Various properties relative to intermittency of space-time fractional SPDEs

• Project/Area Number: 16K05197
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: XIE Bin 信州大学, 学術研究院理学系, 准教授 (50510038)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 確率偏微分方程式 / 無限次元Harnack不等式 / Intermittency / 確率カーン＝ヒリアード方程式 / Hypercontractivity / 特異性 / 確率Chan-Hilliard方程式 / 漸近的強Feller性 / kernel estimate / Harnack inequality / Log-Harnack inequality / 障害物 / 指数的可積分性 / intermittency / 時空ホウイトノイズ / dimension-free / Harnack 不等式 / Brownian sheet / 比較定理 / 確率論

Outline of Final Research Achievements

In this research project, we mainly devoted to various important behaviors relative to the intermittency of singular stochastic partial differential equations. In particular, we proved the intermittent property of the solution of stochastic partial differential equations driven by inhomogeneous space-time white noises. We also studied the ergodicity of some singular stochastic partial differential equations, which include the stochastic Cahn-Hilliard equations with logarithmic potentials as an important example.

Academic Significance and Societal Importance of the Research Achievements

非適切(ill-posed)な確率偏微分方程式の解の性質を考察すべき問題が多く残されている．本研究ではある種の特異性を持つ確率偏微分方程式の解の長時間にわたる振る舞い，特にintermittencyや半群の超圧縮性の研究を行ったので，非適切な確率偏微分方程式についてのエルゴード理論の確立に寄与する意義がある．また，得られた研究成果は多くの応用分野での応用が期待できる．さらに，共著の書籍「確率偏微分方程式」を出版することにより，確率偏微分方程式論の研究の普及や発展に貢献する．

