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MATEMATICAL ANALYSIS OF INFINITE DIMENSIONAL STOCHASTIC MODELS

Research Project

Project/Area Number 13640103
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 Institute of Technology

Principal Investigator

SHIGA Tokuzo  TOKYO INSTITUTE OF TECHNOLOGY, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (60025418)

Co-Investigator(Kenkyū-buntansha) 角 大輝  東京工業大学, 大学院・理工学研究科, 助手 (40313324)
SHIRAI Tomoyuki  Kanazawa University, Faculty of Science, Associate Professor, 理学部, 助教授 (70302932)
MORITA Takehiko  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00192782)
NOMURA Yuji  TOKYO INSTITUTE OF TECHNOLOGY, Graduate School of Science and Engineering, Assistant Professor, 大学院・理工学研究科, 助手 (40282818)
HARA Takashi  Nagoya University, Graduate School of Mathematics, Associate Professor, 多元数理研究科, 助教授 (20228620)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsStrong Disorder / Anderson Localization / Fermion Random Field / Critical Phenomena / アンダーソン局在 / フェルミオン確率場 / 臨界現象 / リヤプノフ指数 / 向きをもつポリマーモデル / Strong disorder / フェルミオンランダム場 / 条件付極限定理 / Bessel meander
Research Abstract

We performed this research project on "Mathematical Analysis of Infinite Dimensional Stochastic Models", and obtained the following results.
1.Asymptotics of Lyapunov exponent of Anderson model : It is established that the discrete heat equation with space-time white noise potential over the discrete Euclidean space, so salled "Parabolic Anderson model" has a Lyapunov exponent exhibiting the exponential order of the solution, which is independent of initial conditions. Morever we developed asymptotic analysis of the exponent with respect to the coupling constant and obtained precise asymptotic formula.
2.Directed Polymers in Random environment : The theory of directed polymers in random environments is still underdeveloping, particularly in low dimensional cases. We established equivalence of local property of the sample path and some decay order of the random partition functions.
3.One of co-workers of this project investigated spectrum structure of random Schroedinger operator over line graph and proved "Anderson localization", another one investigated ergodic properties of the shift dynamics associated with fermion random fields. The other one contributed to the rigorous theory of critical phenomena for stochastic statistical mechanics in establishing presice asymptotic order of two point functions in self-avoiding random walk models and related models.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (41 results)

All Other

All Publications (41 results)

