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Stochastic Processes and Stochastic Analysis on Disordered Media

Research Project

Project/Area Number 22H00099
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionWaseda University

Principal Investigator

熊谷 隆  早稲田大学, 理工学術院, 教授 (90234509)

Co-Investigator(Kenkyū-buntansha) 楠岡 誠一郎  京都大学, 理学研究科, 教授 (20646814)
Croydon David  京都大学, 数理解析研究所, 准教授 (50824182)
福島 竜輝  筑波大学, 数理物質系, 教授 (60527886)
木上 淳  京都大学, 情報学研究科, 教授 (90202035)
Project Period (FY) 2022-04-01 – 2027-03-31
Project Status Granted (Fiscal Year 2024)
Budget Amount *help
¥41,470,000 (Direct Cost: ¥31,900,000、Indirect Cost: ¥9,570,000)
Fiscal Year 2025: ¥10,140,000 (Direct Cost: ¥7,800,000、Indirect Cost: ¥2,340,000)
Fiscal Year 2024: ¥7,410,000 (Direct Cost: ¥5,700,000、Indirect Cost: ¥1,710,000)
Fiscal Year 2023: ¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000)
Fiscal Year 2022: ¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000)
Keywords確率論 / 複雑系 / 数理物理 / 解析学 / ポテンシャル論
Outline of Research at the Start

確率論的ポテンシャル論を基礎にした汎用性の高い手法を編み出すことで、フラクタルやランダム媒質を典型例とする複雑な系の上の確率過程、確率解析を発展させ、関連分野への応用を進める。共形不変なスケール極限を持つ確率モデルの上のダイナミックスの解析や非局所作用素のhomogenization(均質化)理論の発展など、現代確率論の大きな潮流であるrandom geometryの研究対象となる空間上の解析学を展開することを目的とする。複雑な系の動的理論を総合的に扱うことで、ランダム媒質が作り出す様々な新奇現象を体系的に捉え、複雑な系の上の確率過程と確率解析の研究に新展開を与える。

Outline of Annual Research Achievements

1。熊谷は、S. Andres氏、D. Croydon氏との共同研究で、異常拡散現象を引き起こすBouchaudのtrap modelと呼ばれる、離散ランダムウォークの時間変更に関するモデルについて、熱核の対角評価の振動が(ランダム媒質に関して)確率1で起こることを、熱核に関する重複対数の定理を示すことで証明した。振動の振幅の度合いはtrap modelのパラメータに依存するが、この共同研究ではパラメータに応じた振幅を詳細に解明した。
2。楠岡は、特異確率偏微分方程式と従来の確率微分方程式の理論における細かな違いが解の一意性やマルコフ性に与える影響について、具体例をもって考察したものを論文としてまとめた。また、指数確率量子場測度の研究について、査読者によるコメントを踏まえ、平均化による近似に対しても同様の構成の手法が適用できることを論文に追記した。
3。木上は、距離空間の上に local regular Dirichlet 形式が得られるための十分条件を用いて、多角形をベースとした局所対称性を持つ自己相似集合上に拡散過程の構成できるための十分条件を得た。
4。福島とCroydonは、S. Junk氏との共同研究で、前年度に引き続きMott variable range hoppingというランダム媒質中のランダムウォークの研究を行った。局在が起こる点の間の有効抵抗がheavy-tailedの場合に劣拡散的なスケール極限を決定した研究に続いて、blockingがextremalな場合の局在が起こることを証明した。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

初年度から代表者、分担者が積極的に国際共同研究を展開し、着実に成果を積み上げているため。コロナによる移動制限が大きく緩和し、22年度の夏には海外渡航も比較的容易になり、代表者、分担者が海外渡航を行い、また海外の有力研究者も来日して、対面による密度の濃い議論を展開することができた。

Strategy for Future Research Activity

引き続き、対面による国際共同研究を本研究の軸としていく。
研究の目的に記した3つの課題を中心に、代表者、分担者がそれぞれの持ち味を生かした研究を展開する。特に、飛躍型確率過程(非局所作用素)のランダム媒質への応用については、いくつかのグループによる競合も起こっているので、これまでの研究の蓄積を武器にこの1-2年でより精密な解析を目指す。

Report

(2 results)
  • 2022 Comments on the Screening Results   Annual Research Report
  • Research Products

    (26 results)

All 2023 2022 Other

All Int'l Joint Research (5 results) Journal Article (10 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 10 results,  Open Access: 4 results) Presentation (8 results) (of which Int'l Joint Research: 8 results,  Invited: 8 results) Remarks (1 results) Funded Workshop (2 results)

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

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Fujian Normal University/Shianghai Jiao Tong University(中国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Seoul National University(韓国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Cornell University/University of Washington(米国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Vietnam Academy of Sci. and Tech.(ベトナム)

    • Related Report
      2022 Annual Research Report
  • [Journal Article] Scaling limit for random walk on the range of random walk in four dimensions2023

    • Author(s)
      D. A. Croydon and D. Shiraishi
    • Journal Title

      Annales de l'institut Henri Poincare (B) Probabilites et Statistiques

      Volume: 59 Issue: 1 Pages: 166-184

    • DOI

      10.1214/22-aihp1243

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Spectral dimension of simple random walk on a long-range percolation cluster2022

