Stochastic analysis and the Hormander type operators on infinite dimensional spaces

• Project/Area Number: 20740076
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Okayama University
• Principal Investigator: KAWABI Hiroshi Okayama University, 大学院・自然科学研究科, 准教授 (80432904)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 確率解析 / マリアヴァン解析 / Rough Path理論 / 微分作用素 / 一意性問題 / Riesz変換 / 経路空間 / Gibbs測度 / 一意性定理 / rough path理論

Research Abstract

I studied some properties on the the Hormander type operators on infinite dimensional spaces by applying Malliavin calculus, rough path analysis, SPDEs, and Dirichlet form. In particular, I proved strong uniqueness of Dirichlet operators for Gibbs measures on an infinite volume path space, and constructed a unique strong solution for the corresponding SPDE via the Dirichlet form approach. Besides, I studied precise asymptotic behavior of some functional integrations for a class of infinite dimensional stochastic processes which include the Brownian motion on loop spaces.

Research Products

• [Journal Article] Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): S.Albeverio, H.Kawabi, M.Rockner
  • Journal Title: Isaac Newton Institute Preprint Series NI10013-SPD
• [Journal Article] Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): Sergio Albeverio, Hiroshi Kawabi, Michael Roeckner
  • Journal Title: Issac Newton Institute Preprint Series Volume: NI10013-SPD
• [Journal Article] Topics on diffusion semigroups on a path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Journal Title: Proceedings of RIMS Workshop on Stochastic Analysis and Applications RIMS Kokyuroku Bessatsu B6 Pages: 153-165
  • Peer Reviewed
• [Journal Article] On the Laplace-type asymptotics and the stochastic Taylor expansion for Ito functional of Brownian rough paths (2008)
  • Author(s): Y.Inahama, H.Kawabi
  • Journal Title: Proceedings of RIMS Workshop on Stochastic Analysis and Applications RIMS Kokyuroku Bessatsu B6 Pages: 139-152
  • Peer Reviewed
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2011)
  • Author(s): 河備浩司
  • Organizer: 筑波大学解析セミナー
  • Place of Presentation: 筑波大学数理物質科学研究科
  • Year and Date: 2011-02-23
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2011)
  • Author(s): 河備浩司
  • Organizer: 筑波大学解析セミナー
  • Place of Presentation: 筑波大学大学院数理物質科学研究科
  • Year and Date: 2011-02-23
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2011)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「Markov過程と関連する話題」
  • Place of Presentation: 関西大学システム理工学部
  • Year and Date: 2011-01-29
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2011)
  • Author(s): H.Kawabi
  • Organizer: 科研費シンポジウム「Probability and PDE 2011」
  • Place of Presentation: 広島大学大学院工学研究科
  • Year and Date: 2011-01-12
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2011)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「Probabiity and PDE 2011」
  • Place of Presentation: 広島大学大学院工学研究科
  • Year and Date: 2011-01-12
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「確率解析とその周辺」
  • Place of Presentation: 岡山大学理学部
  • Year and Date: 2010-11-13
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): 河備浩司
  • Organizer: 東京確率論セミナー
  • Place of Presentation: 東京工業大学大学院理工学研究科
  • Year and Date: 2010-11-01
• [Presentation] Strong uniqueness of diffusions to Gibba measures on a path space with exponential interactions (2010)
  • Author(s): 河備浩司
  • Organizer: 東京確率論セミナー
  • Place of Presentation: 東京工業大学大学院理工学研究科
  • Year and Date: 2010-11-01
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「第13回広島解析セミナー」
  • Place of Presentation: 広島大学大学院工学研究科
  • Year and Date: 2010-08-23
• [Presentation] Strong uniqueness of diffusions to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「第13回広島応用解析セミナー」
  • Place of Presentation: 広島大学大学院工学研究科
  • Year and Date: 2010-08-23
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): H.Kawabi
  • Organizer: Welsh Probability Seminar
  • Place of Presentation: Mathematics Department, Swansea University
  • Year and Date: 2010-05-27
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: Welsh Probability Seminar
  • Place of Presentation: Swansea University
  • Year and Date: 2010-05-27
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): H.Kawabi
  • Organizer: Stochastic Analysis Seminar
  • Place of Presentation: School of Mathematics, Loughborough University
  • Year and Date: 2010-05-12
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: Stochastic Analysis Seminar
  • Place of Presentation: Loughborough University
  • Year and Date: 2010-05-12
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): H.Kawabi
  • Organizer: (臨時) Stochastic Analysis Seminar
  • Place of Presentation: School of Mathematics, University of Manchester
  • Year and Date: 2010-05-10
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: (臨時)Stochastic Analysis Seminar
  • Place of Presentation: University of Manchester
  • Year and Date: 2010-05-10
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): H.Kawabi
  • Organizer: 岡山解析・確率セミナー兼数理物理セミナー
  • Place of Presentation: 岡山大学理学部
  • Year and Date: 2010-04-16
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2010)
  • Author(s): H.Kawabi
  • Organizer: Workshop : Rough Paths, SPDEs and Related Topics
  • Place of Presentation: Isaac Newton Institute for Mathematical Sciences, University of Cambridge
  • Year and Date: 2010-04-06
