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Asymptotic properties of heat kernels and their applications

Research Project

Project/Area Number 11640163
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionNagoya University

Principal Investigator

ICHIHARA Kanji  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00112293)

Co-Investigator(Kenkyū-buntansha) MIYAKE Masatake  Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70019496)
CHIYONOBU Taizo  Nagoya University, Graduate School of Mathematics, Research Associate, 大学院・多元数理科学研究科, 助手 (50197638)
OSADA Hirofumi  Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20177207)
SUGIURA Makoro  Ryukyu University, Focalty of Science, Associate Professor, 理学部, 助教授 (70252228)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsheat kernel / harmonic transform / curvature / pinned process / large deviation / covering space / Gibbs measure / Serpinski carpet / 負曲率リーマン多様体 / ラプラシアン / スペクトル / ブラウン運動 / Labachevsky空間 / pinned process
Research Abstract

Ichihara has studied about some relationships between the spectrums of a second order elliptic partial differential operator and the asymptotic properties of the associated heat kernel. First, the exact decay order of the heat kernels in time has been given for a class of complete, simply-connected, negatively curved Riemannian manifolds with asymptotically constant negative sectional curvature. In particular, it has been clarified that there exists an intimate connection between the decay part of polynomial order of the heat kernel and a term of the perturbation of constant negative curvature. Furthermore large deviation principles of the Donsker-Varadhan type have been established for a class of diffusion processes with strong transience property. It is well known that the usual large deviation for a Markov process does not hold when the infimum of the spectrums for its infinitesimal generator ×(-1) is positive. Ichihara has shown that it is still possible to get a nice large deviation principle so far as the pinned process is concerned if an appropriate rate function is chosen.
Osada has constructed a Gibbs measure on the path space under the existence of interaction potential and has proved its uniquness in a class of Gibbs measures. Making use of the theory of Dirichlet spaces, a diffusion process having the above Gibbs measure as an invariant probabilty measure has been constructed on the path space. Furthermore Osada has constructed diffusion processes on the Sierpinski Carpet and has given estimates for their transition densities.
Chiyonobu has carried out precise estimates for integrations of functionals in an infinite dimensional space, which appear in statistical mechanics.

Report

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

    (21 results)

All Other

All Publications (21 results)

  • [Publications] 市原完治: "Long time asymptotics for heat kernels on negatively curved Riemannian manifolds"Preprint series in Math.Sci.'School of Informatics and Sciences and Graduate School of Human Inf.,Nagoya Univ.. 3. 1-30 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 市原完治: "Large deviation for pinned covering diffusion"Preprint series in Math.Sci.,School of Informatics and Sciences and Graduate School of Human Inf.,Nagoya Univ.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 長田博文: "A family of diffusion processes on Sierpinski carpets"Probability Theory and Related Fields. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 長田博文: "Tagged particles of interacting Brownian motions with skew symmetric drifts"Monte Carlo Methods and Applications . (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 千代延大造: "A limit formula for a class of Gibbs measures with long range pair interactions"J.Math.Sci.Univ.Tokyo. 7. 463-486 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 三宅正武: "Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain"Annales Polonici Math.. LXXIV. 215-228 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kanji Ichihara: "Long time asymtotics for heat kemels on negatively curved Riemannian manifolds."Preprint series in Math.Sci., School of Informatics and Sciences and Graduate School of Human Informatics ; Nagoya University. 3. 1-30 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kanji Ichihara: "Flarnack's inequality on groups. Trends in contemporary infinite dimensional analysis and quantum probability (edited by L.Accarde et al)"Publ.of Italian school of east asian studies in Kyoto. 177-188 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kanji Ichihara: "Large deviation for pinned covering diffusion."Preprint series in Math.Sci., School of Informatics and Sciences and Graduate School of Human Informatics, Nagoya University. (to appear)..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hirofumi Ssada: "A family of diffusion processes on Sierpinski carpets."Probability Theory and Related Fields. (to appear)..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hirofumi Osada: "Togged particles of interacting Brownian motions with skew symmetric drifts."Monte Carlo Methods and Applications. (to appear)..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Taizo Chiyonobu: "A limit formula for a class of Gibbs measures with long range pair interactions."J.Math.Sci.Univ.Tokyo. 7. 463-486 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masatake Miyake: "Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain"Annales Polonici Math.. LXXIV. 215-228 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 市原完治: "Long time asymptotics for heat kernels on negatively curved Riemannian manifolds"Preprint series in Math.Sci.,School of Informatics. and Sciences and Graduate School of Human Inform.,Nagoyallniv.. 3. 1-30 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 市原完治: "Large deviation for pinned covering diffusion "Preprint series in Math.Sci.,School of Informatics and Dciences and Groduate School of Human-Intorm,NagoyaUniv. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 長田博文: "A family of diffusion processes on Sierpinski carpets"Probability Theory and Related Fields. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 長田博文: "Tagged particles of interacting Brownian motions with skew symmetric drifts"Monte Carlo Methods and Applications. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 千代延大造: "A limit formula for a class of Gibbs measures with long range pair interactions "J.Math.Sci.Univ.Tokyo. 7. 463-486 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 三宅正武: "Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain"Annales Polonici Math.. LXXIV. 215-228 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 市原完治: "Harnack's inequality on groups"special volume in honor of the 70th birthday of T.Hida,World Scientific Publishing Co.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 長田博文: "Gibbs measures relative to Brownian motion"Annals of Probability. 27. 1183-1207 (1999)

    • Related Report
      1999 Annual Research Report

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

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