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Stochastic analysis on stochastic generalized Cahn-Hilliard equations

Research Project

Project/Area Number 20K03627
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionShinshu University

Principal Investigator

謝 賓  信州大学, 学術研究院理学系, 教授 (50510038)

Project Period (FY) 2020-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2023: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords保存系 / 良設定問題 / 後退確率微分方程式 / 特異係数 / 輸送型ノイズ / 特異型 / 定量的 / anisotropic / nonlinear growth / Backward SDEs / one-side monotonicity / Navier-Stokes / 確率偏微分方程式 / 確率Cahn-Hilliard 方程式 / 絶対連続性 / 非退化 / 大域解 / 定常解 / KPZ / Malliavin解析 / Cahn-Hilliard equation / Random enviroment / パラコントロール / quasilinear / 緊密性 / 確率解析 / 密度関数
Outline of Research at the Start

確率Cahn-Hilliard方程式は保存則をもつ非線形確率偏微分方程式の例の一つとして注目され,様々な研究分野において盛んに研究されている.本研究は,無限次元空間上の確率解析理論に基づき,確率Cahn-Hilliard方程式や確率Cahn-Hilliard/Allen-Cahn 方程式を含むより一般の確率Cahn-Hilliard方程式に関する多彩な確率的性質についての研究を行い,非線形確率偏微分方程式論構築を目指すことである.特に,一般化された確率Cahn-Hilliard方程式に付随するマルコフ過程の長時間にわたる振る舞いについて研究を行う.

Outline of Annual Research Achievements

本年度では,保存系の確率偏微分方程式の解に確率的性質,非線形確率偏微分方程式の解の長時間のふるまい,臨界条件を満たす後退確率微分方程式の良設定問題等の研究に取り組んだ.
質量保存の法則にしたがう乗法的ノイズが加えられた確率偏微分方程式に対して,特異な係数を持つ場合の良設定問題等を調べた.このような確率微分方程式はさまざまな分野に応用されている.良設定問題に関する研究を行うためにノイズの構成に工夫した.また,昨年度に引き続きエルゴード性に関わる解の絶対連続やFreidlin-Wentzell型大偏差原理等の研究に取り組んできた.さらに,特異型の確率Cahn-Hilliard 方程の解の構成にも力を入れた.しかし,予測した結論に至らなかったので,引き続き研究する予定である.
非線形確率偏微分方程式の解の長時間のふるまいに関する研究を行った.適切な条件の下で,輸送型ノイズが加わった2次元非線形確率偏微分方程式が決定的な偏微分方程式への定量的な収束率を調べることをし,確率解析的観点より部分的な研究成果をあげた.
ある種の臨界条件の下で,後退確率微分方程式についての良設定問題の研究を行い,解の存在,一意性を中心に調べた.後退確率微分方程式は数学をはじめ,数理ファイナンス,物理学等に幅広く用いられている.本研究課題に取り組む際,確率偏微分方程式への応用を念頭に置き,終端値に積分可能性の意味で臨界条件を課すことで,後退確率微分方程式の解の存在と一意性が成り立つことを示した.また,このような後退確率微分方程式が確率偏微分方程式にどのように応用できるかを調べ始めた.

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

今年度は,引き続き一般された確率Cahn-Hilliard方程式についてのエルゴード性,解の絶対連続性や,特性係数と乗法的ホワイトノイズを持つ保存系の確率偏微分方程式の良設定問題を考えた上,特異型の保存系の確率偏微分方程式の解の構成を考察した.また,非線形確率偏微分方程式の解のふるまいおよび後退確率微分方程式についての研究も進めた.これらの研究により,今年度も研究を進めることができた.一方,本研究課題の重要なテーマである保存系の確率偏微分方程式についての大偏差原理の研究と特異型の確率偏微分方程式の研究は進展したものの,部分的な成果を得るにとどまってしまった.そのため,総合して「やや遅れている」とした.

Strategy for Future Research Activity

本年度の研究状況に基づき,質量保存の法則にしたがう確率偏微分方程式に関する大偏差原理等の研究を優先的に進める予定である.また,来年度は確率偏微分方程式に関する最新の動向に注目し,保存則系の確率微分方程式や輸送型ノイズが加わった2次元非線形確率偏微分方程式についての研究も積極的に取り組む予定である.