  • [Publications] M.Cranston, T.S.Mountford, T.Shiga: "Lyapunov Exponents for the Parabolic Anderson Model."Acta Mathematica Universite Comenianae. 71. 163-188 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Comets, T.Shiga, N.Yoshida: "Directed Polymers in Random Environment : Path Localization and Strong Disorder."Bernoulli. 9. 1-19 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with certain Fredholm determinants II : fermion shifts and their ergodic and Gibbs properties"Annals of Probability. 31. 1533-1564 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Nakano, Y.Nomura: "Random magnetic fields on line graphs"Journal of Mathematical Physics. 44. 4988-5002 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hara, R.van der Hofstad, G.Slade: "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models."Annals of Probability. 31. 349-408 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Cranston, T.Mountford, T.Shiga: "Lyapunov Exponents for the Parabolic Anderson Model"Acta Mathematica Universitatis Comenianae. 71. 161-186 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Comets, T.Shiga, N.Yoshida: "Directed Polymers in Random Environment : Path Localization and Strong Disorder"Bernoulli. 9. 1-19 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Z.Li, T.Shiga, M.Tomisaki: "A Conditional Limit Theorem for Generalized Diffusion Processes"J.Math.Kyoto Univ.. 43. 567-583 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Comets, T.Shiga, N.Yoshida: "Probabilistic Analysis of Directed Polymers in a Random Environment : a Review"Advanced studies in pure Mathematics. 39. 115-142 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Morita: "Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse"J.Math.Soc.Japan. 56(to appear). (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Morita: "Piecewise C^2 perturbation of Lasota-Yorke maps and their ergodicproperties"Osaka J.Math. 40. 207-233 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hara, T.Hattori, H.Watanabe: "Triviality of hierarchical Ising in four dimensions"Commun.Math.Phys.. 220. 13-40 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hara, R.van der Hofstad, G.Slade: "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models"Ann.Prob.. 31. 349-408 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.Higuchi, T.Shirai: "Weak Bloch property for the discrete magnetic Schr\"{o}dinger operators"Nagoya Math.J.. 161. 127-154 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shirai, H.J.Yoo: "Glauber dynamics for fermion point processes"Nagoya Math.J.{\bf 168}. 139-166 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shirai: "Long time behavior of the transition probability of a random walk with drift on an abelian covering graph"Tohoku Math.J.. 55. 255-269 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with certain Fredholm determinants I : fermion, Poisson and boson point processes"J.Funct.Anal.. 205. 414-463 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with certain Fredholm determinants II : fermion shifts and their ergodic and Gibbs properties"Ann.Probab.. 31. 1533-1564 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.Higuchi, T.Shirai: "Isoperimetric constants of $(d,f)$-regular planar graphs"Interdiscip.Inform.Sci.. 9. 221-228 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Klopp, Shu Nakamura, F.Nakano, Y.Nomura: "Anderson localization for 2D discrete Schroedinger operators with random magnetic fields"Ann.Henri Poincare. 4. 795-811 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Nakano, Y.Nomura: "Random magnetic fields on line graphs"J.Math.Phys.. 44. 4988-5002 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Sumi: "Dynamics of sub-hyperbolic and semi-hyperbolic rational semi-groups and skew products"Ergod.Th.and Dynam.Sys.. 21. 563-603 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Sumi: "The dimensions of Julia sets of expanding rational semgroups"Kodai Mathematical Journal. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Sugawa, R.Stankewitz, H.Sumi: "Some counterexamples in dynamics of rational semigroups"Annales Academia Scientiarum Fennica Mathematica. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Cranston, T.S.Mountford, T.Shiga: "Lyapunov Exponents for the Parabolic Anderson Model with Levy noise."Probability Theory and Related Fields. (掲載予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Comets, T.Shiga, N.Yoshida: "Directed Polymers in Random Environment : Path Localization and Strong Disorder."Bernoulli. 9. 1-19 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Z.Li, T.Shiga, M.Tomisaki: "A Conditional Limit Theorem for Generalized Diffusion Processes."Journal of Mathematics of Kyoto University. 43. 567-583 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Nakano, Y.Nomura: "Random magnetic fields on line graphs."Journal of Mathematical Physics. 44. 4988-5002 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with certain Fredholm determinants II : fermion shifts and their ergodic and Gibbs properties."Annals of Probability. 31. 1533-1564 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Hara, R.van der Hofstad, G.Slade: "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models."Annals of Probability. 31. 349-408 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.N.Ethier, T.Shiga: "Fleming-Viot process with unbounded selection, II"Proceeding on Markov Processes and Controlled Markov Chains Kluwer Acad. Pub., eds Z.Hou, J.A.Filar and A.Chen. 123. 234-240 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Cranston, T.S.Mountford, T.Shiga: "Lyapunov Exponents for the Parabolic Anderson Model"Acta Mathematica Univasitatis Comenianae. 71. 161-184 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shirai, H.J.Yoo: "Glauber dynamics for fermion point processes"Naoya Mathematical Journal. 168. 139-166 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with certain Fredholm determinants II : Fermion shifts and their ergodic and Gibbs properties"Annals of Probability. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Comets, T.Shiga, N.Yoshida: "Directed Polymers in Random Environment : Path Localization and Strong Disorder"Bernoulli. 9. 1-19 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Z.Li, T.Shiga, M.Tomisaki: "A Conditional Limit Theorem for Generalized Diffusion Processes"Journal of Mathematics of Kyoto University. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shiga, A.Shimizu, T.Soshi: "Passage-time moments for positively recurrent Markov chains"Nagoya Mathematical Journal. 162. 165-185 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yu.Higuchi, T.Shirai: "Weak Bloch property for the discrete magnetic Schrodinger operators"Nagoya Mathematical Journal. 161. 127-154 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Shirai, H.J.Yoo: "Glauber dynamics for fermion point processes"Naoya Mathematical Journal. (掲載予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Sumi: "Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products"Ergodic theory and dynamical systems. 21. 563-603 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hara, T.Hattori, H.Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commucation in Nathematical Physics. 220. 13-40 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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