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Developments of Analysis on Fractals

Research Project

Project/Area Number 10440029
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

HINO Masanod  Kyoto University, Graduate School of Informatics, Ledurer, 情報学研究科, 講師 (40303888)

Co-Investigator(Kenkyū-buntansha) TAKAHASHI Yoichiro  Kyoto University, Res.Inst.Math.Sci., Professor, 数理解析研究所, 教授 (20033889)
KIGAMI Jun  Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (90202035)
KUMAGAI Takashi  Kyoto Univ., Graduate School of Informatics, Associate Professor, 情報学研究科, 助教授 (90234509)
SHINODA Masato  Nara Women's University, Fac.of Sci., Lecturer, 理学部, 講師 (50271044)
MATSUMOTO Hiroyuki  Nagoya Univ., School of Info.and Sci., Associate Professor, 情報文化学部, 助教授 (00190538)
杉浦 誠  名古屋大学, 多元数理科学研究科, 助手 (70252228)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥7,800,000 (Direct Cost: ¥7,800,000)
Fiscal Year 2000: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1999: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1998: ¥2,400,000 (Direct Cost: ¥2,400,000)
KeywordsFractal / Diffusion Process / Stochastic Process / Multi-fractal / Hausdorff dimension / Heat kernel / Limit theorem / Besov space / 固有値分布 / ランダムウォーク
Research Abstract

We have established the following results throughout this project.
1. On spectra of self-adjoint operators on fractals
We study the short time asymptotics for heat kernels when the self-similarity of the measure and that of the diffusion process are different. In this case, the asymptotics highly depend on the initial points and there is a multi-fractality. We compute the Hausdorff dimension and the paper will appear. Further, we define a family of quasi-distances and show that the Hausdorff dimension can be expressed in a simple way using some special quasi-distance.
2. On stochastic processes on random fractals
(1) We construct a diffusion process and study its heat kernel properties on a homogeneous random Sierpinski carpet.
(2) We study a short time heat kernel asymptotics for a diffusion process on a random recursive Sierpinski gasket, which does not have spatial symmetries.
These results are in our paper that have already appeared.
3. Stochastic analysis on fractals
(1) We construct a diffusion process on a (Euclidean) space where countable numbers of disordered media (such as fractals) are embedded.
We apply a trace theory of Besov spaces for the proof. The paper has already appeared and we are now working on large deviations for the process.
(2) Concerning the "stochastic analysis for stochastic processes with rough paths" studied by T.Lyons, we have not obtained a new result so far. It is one of the future problem to study the detailed properties of the stochastic differential equations established by him.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (58 results)

All Other

All Publications (58 results)