Report

(4 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (26 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (3 results) Journal Article (6 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 6 results,  Open Access: 2 results) Presentation (16 results) (of which Int'l Joint Research: 13 results,  Invited: 15 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of Arizona(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Centrale Supelec(フランス)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of Arizona(米国)

    • Related Report
      2020 Research-status Report
  • [Journal Article] One dimensional BSDEs with critical integrable terminal values and infinite time horizon2024

    • Author(s)
      Adachi Kazuki、Xie Bin
    • Journal Title

      Systems & Control Letters

      Volume: 187 Pages: 105779-105779

    • DOI

      10.1016/j.sysconle.2024.105779

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs2022

    • Author(s)
      Funaki Tadahisa、Xie Bin
    • Journal Title

      Stochastics and Partial Differential Equations: Analysis and Computations

      Volume: 10 Issue: 3 Pages: 858-897

    • DOI

      10.1007/s40072-022-00243-z

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications2022

    • Author(s)
      Xie Bin
    • Journal Title

      Stochastics and Partial Differential Equations: Analysis and Computations

      Volume: 10 Issue: 2 Pages: 419-445

    • DOI

      10.1007/s40072-021-00203-z

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotics of PDE in random environment by paracontrolled calculus2021

    • Author(s)
      Funaki Tadahisa、Hoshino Masato、Sethuraman Sunder、Xie Bin
    • Journal Title

      Annales de l'Institut Henri Poincar?, Probabilit?s et Statistiques

      Volume: 57 Issue: 3 Pages: 1702-1735

    • DOI

      10.1214/20-aihp1129

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises2020

    • Author(s)
      L. Goudenege, B. Xie
    • Journal Title

      Journal of Differential Equations

      Volume: 269 Issue: 9 Pages: 6988-7014

    • DOI

      10.1016/j.jde.2020.04.047

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A note on random strings with values in a bounded convex domain2020

    • Author(s)
      B. Xie
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B79 Pages: 199-213

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Global solvability of singular quasilinear stochastic PDEs2024

    • Author(s)
      B. Xie
    • Organizer
      Interacting Particle Systems and Stochastic Analysis
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 6.Global solvability and stationary solutions of singular quasilinear stochastic PDEs2023

    • Author(s)
      B. Xie
    • Organizer
      Probaiblity seminar, Academy of Mathematics and Systems Science, CAS
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scalar backward stochastic differential equations2023

    • Author(s)
      B. Xie
    • Organizer
      The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global solvability and stationary solutions of singular quasilinear SPDEs2023

    • Author(s)
      B. Xie
    • Organizer
      Stochastic Analysis, RIMS, Kyoto University
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Reflected stochastic partial differential equations driven by space-time white noise2023

    • Author(s)
      B. Xie
    • Organizer
      The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic partial differential equations with reflecting walls2023

    • Author(s)
      B. Xie
    • Organizer
      Workshop on Reaction-Diffusion Equations and Related Stochastic Topics, Waseda University
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global solvability and stationary solutions of singular quasilinear stochastic PDEs2022

    • Author(s)
      Bin XIE
    • Organizer
      Open Japanese-German conference on stochastic analysis and applications
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Study on the asymptotic behavior of a quasilinear PDE in random. environment2022

    • Author(s)
      Bin XIE
    • Organizer
      Seminar at Department of Fundamental Mathematics, Beihang University
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Convergence to stationary solutions in singular quasilinear stochastic PDEs2021

    • Author(s)
      Bin Xie
    • Organizer
      The 19th Symposium Stochastic Analysis on Large Scale Interacting Systems
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Global solvability of a quasilinear SPDE2021

    • Author(s)
      Bin Xie
    • Organizer
      2021年度 確率解析シンポジウム
    • Related Report
      2021 Research-status Report
  • [Presentation] Solvability and convergence of solutions corresponding to a quasilinear SPDE in random environment2021

    • Author(s)
      Bin Xie
    • Organizer
      PDEs and Probability Theory - beyond boundaries
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior of a quasilinear PDE in random environment by paracontrolled calculus2021

    • Author(s)
      Bin Xie
    • Organizer
      The 2nd Workshop on Stochastic Dynamics
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptoticity of a quasilinear PDE in random environment2020

    • Author(s)
      B. Xie
    • Organizer
      Webinar on stochastic analysis, CAS, China
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The asymptotic behavior of a quasilinear PDE in random environment2020

    • Author(s)
      B. Xie
    • Organizer
      APWW Workshop, Tsinghua University, China
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Ergodicity of stochastic Cahn-Hilliard equation with the logarithmic free energy2020

    • Author(s)
      B. Xie
    • Organizer
      Webinar on Probability Seminar, Hunan University of Technology, China
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The asymptotic behavior of a quasilinear PDE in random environment2020

    • Author(s)
      B. Xie
    • Organizer
      ReaDiNet 2020: An online conference on mathematical biology, France
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Shinshu summer workshop on probability theory2023

    • Related Report
      2023 Research-status Report

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Published: 2020-04-28   Modified: 2024-12-25  

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