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Potential theory for non-local operators

Research Project

Project/Area Number 14F04021
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

熊谷 隆  京都大学, 数理解析研究所, 教授 (90234509)

Co-Investigator(Kenkyū-buntansha) WANG JIAN  京都大学, 数理解析研究所, 外国人特別研究員
WANG Jian  京都大学, 数理解析研究所, 外国人特別研究員
Project Period (FY) 2014-04-25 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2016: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2015: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2014: ¥300,000 (Direct Cost: ¥300,000)
Keywords飛躍型確率過程 / 非局所作用素 / ポテンシャル論
Outline of Annual Research Achievements

今年度も京都大学数理解析研究所においてWang氏と定期的にインフォーマルなセミナーを行い、また海外から研究者を招聘してWang氏を交えて議論を行うことで、以下の進展が得られた。
1)広い範疇のsubordinatorから作られる一般化された分数冪時間微分熱方程式の弱解の存在と一意性、そして解の確率論的表現について解析した。さらにこのような分数冪時間微分熱方程式の熱核の精密な上下評価を導出した。この結果は、Wang氏とZ.Q. Chen氏、P. Kim氏との共著論文にまとめ、雑誌に掲載予定である。
2)長距離の飛躍を持つランダムコンダクタンスモデルで、特にコンダクタンスの形が安定過程型の場合に、コンダクタンスのある種の可積分条件の下で、媒質のランダム性に関して確率1でスケール極限が安定過程になることを証明した。この研究は、Wang氏と積み上げて来たstable-like processの熱核評価、ハルナック不等式の安定性の研究を発展させる方向であり、ランダムな非局所作用素のhomogenizationの問題にも繋がる重要性を持つ。現在、当該研究の成果をWang氏とX. Chen氏との共著論文として執筆中である。
3)確率過程のthick pointの大偏差原理に関するDembo氏、Wang氏との共同研究は、技術的問題点が依然完全にはクリアできていない状況であるが、係数が周期関数かつ連続関数である場合には、divergence formの枠内で議論が進みうるという手応えを得ている。さらに議論を進め、共同研究の形にまとめることを目標としている。
Wang氏とは今後も継続的に共同研究を行うことで合意を得ている。本研究で得られた安定性理論をさらに発展させて行きたいと考えている。

Research Progress Status

29年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

29年度が最終年度であるため、記入しない。

Report

(4 results)
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (20 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (5 results) Journal Article (6 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 6 results,  Acknowledgement Compliant: 1 results) Presentation (8 results) (of which Int'l Joint Research: 4 results,  Invited: 8 results) Remarks (1 results)

  • [Int'l Joint Research] Stanford University/University of Washington(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Fujian Normal University/Shianghai Jiao Tong University(China)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Seoul National University(韓国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Fujian Normal University(中国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] University of Washington(米国)

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Heat kernel estimates for time fractional equations2018

    • Author(s)
      Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang
    • Journal Title

      Forum. Math.

      Volume: 掲載予定

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Elliptic Harnack inequalities for symmetric non-local Dirichlet forms2018

    • Author(s)
      Z.-Q. Chen, T. Kumagai and J. Wang
    • Journal Title

      J. Math. Pures Appl.

      Volume: 掲載予定

    • NAID

      120007147052

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Laws of the iterated logarithm for symmetric jump processes2017

    • Author(s)
      Panki Kim, Takashi Kumagai and Jian Wang,
    • Journal Title

      Bernoulli

      Volume: 掲載予定

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] L^p-Wasserstein distance for stochastic differential equations driven by Levy processes2016

    • Author(s)
      Jian Wang
    • Journal Title

      Bernoulli

      Volume: 22 Issue: 3 Pages: 1598-1616

    • DOI

      10.3150/15-bej705

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Functional inequalities for stable-like Dirichlet forms.2015

    • Author(s)
      Feng-Yu Wang and Jian Wang
    • Journal Title

      J. Theoret. Probab.

      Volume: 28 Issue: 2 Pages: 423-448

    • DOI

      10.1007/s10959-013-0500-5

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Linear evolution equations with cylindrical Levy noise: gradient estimates and exponential ergodicity.2015

    • Author(s)
      Jian Wang
    • Journal Title

      Stoch. Anal. Appl.

      Volume: 33 Issue: 2 Pages: 306-330

    • DOI

      10.1080/07362994.2014.989330

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Presentation] Heat kernel estimates for time fractional equations2017

    • Author(s)
      T. Kumagai
    • Organizer
      13th Workshop on Markov Processes and Related Topics at Wuhan
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Heat kernel estimates and Harnack inequalities for symmetric non-local Dirichlet forms and their applications2017

    • Author(s)
      Jian Wang
    • Organizer
      Stochastic Analysis and its Applications
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Intrinsic ultracontractivity of symmetric pure-jump processes on unbounded domains2017

    • Author(s)
      Jian Wang
    • Organizer
      Dirichlet forms and their geometry
    • Place of Presentation
      東北大学
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2016

    • Author(s)
      Takashi Kumagai
    • Organizer
      SPDE and Related Fields
    • Place of Presentation
      Bielefeld大学
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      International Conference on Stochastic Analysis and Related Topics
    • Place of Presentation
      Wuhan University
    • Year and Date
      2015-08-05
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Stability of parabolic Harnack Inequality for Jump Processes on Metric Measure Spaces2015

    • Author(s)
      Jian Wang
    • Organizer
      11th Workshop on Markov Processes and Related Topics
    • Place of Presentation
      Shanghai Jiao Tong University
    • Year and Date
      2015-07-28
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Detailed heat kernel estimates and laws of the iterated logarithm2015

    • Author(s)
      熊谷隆
    • Organizer
      神戸Workshop 格子上の確率解析とその周辺
    • Place of Presentation
      神戸大学
    • Year and Date
      2015-03-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Coupling by Reflection and Holder Regularity for Non-Local Operators of Variable Order2014

    • Author(s)
      Jian Wang
    • Organizer
      WORKSHOP ON DIRICHLET FORMS & STOCHASTIC ANALYSIS 2014
    • Place of Presentation
      TU-Dresden, ドイツ
    • Year and Date
      2014-10-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Remarks] Takashi Kumagai's home page

    • URL

      http://www.kurims.kyoto-u.ac.jp/~kumagai/kumpre.html

    • Related Report
      2017 Annual Research Report 2016 Annual Research Report 2015 Annual Research Report 2014 Annual Research Report

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Published: 2015-01-22   Modified: 2024-03-26  

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