  • [Publications] M.Hino: "Convergence of non-symmetric forms."Journal of Mathematics of Kyoto University. 38. 329-341 (1998)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Hino: "On short time asymptotic behavior of some symmetric diffusions on general state spaces."Potential Analysis. (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] M.Hino: "On the space of BV functions and a related stochastic calculus in infinite dimensions."Journal of Functional Analysis. (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] T.Kumagai: "Transition density estimates for diffusion processes on post critically finite self-similar fractals."Proc.London Math.Soc.. 78. 431-458 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets."J.Math.Soc.Japan. 159. 373-408 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Brownian motion penetrating fractals-An application of the trace theorem of Besov spaces-."J.Funct.Anal.. 170. 69-92 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gasket."Stoch.Proc.Their Appl.. 92. 61-85 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions."Maths.Proc.Cam.Phil.Soc.. (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] Y.Takahashi: "Random point fields associated with certain Fredholm determinants."Proceeding of the Second International ISAAC Congress,. (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] J.Kigami: "Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets."J.Funct.Anal.. 156. 170-198 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets."J.Math.Sci.Univ.Tokyo. 7. 27-33 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Green's function on fractals."Fractals. 8. 385-402 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: " On Bougerol and Dufresne's identities for exponential Brownian functionals."Proc.Japan Acad.Ser.A. 74. 152-155 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "Brownian motion on the hyperbolic plane and Selberg trace. formula."J.Func.Anal.. 163. 63-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part I : A time inversion approach."Nagoya Math.J.. 159. 125-166 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space."J.Math.Soc.Japan. 52. 269-292 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Shinoda: "Lower estimate for the critical line of contact processes."The Transactions of Materials Research Society. 159.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 高橋陽一郎: "数学-複雑系の科学と現代思想"青土社. (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Cambridge Univ.Press"Analysis on fractals(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] M.Hino: "Convergence of non-symmetric forms."Journal of Mathematics of Kyoto University. 38. 329-341 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Hino: "Exponential decay of positivity preserving semigroups 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
      2000 Final Research Report Summary
  • [Publications] M.Hino: "On short time asymptotic behavior of some symmetric diffusions on general state spaces."Potential Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Hino: "On the space of BV functions and a related stochastic calculus in infinite dimensions."Journal of Functional Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Transition density estimates for diffusion processes on post critically finite self-similar fractals."Proc.London Math.Soc.. 78. 431-458 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets."J.Math.Soc.Japan. 52. 373-408 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Brownian motion penetrating fractals-An application of the trace theorem of Besov spaces-."J.Funct.Anal. 170. 69-92 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gasket."Stoch.Proc.Their Appl.. 92. 61-85 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions."Maths.Proc.Cam.Phil.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Takahashi: "Random point fields associated with certain Fredholm determinants."Proceeding of the Second International ISAAC Congress. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets."J.Funct.Anal.. 156. 170-198 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets."J.Math.Sci.Univ.Tokyo. 7. 27-33 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Green's function on fractals."Fractals. 8. 385-402 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "On Bougerol and Dufresne's identities for exponential Brownian functionals."Proc.Japan Acad.Ser.A. 74. 152-155 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "Brownian motion on the hyperbolic plane and Selberg trace formula."J.Func.Anal.. 163. 63-110 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part I : A time inversion approach."Nagoya Math.J.. 159. 125-166 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Matsumoto: "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space."J.Math.Soc.Japan. 52. 269-292 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Shinoda: "Lower estimate for the critical line of contact processes."The Transactions of Materials Research Society of Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Takahashi: "Mathematics-disordered media and contemporary thought (in Japanese)"Seido-sha. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Kigami: "Cambridge Univ.Press, (to appear)"Analysis on fractals..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Hino: "On short time asymptotic behavior of some symmetric diffusions on general state spaces."Potential Analysis. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kumagai: "Stochastic processes on fractals and related topics."Sugaku Expositions,Amer.Math.Sac.. 3. 55-71 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kumagai: "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gasket."Stoch.Proc.Their Appl.. 92. 61-85 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] J.Kigami: "Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets."J.Math.Sci.Univ.Tokyo. 7. 27-33 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] J.Kigami: "Green's function on fractals."Fractals. 8. 385-402 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Matsumoto: "An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part I : A time inversion approach."Nagoya Math.J.. 159. 125-166 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] J.Kigami: "Cambridge Univ.Press"Analysis on fractals(発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Hino: "Exponential decay of positivity preserving semigroups on U^p"Osaka Journal of Math.. (発売予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] J.Kigami: "Markov property of kusuoka-Zhou's Divichlet forms on self-similar sets"Tokyo Journal of Math.. (発売予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Kumagai: "Transition density estimates for diffusion processes on post critically finite self-similar fractals"Proc. Lodon Math. Soc.. 78-3. 431-458 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Kumagai: "Large deviations for Brownian motion on the Sierpinski gasket"Stoch, Proc.Their Appl.. 85. 225-235 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Mathumoto: "Brownian motion on the Hyperblic plane and Selberg trace formula"J. Funct. Analy.. 163. 63-110 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Matyumoto: "A version of Pitman's 2M-X theorem for geometric Brownian motions"C. R. Acad. Sci. Paris : Serie I. 328. 1067-1074 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Matsumoto: "On Beugerol and Dufreshe's identities for exponeutial Brownian functionals." Proceedings of Japan Academy Ser A. 74. 152-155 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Matsumoto: "Brownian motion on the Hyperbdic plane and Selberg trace formula." Journal of Functional Analysis. (発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Hino: "Convergence of non-symmetric forms." Journal of Math. Kyoto University. 35. 717-734 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Hino: "Existence of invariant measures for diffusion processes on a Wiener space" Osaka Journal of Math.35. 717-734 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] J.Kigami: "Distributions of localized eigenvalues of Laplacians on Post Critically Finite self-similar sets." Journal of Functional Analysis. 156. 170-198 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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