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Multilateral research of stochastic analysis in infinite dimensional spaces

Research Project

Project/Area Number 11440045
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

SHIGEKAWA Ichiro  Graduate School of Science, Kyoto University, Professor, 大学院・理学(系)研究科(研究院), 教授 (00127234)

Co-Investigator(Kenkyū-buntansha) HINO Masanori  Graduate School of Informations, Kyoto University, Lecturer, 大学院・情報学研究科, 講師 (40303888)
NOMURA Takaaki  Graduate School of Science, Kyoto University, Associate Professor, 大学院・理学(系)研究科(研究院), 助教授 (30135511)
YOSHIDA Nobuo  Graduate School of Science, Kyoto Univesity, Lecturer, 大学院・理学(系)研究科(研究院), 講師 (40240303)
AIDA Shigeki  Graduate School of Engineering, Associarte Prefessor scinece, Osaka University, 大学院・基礎工学研究科, 助教授 (90222455)
UEKI Naomasa  Graduate School of Human and Enviornmental Studies, Kyoto University, Associate Professor, 大学院・人間環境学研究科, 助教授 (80211069)
高信 敏  金沢大学, 理学部, 助教授 (40197124)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥5,500,000 (Direct Cost: ¥5,500,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1999: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordssemigroup domination / interturning property / square field operator / Littlewood-Paley ineguality / multiplier / second fundamental form / Hodge-Kodaira opoerator / logarithmic Sobolev ineguality / Littlewood-Paley不等式 / ブラウン運動 / 半群の比較 / Schrodinger作用素 / 生成作用素の定義域 / Riesz変換 / Brown運動 / Dirichlet形式 / 反射壁Brown運動 / 基本解 / 準楕円性
Research Abstract

We have accomplished the research of the semigroup domination and the intertwining property of semigroups. The semigroup domination stands for that |T^^→_tu|【less than or equal】T_t|u| holds for semigroup T^^→_t and T_t. Here u are supposed to be a vector valued function. Typical example is a differential form on a Riemannian manifold. The characterization in terms of the generator is known. We give here a sufficient condition in terms of square field operator. Crucial assumptions are the positivity and the locality. Intertwining property is the following property: for two generator L, L^^→, it holds that DL = L^^→D + R. We give necessary and sufficient conditions in terms of the semigroup and the resolvents. Our aim has been to reconstruct the.Bakry-Emery T_2 theory but we have succeeded in including diffusion processes with boundary condition and it gives a generalization of Bakry-Emery T_2 theory.
For application, we can make use of them to show the Littlewood-Paley inequality and the L^p multiplier theorem. In fact, we have considered the Brownian motion on an Riemannian manifold with boundary and show the Littlewood-Paley inequality for it under the assumption of positivity of the second fundamental form of the boundary. Making use of this, we can show the L^p boundedness of the Riesz transformation. L^p multiplier theorem is to give an sufficient condition on φ and A so that φ(A) is a bounded operator on L_p. Stein showed this for a generator of a symmetric diffusion process with φ being of Laplace transform type. We have shown that the same result holds for the Hodge-Kodaira operator on a Riemannian manifold.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (29 results)

All Other

All Publications (29 results)

  • [Publications] Ichiro Shigekawa: "Semigroup domination on a Riemannian manifold with boundary"Acta Applicandae Math.. 63. 385-410 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "The domein of a generator and the intertwining property"Stochastics in Finite and Infinite Dimensions. 401-410 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "Littlewood-Paley inequality for a diffusion sastisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary"Osaka J. Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nobuo Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probab. Th. Rel. Fielsds. 115. 1-40 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Masanori Hino: "Exponential decay of positivity preserving semigroups on L^p"Osaka J. Math.. 37. 603-624 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Naomasa Ueki: "Asymptotic expansion of stochastic osillatory integrals with rotation invariance"Ann. Inst. H. Poincare Probab. Statist.. 35. 417-457 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "Semigroup domination on a Riemannian manifold with boundary"Acta Applicandae Math.. 63. 385-410 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "The domain of a generator and the intertwining property"Stockastics in Finite.and Infinite Dimensions,ed. by T. Hida et al, Birkhaeauser, Boston. 401-410 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemanhian manifold with boundary"Osaka J. Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nobuo Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probab. Th. Rel. Fields 1. 151. 1-40 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Masanori Hino: "Exponential decay of podsitivity preserving semi-groups on L^p"Osaka Journal of Mathematics. 37. 603-624 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Naomasa Ueki: "Asymptotic expansion of stochastic oscillatory integrals with rotation invariance"Ann. Inst. H. Poincare Probab. Statist. 35. 417-457 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "Semigroup domination on a Riemannian manifold with boundary"Acta Applicandae Math.. 63. 385-410 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Ichiro Shigekawa: "The domain of a generator and the intertwining property"Stochastics in Finite and Infinite Dimensions, ed. by T. Hida et al. Birkhauser, Boston. 401-410 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Ichiro Shigekawa: "Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary"Osaka J. Math.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Nobuo Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probab. Th. Rel. Fields. 115. 1-40 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masanori Hino: "Exponential decay of positivity preserving semigroups on $L^∧p$, Osaka Journal of Mathematics, Vol.37(2000), 603-624."Osaka Journal of Mathematics. 37. 603-624 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Naomasa Ueki: "Asymptotic expansion of stochastic oscillatory integrals with rotation invariance"Ann. Inst. H. Poincare Probab. Statist.. 35. 417-457 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] Ichiro Shigekawa: "The domain of a generator and the intertwining property"Stochastics in cimte and infinite dimensions. 401-410 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Ichiro Shigekawa: "Semigroup domination on a Riemannian manifold with boundary (to appear)"Acta Applicandae Mathematicae. (200)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shigeki Aida: "Stochastic analysis on loop spaces"Sugaku exposrtions. 13・2. 197-214 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Nomasa Ueki: "Simple examples of Lifschitz tails in Gaussian random magnetic fields"Annales Henri Pomcare. 1. 473-498 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Nobuo Yoshida: "Application of log : Soberly inequality to the stochastic dynamics of unbounded spin systems on the lattice"Journal of Functional Analysis. 173. 74-102 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Ichiro Shigekawa: "Semigroup domination on a Riemannian manifold boundary"To appear in the special volume of Takeyuki Hida.

    • Related Report
      1999 Annual Research Report
  • [Publications] Masanori Hino: "Eeponential decay of positivity preserving semigrops on L^p"To appear in Osaka Journal of Mathematics.

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Sugita,Satoshi Takanobu: "A limit theorem for Weyl transformation in infinite dimensional torus and central limit theorem for correlated multiple Wiener integrals"To appear in Journal of Math. Sci. Univ. Tokyo.

    • Related Report
      1999 Annual Research Report
  • [Publications] Shigeki Aida: "On the irreducibility of Dirichlet forms on domains in infinite dimensional spaces"To appear in Osaka Journal of Mathematics.

    • Related Report
      1999 Annual Research Report
  • [Publications] Naomasa Ueki: "Asymototic expansion of stochastic osoillatory integrals with rotation in varianc"Ann. Inst. H. Poincare Probab. Statist.. 35. 417-457 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Nobuyuki Ikeda,Hiroyuki Matsumoto: "Brownian motion on the Hyperbolic plane and Selberg trace formula"Journal of Funct. Anal. 163. 63-110 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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