Research Products

• [Int'l Joint Research] CNRS FR3487(フランス)
• [Int'l Joint Research] University of Kansas(米国)
• [Int'l Joint Research] Chinese Academy of Sciences(中国)
• [Journal Article] A note on random strings with values in a bounded convex domain (2020)
  • Author(s): B. Xie
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B79 Pages: 199-213
  • Peer Reviewed
• [Journal Article] Comparison theorem and correlation for stochastic heat equations driven by Levy space-time white noises (2019)
  • Author(s): M. Niu and B. Xie
  • Journal Title: Discrete & Continuous Dynamical Systems - B Volume: 24 Issue: 7 Pages: 2989-3009
  • DOI: 10.3934/dcdsb.2018296
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hypercontractivity for space-time white noise driven SPDEs with reflection (2019)
  • Author(s): Xie Bin
  • Journal Title: Journal of Differential Equations Volume: 266 Issue: 9 Pages: 5254-5277
  • DOI: 10.1016/j.jde.2018.10.022
  • Peer Reviewed
• [Journal Article] Wang's Harnack inequalities for space-time white noises driven SPDEs with two reflecting walls and their applications (2019)
  • Author(s): M.Niu, B. Xie
  • Journal Title: J. Math. Anal. Appl. Volume: 469 Issue: 2 Pages: 568-593
  • DOI: 10.1016/j.jmaa.2018.09.029
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Intermittency for stochastic partial differential equations driven by strongly inhomogeneous space-time white noises (2018)
  • Author(s): Bin XIE
  • Journal Title: Journal of Differential Equations Volume: 264 Issue: 2 Pages: 1050-1079
  • DOI: 10.1016/j.jde.2017.09.028
  • Peer Reviewed
• [Journal Article] Harnack inequalities for the stochastic partial differential equations with reflecting walls and their application (2018)
  • Author(s): 謝 賓
  • Journal Title: 共同研究リポート４０２ Volume: 402 Pages: 104-111
• [Journal Article] Intermittency and noise excitation for stochastic heat equations driven by various noise (2017)
  • Author(s): Bin XIE
  • Journal Title: 統計数理研究所共同研究リポート Volume: 385
  • Acknowledgement Compliant
• [Journal Article] Some effects of the noise intensity upon non-linear stochastic heat equations on [0,1] (2016)
  • Author(s): Bin XIE
  • Journal Title: Stochastic Process. Appl. Volume: 126
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] Some properties of stochastic parabolic partial differential equations (2019)
  • Author(s): 謝賓
  • Organizer: Probability Seminar, Jiangsu Normal University, China
  • Invited
• [Presentation] Ergodic properties of stochastic partial differential equations with reflecting walls (2019)
  • Author(s): 謝賓
  • Organizer: 2019 Spring Probability Workshop, Taiwan
  • Int'l Joint Research / Invited
• [Presentation] Dimension-free Harnack inequalities for stochastic Cahn-Hilliard equation with logarithmic nonlinearity (2019)
  • Author(s): 謝賓
  • Organizer: Japanese-German Open Conference on Stochastic Analysis 2019
  • Int'l Joint Research / Invited
• [Presentation] Ergodic properties of stochastic partial differential equations with reflections (2019)
  • Author(s): 謝賓
  • Organizer: SEMINAIRE-GROUPE DE TRAVAIL PROBABILITES - STATISTIQUES - CONTROLE, France
  • Invited
• [Presentation] Ergodic property of stochastic Cahn-Hilliard equation with the logarithmic free energy (2019)
  • Author(s): 謝賓
  • Organizer: ReaDinet 2019: Mathematical Analysis for Biology and Ecology, France
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic behavior of reflected SPDEs with singular potentials driven by additive noises (2019)
  • Author(s): 謝賓
  • Organizer: 2019年度 確率解析シンポジウム
• [Presentation] Intermittency and noise excitation for stochastic partial differential equations (2019)
  • Author(s): B. Xie
  • Organizer: Probability Seminiar, CAS
  • Invited
• [Presentation] Dimension-free Harnack inequalities for the Markov semigroup generated by reflected SPDEs (2018)
  • Author(s): B. Xie
  • Organizer: 17th workshop on Stochastic Analysis on Large Scale Interacting Systems
  • Invited
• [Presentation] On the reflected stochastic partial differential equation driven by space-time white noise (2018)
  • Author(s): B. Xie
  • Organizer: Seminar at Geometric Measure and related fields, BUAA
  • Invited
• [Presentation] On the space-time white noise dirven SPDE with reflection (2018)
  • Author(s): B. Xie
  • Organizer: Workshop on Stochastic systems: their analysis, geometry and perturbation
  • Invited
• [Presentation] Harnack inequalities for reflected SPDEs and their applications (2018)
  • Author(s): B. Xie
  • Organizer: The 12th AIMS International Conference on Dynamical Systems, Differential Equations, and Applications
  • Int'l Joint Research / Invited
• [Presentation] Hypercontractive property for reflected SPDE driven by space-time white noise (2018)
  • Author(s): B. Xie
  • Organizer: 2018 IMS Asia Pacific Rim Meeting in Singapore
  • Int'l Joint Research / Invited
• [Presentation] Harnack inequalities for the stochastic partial differential equations with reflecting walls and their applications (2017)
  • Author(s): 謝 賓
  • Organizer: 共同研究集会「無限分解可能過程に関連する諸問題」
• [Presentation] The fluctuations for the evolutional models of two-dimensional Young diagrams (2017)
  • Author(s): 謝 賓
  • Organizer: 確率・ 統計・ 行列ワークショップ 松本 2017
  • Invited
• [Presentation] Dimension-free Harnack inequalities for the stochastic partial differential equation with reflection (2017)
  • Author(s): Bin XIE
  • Organizer: 16th International symposium "Stochastic Analysis on Large Scale Interacting Systems
  • Int'l Joint Research / Invited
• [Presentation] Log-Harnack inequality for reflected stochastic partial differential equations with multiplicative noise (2017)
  • Author(s): Bin XIE
  • Organizer: 2017 IMS-China International Conference on Statistics and Probability
  • Int'l Joint Research / Invited
• [Presentation] Intermittency of Stochastic Partial Differential Equations (2016)
  • Author(s): Bin XIE
  • Organizer: The 10th ICSA international conference
  • Place of Presentation: Shanghai Jiao Tong University, China
  • Year and Date: 2016-12-19
  • Int'l Joint Research / Invited
• [Presentation] Intermittency and noise excitation for stochastic heat equations driven by various noises (2016)
  • Author(s): 謝 賓
  • Organizer: 共同研究集会「無限分解可能過程に関連する諸問題」
  • Place of Presentation: 統計数理研究所
  • Year and Date: 2016-12-08
• [Presentation] Intermittency for stochastic partial differential equations driven by inhomogeneous space-time noises (2016)
  • Author(s): Bin XIE
  • Organizer: 15th workshop on Stochastic Analysis on Large Scale Interacting Systems
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2016-11-02
  • Int'l Joint Research / Invited
• [Presentation] Intermittent property of parabolic stochastic partial differential equations (2016)
  • Author(s): 謝 賓
  • Organizer: 東京確率セミナー
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2016-07-25
  • Invited
• [Presentation] On intermittency of stochastic heat equation with Dirichlet boundary condition (2016)
  • Author(s): Bin XIE
  • Organizer: Workshop on stochastic differential equations and stochastic processes
  • Place of Presentation: Nankai University，China
  • Year and Date: 2016-04-27
  • Int'l Joint Research / Invited
• [Book] 確率偏微分方程式 (2019)
  • Author(s): 舟木 直久，乙部 厳己，謝 賓
  • Total Pages: 335
  • Publisher: 岩波書店
  • ISBN: 9784000298254
• [Funded Workshop] Workshop on “Stochastic partial differential equations and related topics" (2018)