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Study of path-behaviors of Biharmonic Psuedo Process with boundaries

Research Project

Project/Area Number 10640128
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 InstitutionTOKYO METOROPOLITAN UNIVERSITY

Principal Investigator

NISHIOKA Kunio  Tokyo Metropolitan Univ., Dept of Math., AP, 理学研究科, 助教授 (60101078)

Co-Investigator(Kenkyū-buntansha) HIRATA Masaki  Dept of Math. AP, 理学研究科, 助手 (70254141)
SUMI Naoya  Dept of Math. AP, 理学研究科, 助手 (50301411)
AOKI Nobuo  Dept of Math.. P, 理学研究科, 教授 (60087020)
SATO Sadao  Tokyo Electric Univ., Dept of Inform. AP, 理工学部, 助教授 (10170747)
SHIMURA Michio  Toho Univ., Dept of Math.., P (90015868)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsbiharmonic operator / monopole / dipole / absorbing boundary / sticky boundary / reflecting boundary / Poisson分布 / 無限次元力学系 / 弱い意味での確率積分 / Girsanovの公式 / 4階非線形偏微分方程式 / 確率解 / カオス
Research Abstract

-△ィイD12ィエD1 is called a biharamonic operator, which we often encounter in the theory of elasticity and fluid dynamics. Following pioneer works of Hochberg (1978) and Funaki (1979), many attempts are done to study the operator from the view point of probability theory.
We consider 'particles' whose 'transition probability density' is the fundamental solution of ∂ィイD2tィエD2u = -△ィイD12ィエD1u, though it may have negative values. We call orbits of these 'particles' biharmonic pseudo process, or BPP in short. Our aim is to study relations between analytic boundary conditions of -△ィイD12ィエD1 and behaviors of BPP.
By Nishioka (1997), we know that BPP behave as a mixture of two sorts of particles which we call 'monopoles' and 'dipoles'. So we need two boundary conditions to solve a boundary value problem related to -△ィイD12ィエD1, since each boundary condition controls behavior of each sort of particles at the boundary. We set up one of 'absorbing', 'sticking', and various 'reflecting' barriers on monopoles and dipoles one by one. Then their combinations solve various kinds of boundary value problems to -△ィイD12ィエD1.

Report

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

    (28 results)

All Other

All Publications (28 results)

  • [Publications] K. Nishioka: "The firsthitting time and place of a half-line by a biharmonic pseudo prosess"Japan. J. Math.. 23. 235-280 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Aoki, Y. M. Chung, K. Tomioka: "Topological properties of differentiable maps derived from 2-toral endomorphisms"Topolog. Appl.. 82. 105-123 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Sumi: "A class of differentiable toral maps which are topologically mixing"proc. Amer. Math. Soc.. 127. 915-924 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Sumi: "Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures"Trans. Amer. Math. Soc.. 351. 3373-3385 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Sato and M. Yor: "Computations of moments for discounted Brownian additive functionals"J. Math. Kyoto Univ.. 38. 475-486 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Hirata, B. Sassol, S. Vaienti: "Statistics of Return Times : A General Framework and New Applications"Comm. Math. Phys.. 206. 33-55 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 岡部恒春,西岡國雄,et al: "数学英和小辞典"講談社サイエンティフィック. 294 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Nishioka: "Boundary conditions to One-dimensional biharmonic pseudo process"preprint.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Nishioka , (with T. Okabe): "Handy Dictionary of Mathematics"Kodan-shya Scientific. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Sumi: "Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures"Trans. Amer. Math. Soc.. 351. 3373-3385 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Sumi: "A class of differentiable toral maps which are topologically mixing"Proc. Amer. Math. Soc. 127. 915-924 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Hirata (with B. Saussol and S. Vaienti): "Statistics of return times: A general frame work and new application"Comm. Math. Phys.. 206. 33-35 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Hirata (with B. Saussol and S. Vaienti): "On the distribution of the return times of the dynamical systems by piecewise monotone transformations"preprint.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Aoki: "Differentiable maps having hyperbolic sets"Topology Appl.. 82. 15-48 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Nishioka: "The first hitting time and place of a half-line by a biharmonic pseudo process"Japan. J. Math.. 23. 235-280 (1997)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Aoki, Y.M. Chung, K.Tomioka: "Topological properties of differentiable maps derived from 2-toral endomorphisms"Topolog. Appl.. 82. 105-123 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Sumi: "A class of differentiable toral maps which are topologically mixing"Proc. Amer. Math. Soc.. 127. 915-924 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Sumi: "Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures"Trans. Amer. Math. Soc.. 351. 3373-3385 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Sato and M.Yor: "Computations of moments for discounted Brownian additive functionals"J. Math. Kyoto Univ.. 38. 475-486 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Hirata, B.Sassol, S.Vaienti: "Statistics of Return Times : A General Framework and New Applications"Comm. Math. Phys.. 206. 33-55 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 岡部恒春,西岡國雄,et al: "数学英和小辞典"講談社サイエンティフィック. 294 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Nishioka: "The first hitting time and place of a half-line by a biharmonic pseudo process" Japan.J.Math.23. 235-280 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Nishioka: "Monopoles and dipoles in a biharmonic pseudo process" Proc.Japan Acad.72A. 47-50 (1996)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Sumi: "A class of toral differentiable maps which are topologically mixing" Proc.Amer.Math.Soc.to appear. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Sumi: "Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures" Trans.Amer.Math.Soc.to appear. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Sato and M.Yor: "Computations of moments for discounted Brownian additive functionals" J.Math.Kyoto.Univ.38. 475-486 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Shimura: "Exit probability of two-dimensional random walk from the quadrant" Proc.Japan Acad.to appear. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 岡部恒春,西岡國雄, et al: "数学英和小辞典" 講談社サイエンティフィック(出版予定), (1999)

    • Related Report
      1998 Annual Research Report

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

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