

    • Author(s)
      河備 浩司
    • Organizer
      奈良女子大学理学部数学教室談話会
    • Place of Presentation
      奈良女子大学
    • Year and Date
      2014-11-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Long time behavior of non-symmetric random walks on crystal lattices and related variational problems2014

    • Author(s)
      河備 浩司
    • Organizer
      研究集会「The Ninth Meeting on Probability and PDE」
    • Place of Presentation
      津田塾大学
    • Year and Date
      2014-11-13
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Uniqueness of Dirichlet forms related to stochastic quantization under exponential interaction in two-dimensional finite volume2014

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Bielefeld Stochastic Afternoon
    • Place of Presentation
      University of Bielefeld
    • Year and Date
      2014-10-29
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Long time behavior of non-symmetric random walks on crystal lattices2014

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Workshop on Dirichlet Form and Stochastic Analysis
    • Place of Presentation
      TU-Dresden
    • Year and Date
      2014-10-27
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Uniqueness of Dirichlet forms related to stochastic quantization under exponential interaction in two-dimensional finite volume2014

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Workshop: Dirichlet Form Theory and its Applications
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach
    • Year and Date
      2014-10-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] A biased introduction to stochastic quantization equations and related topics2014

    • Author(s)
      河備 浩司
    • Organizer
      RIMS研究集会「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
      2014-09-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Stationary phase for rough differential equations2014

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Swansea-Japanese workshop on Stochastic Analysis
    • Place of Presentation
      Swansea University
    • Year and Date
      2014-09-09
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Weak convergence of laws of non-symmetric random walks on crystal lattices2014

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      UK-Japan Stochastic Analysis School
    • Place of Presentation
      University of Warwick
    • Year and Date
      2014-09-05
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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