    • Author(s)
      V. H. Can, D. A. Croydon and T. Kumagai
    • Journal Title

      Electronic Journal of Probability

      Volume: 27 Issue: none Pages: 1-37

    • DOI

      10.1214/22-ejp783

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Heat kernel bounds for nonlocal operators with singular kernels2022

    • Author(s)
      M. Kassmann, K.-Y. Kim and T. Kumagai
    • Journal Title

      J. Math. Pures Appl

      Volume: 164 Pages: 1-26

    • DOI

      10.1016/j.matpur.2022.05.017

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Long range random walks and associated geometries on groups of polynomial growth2022

    • Author(s)
      Chen Zhen-Qing、Kumagai Takashi、Saloff-Coste Laurent、Wang Jian、Zheng Tianyi
    • Journal Title

      Annales de l'Institut Fourier

      Volume: 72 Issue: 3 Pages: 1249-1304

    • DOI

      10.5802/aif.3515

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Heat kernels for reflected diffusions with jumps on inner uniform domains2022

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

      Trans. Amer. Math. Soc

      Volume: 375 Issue: 10 Pages: 6797-6841

    • DOI

      10.1090/tran/8678

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Heat kernel estimates for general symmetric pure jump Dirichlet forms2022

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

      Ann. Sc. Norm. Super. Pisa Cl. Sci.

      Volume: ー Pages: 6797-6841

    • DOI

      10.2422/2036-2145.202001_012

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Two-Sided Heat Kernel Estimates for Symmetric Diffusion Processes with Jumps: Recent Results2022

    • Author(s)
      Chen Zhen-Qing、Kim Panki、Kumagai Takashi、Wang Jian
    • Journal Title

      Dirichlet forms and related topics, 63--83, Springer Proc. Math. Stat.

      Volume: 394 Pages: 63-83

    • DOI

      10.1007/978-981-19-4672-1_5

    • ISBN
      9789811946714, 9789811946721
    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stochastic quantization associated with the exp(Φ)_{2}-quantum field model driven by space- time white noise on the torus in the full L_{1}-regime2022

    • Author(s)
      Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
    • Journal Title

      Probability Theory and Related Fields

      Volume: - Issue: 1-2 Pages: 391-447

    • DOI

      10.1007/s00440-022-01126-z

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Number of paths in oriented percolation as zero temperature limit of directed polymer2022

    • Author(s)
      Fukushima Ryoki、Junk Stefan
    • Journal Title

      Probability Theory and Related Fields

      Volume: 183 Issue: 3-4 Pages: 1119-1151

    • DOI

      10.1007/s00440-022-01130-3

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Biased random walk on supercritical percolation: anomalous fluctuations in the ballistic regime2022

    • Author(s)
      Adam M. Bowditch and David A. Croydon
    • Journal Title

      Electronic Journal of Probability

      Volume: 27 Issue: none Pages: 1-22

    • DOI

      10.1214/22-ejp794

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Spectral dimension of simple random walk on a long-range percolation cluste2022

    • Author(s)
      Takashi Kumagai
    • Organizer
      Ninth Bielefeld-SNU Joint Workshop in Mathematics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic homogenization of non-symmetric jump-type processes with drifts2022

    • Author(s)
      Takashi Kumagai
    • Organizer
      From Dirichlet Forms to Wasserstein Geometry, HCM Conference
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Anomalous diffusions and time fractional differential equations2022

    • Author(s)
      Takashi Kumagai
    • Organizer
      Geometry, Stochastics and Dynamics UK-Japan at Imperial College
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic homogenization of non-symmetric jump-type processes2022

    • Author(s)
      Takashi Kumagai
    • Organizer
      Random media and large deviations Courant Institute
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction of a non-Gaussian and rotation-invariant $\Phi^4$-measure and associated flow on $\mathbb{R}^3$ through stochastic quantization2022

    • Author(s)
      Seiichiro Kusuoka
    • Organizer
      Geometry, Stochastics \& Dynamics -Celebrating 20 years of UK-Japan Winter Schools
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Conductive homogeneity of compact metric spaces and construction of p-energy2022

    • Author(s)
      Jun Kigami
    • Organizer
      2nd Joint Congress of Mathematics of AMS-EMS-SMF
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Number of paths in oriented percolation as zero temperature limit of directed polymer2022

    • Author(s)
      Ryoki Fukushima
    • Organizer
      Open German-Japanese conference on stochastic analysis and applications, University of Munster
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Sub-diffusive scaling regimes for one-dimensional Mott variable-range hopping.2022

    • Author(s)
      David Croydon
    • Organizer
      Mathematisches Forschungsinstitut Oberwolfach
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] Takashi Kumagai's home page

    • URL

      https://t-kumagai.w.waseda.jp

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] BIRS Workshop (Banff, Canada), Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities2022

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] Oberwolfach Workshop (Germany), Heat Kernels, Stochastic Processes and Functional Inequalities2022

    • Related Report
      2022 Annual Research Report

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Published: 2022-04-19   Modified: 2025-04-17  

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