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: Workshop : Rough Paths, SPDEs and Related Topics
  • Place of Presentation: Isaac Newton Institute, University of Cambridge
  • Year and Date: 2010-04-06
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): H.Kawabi
  • Organizer: Stochastic Analysis Seminar
  • Place of Presentation: Mathematical Institute, University of Warwick
  • Year and Date: 2010-03-17
• [Presentation] On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: Stochastic Analysis Seminar
  • Place of Presentation: University of Warwick
  • Year and Date: 2010-03-17
• [Presentation] On Dirichlet operators of of exp(\phi)_{1}-quantum fields in infinite volume (2010)
  • Author(s): H.Kawabi
  • Organizer: First CIRM-HCM joint meeting: Stochastic Analysis, SPDEs, Particle Systems, Optimal Transport, Levico Terme
  • Place of Presentation: Trento
  • Year and Date: 2010-01-26
• [Presentation] On Dirichlet operators of exp(phi)_{1}-quantum fields in infinite volume (2010)
  • Author(s): Hiroshi Kawabi
  • Organizer: The 1st CIRM-HCM Joint Meeting on "Stochastic analysis, SPDE' s, Particle systems, Optimal transport"
  • Place of Presentation: Levico Terme(Trento)
  • Year and Date: 2010-01-26
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2009)
  • Author(s): H.Kawabi
  • Organizer: Oberseminar Stochastische Prozesse
  • Place of Presentation: Mathematical Institute, University of Leipzig
  • Year and Date: 2009-11-05
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2009)
  • Author(s): Hiroshi Kawabi
  • Organizer: Oberseminar Stochastische Prozesse
  • Place of Presentation: Universitat Leipzig
  • Year and Date: 2009-11-05
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2009)
  • Author(s): H.Kawabi
  • Organizer: Forschungsseminar Stochastische Analysis und Stochastik der Finanzmarkte
  • Place of Presentation: Institute of Mathematics, TU-Berlin
  • Year and Date: 2009-11-02
• [Presentation] Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths (2009)
  • Author(s): Hiroshi Kawabi
  • Organizer: Forschungsseminar Stochastische Analysis und Stochastik der Finanzmarkte
  • Place of Presentation: Technische Universitat Berlin
  • Year and Date: 2009-11-02
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2009)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「確率論とPDE」
  • Place of Presentation: 広島大学大学院工学研究科
  • Year and Date: 2009-01-20
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2009)
  • Author(s): Hiroshi Kawabi
  • Organizer: 研究集会「確率論とPDE」
  • Place of Presentation: 広島大学
  • Year and Date: 2009-01-20
• [Presentation] Riesz transforms and stoanalysis (A survey talk) (2008)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「仙台放物型・楕円型方程式研究集会」
  • Place of Presentation: 東北大学大学院理学研究科
  • Year and Date: 2008-11-08
• [Presentation] Riesz transforms and Stochastic Analysis (A survey talk) (2008)
  • Author(s): 河備浩司
  • Organizer: 科研費シンポジウム「仙台放物型・楕円型方程式研究集会」
  • Place of Presentation: 東北大学
  • Year and Date: 2008-11-07
• [Presentation] The parabolic Harnack inequality and related topics on a path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Organizer: Open Seminar on Stochastics
  • Place of Presentation: Department of Mathematics, TU-Darmstadt
  • Year and Date: 2008-11-04
• [Presentation] The parabolic Harnack inequality and related topics on a path space with Gibbs measures (2008)
  • Author(s): Hiroshi Kawabi
  • Organizer: (臨時)Stochastic analysis semimar
  • Place of Presentation: TU-Darmstadt
  • Year and Date: 2008-11-04
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Organizer: Oberseminar Stochastik, SFB611
  • Place of Presentation: Institute of Applied Mathematics, University of Bonn
  • Year and Date: 2008-10-23
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Organizer: Bielefeld Stochastic Afternoon, SFB701
  • Place of Presentation: Faculty of Mathematics, University of Bielefeld
  • Year and Date: 2008-10-22
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2008)
  • Author(s): Hiroshi Kawabi
  • Organizer: Oberseminar Stochastik, SFB 611
  • Place of Presentation: University of Bonn
  • Year and Date: 2008-10-22
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2008)
  • Author(s): Hiroshi Kawabi
  • Organizer: Bielefeld Stochastic Afternoon, SFB 701
  • Place of Presentation: University of Bielefeld
  • Year and Date: 2008-10-21
• [Presentation] Riesz transforms associated with diffusion operators on a Path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Organizer: 臨時Stochastic Analysis Seminar
  • Place of Presentation: Oxford-Man Institute, University of Oxford
  • Year and Date: 2008-08-27
• [Presentation] Riesz transforms associated with diffusion operators on a path space with Gibbs measures (2008)
  • Author(s): Hiroshi Kawabi
  • Organizer: (臨時)Stochastic analysis semimar
  • Place of Presentation: University of Oxford
  • Year and Date: 2008-08-27
• [Presentation] Riesz transforms on a path space with Gibbs measures (2008)
  • Author(s): H.Kawabi
  • Organizer: The 1^<st> MSJ-SI (Mathematical Society of Japan, Seasonal Institute) Probabilistic Approach to Geometry (Contributed Talk)
  • Place of Presentation: 京都大学時計台記念館国際交流ホール
  • Year and Date: 2008-08-01
• [Presentation] Riesz transforms on a path space with Gibbs measures (2008)
  • Author(s): Hiroshi Kawabi
  • Organizer: The 1-st MSJ-SI, Probabilistic Approach to Geometry
  • Place of Presentation: 東京大学
  • Year and Date: 2008-08-01