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Malliavin calculus for stochastic flows

Research Project

Project/Area Number 15540133
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionOsaka City University

Principal Investigator

KOMATSU Takashi  Osaka City University, Science, Professor, 大学院・理学研究科, 教授 (80047365)

Co-Investigator(Kenkyū-buntansha) KAMAE Tetsuro  Osaka City University, Science, Professor, 大学院・理学研究科, 教授 (80047258)
TAKEUCHI Atsushi  Osaka City Unversity, Science, Res.Assoc., 大学院・理学研究科, 助手 (30336755)
YOSHIDA Msamichi  Osaka City University, Science, Lecturer, 大学院・理学研究科, 講師 (60264793)
DATEYAMA Masahito  Osaka City University, Science, Lecturer, 大学院・理学研究科, 講師 (10163718)
藤井 準二  大阪市立大学, 大学院・理学研究科, 講師 (60117968)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsMalliavin calculus / stochastic flow / stochastic differential equation / infinite particle system / hypoellipticity / Hormander condition / Hibert space / 相互作用
Research Abstract

Certain infinite particle systems with interactions are defined by stochastic differential equations (SDE's) on infinite dimensional spaces. It would be an interesting problem to study on the partial hypoellipticity of infinitesimal generators on infinite dimensional spaces associated with those SDE's. The main work of the present research is the study on the partial hypoellipticity vi the Malliavin calculus for stochastic flows on infinite dimensional spaces defined by SDE's.
Let a_0(x, x^^-) be an R^N-valued smooth function on R^NxR^N, ν(dθ) be a finite measure on a discrete space θ, β_t=(β^θ_t) be a Wiener process, and let μ(dυ) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u∈R^d,
dx^u(t)=(∫a_0(x^u(t), x^υ(t))μ(dυ))dt+∫ν(dθ)a_θ(x^u(t))οdβ^θ_t.
Assume that x^u(0) is smooth in u and the process x(t)=(x^u(t)) takes its values in the Hilbert space H=(L^2(R^d, β(R^d),μ))^N. Let π: H→R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H:
A_0=∫∫μ(du)μ(dυ)a_0(x^u, x^υ)・∂/(∂x^u), A_θ=∫μ(du)a_θ(x^u)・∂/(∂x^u)・
The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (20 results)

All 2004 2003 Other

All Journal Article (18 results) Publications (2 results)

  • [Journal Article] On the partial hypoellipticity of SDE's on Hilbert spaces2004

    • Author(s)
      T.Komatsu
    • Journal Title

      Theory of Stochastic Processes 10(26)

      Pages: 63-73

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generalized Hormander theorem for non-local operators2004

    • Author(s)
      T.komatsu, A.Takeuchi
    • Journal Title

      Recent Developments in Stochastic Analysis and Related topics, World Scientific

      Pages: 234-245

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Two dimensional world with 2k maximal pattern complexity2004

    • Author(s)
      T.Kamae, X.Yu-Mei
    • Journal Title

      Osaka J. Math. 41

      Pages: 257-265

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the partial hypoellipticity of SDE's on Hilbert space2004

    • Author(s)
      T.Komatsu
    • Journal Title

      Theory of Stochastic Processes 10

      Pages: 63-73

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generalized Hormander theorem for non-local operators2004

    • Author(s)
      T.Komatsu, A.Takeuchi
    • Journal Title

      Recent Developments in Stochastic Analysis and Related Topics (Proceedings of the first Sino-German Conference on Stochastic Analysis), World Scientific

      Pages: 234-245

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Two dimensional word with 2k maximal pattern complexity2004

    • Author(s)
      T.Kamae, X.Yu-Mei
    • Journal Title

      Osaka J.Math. 41

      Pages: 257-265

    • NAID

      120005986938

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the partial hypoellipticity of SDE's on Hirbert spaces2004

    • Author(s)
      T.Komatsu
    • Journal Title

      Theory of Stochastic Processes 10(26)

      Pages: 63-73

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Two dimensional word with 2K maximal pattern complexity2004

    • Author(s)
      T.Kamae, X.Yu-Mei
    • Journal Title

      Osaka J.Math. 41

      Pages: 257-265

    • NAID

      120005986938

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Malliavin calculus for Stochastic flows with interaction on Hilbert sp.2003

    • Author(s)
      T.Komatsu
    • Journal Title

      Acta Appl. Math. 78

      Pages: 223-232

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Malliavin calculus for stochastic flows with interaction on Hilbert spaces2003

    • Author(s)
      T.Komatsu
    • Journal Title

      Acta Appl.Math. 78

      Pages: 223-232

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the pinced circle model and the absence of wandering domains for a topological polynomial

    • Author(s)
      M.Yoshida
    • Journal Title

      Indagationes Mathematicae (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Maximal pattern complexity over ? letters

    • Author(s)
      T.Kamae, R.Hui
    • Journal Title

      European J. Combinatorics (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the pinched circle model and the absence of wandering domains for a topological polynomial

    • Author(s)
      M.Yoshida
    • Journal Title

      Indagationes Mathematicae (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Maximal pattern complexity over l letters

    • Author(s)
      T.Kamae, R.Hui
    • Journal Title

      European J.Combinatorics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Numeration systems, fractals and stochastic processes

    • Author(s)
      T.Kamae
    • Journal Title

      Israel J.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generalized Hormander theorem for non-local operators

    • Author(s)
      T.Komatsu, A.Takeuchi
    • Journal Title

      Recent Developments in Stochastic Analysis and Related Topics (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the pinched circle model and the absence of wandering domains for a topological polynomial

    • Author(s)
      M.Yoshida
    • Journal Title

      Indagationes Mathematicae (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Maximal pattern complexity over L letters

    • Author(s)
      T.kamae, R.Hui
    • Journal Title

      European J.Combinatorics (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] T.Komatsu: "On the Malliavin calculus for SDE's on Hilbert spaces"Acta Applicandae Mathematicae. 78. 223-232 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Yoshida: "On the pinched circle model and the absence of wandering domains..."Indagationes Mathematicae. (to appear).

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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