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Aspects of large deviation principles

Research Project

Project/Area Number 13440030
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

TAKAHASHI Youichiro  TAKAHASHI,Youichiro, 数理解析研究所, 教授 (20033889)

Co-Investigator(Kenkyū-buntansha) SHIGEKAWA Ichiro  KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00127234)
HINO Masaomi  KYOTO UNIVERSITY, Graduate School of Informatics, Associate Professor, 大学院・情報学研究科, 助教授 (40303888)
KUMAGAI Takashi  Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (90234509)
HIGUCHI Yasuko  Showa Univ., Coll. of General Education, Associate Professor, 教養部, 講師 (20286842)
SHIRAI Tomoyuki  Kanazawa Univ., Fac. of Sciences, Lecturer, 理学部, 助教授 (70302932)
原 啓介  立命館大学, 理工学部, 講師 (30298715)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥13,100,000 (Direct Cost: ¥13,100,000)
Fiscal Year 2003: ¥5,000,000 (Direct Cost: ¥5,000,000)
Fiscal Year 2002: ¥5,000,000 (Direct Cost: ¥5,000,000)
Fiscal Year 2001: ¥3,100,000 (Direct Cost: ¥3,100,000)
Keywordsstochastic process / large deviation / fermion point process / Fredholm determinant / Wiener functional / エントロピー / 漸近挙動 / 熱核 / ウイーナー汎関数 / 関数力学系
Research Abstract

Large deviation principle is one of the most basic laws in probability theory as well as the law of large numbers and the central limit theorem. We have studied the various features of the large deviation principle, starting from the elucidation of the structure of a few new classes of stochastic processes.
We(Shirai and Takahashi) introduced classes of random point fields or point processes, parameterized by a real number α, including fermion point processes(α= -1) and boson point processes(α= 1). They are associated with Fredholm determinants (to the power 1/α)of symmetric integral operators. We established the existence theorem of such random point fields for α= -1/n(n = 1,2,....) and for α = 2/m(M = 1, 2,...) by constructing them in a probabilistic manner. We also proposed a-statistics generalizing. Fermi and Bose statistics and a conjecture that such random fields exist for other values of a. Moreover, w answered to the question raised by Spohn, Johanson and others affirmatively byy showing that such random point fields do exist even for nonsymmetric integral operators provided that they are transition operators of one dimensional diffusions or birth and death processes. Based upon these facts we have established the large deviation principle in addition to other basic limit theorems and ergodic properties such as the estimates of metric entropy and Bernoulli and other properties. These results are published in Ann. Probability, J. Functional Analysis and, ASPM Series vol. 39. Besides them Shirai published an interesting application to Glauber dynamics.
Other investigators have obtained their results related to the large deviation on their own fields: Higuchi (with Shirai) on the random walk and the Schrodinger operator on infinite graphs, Kumagai on the diffusions on fractals, Shigekawa and Hino on the Wiener space and Hara on quadratic Wiener functionals.

Report

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

    (59 results)

All Other

All Publications (59 results)

  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with Fredholm determinants I : fermion and boson point processes"J.Funct. Analysis. 205. 411-463 (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 Fredholm determination II : fermion shifts and their ergodic property"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] T.Shirai, Y.Takahashi: "Random point fields associated with Fermion and Boson and other statistics"Advanced Studies in Pure Mathematics 39, Math.Society of Japan/American Math. Society. 345-354 (2004)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI, T.SHIRAI: "Weak Bloch property for discrete magnetic Schrodinger operators"Nagoya Math.J.. 161. 127-154 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI: "A remark on exponential growth and the spectrum of the Laplacian"Kodai Math.J.. 24. 42-47 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI: "Combinatorial curvature for planar graphs"J. Graph Theory. 38. 220-229 (2001)

    • Description
      「研究成果報告書概要(和文)」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.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI: "Boundary area growth and the spectrum of discrete Laplacian"Ann.Global Anal.Geom.. 24. 201-230 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI, T.SHIRAI: "Some spectral and geometric properties for infinite graphs"The Proceedings for the 1st JAMS symposium, AMS Contemp.Math.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Fukushima, N., Hino, M.: "On the space of BV functions and a related stochastic calculus in infinite dimensions"J.Funct.Anal.. 183. 245-268 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hino, M.: "On short time asymptotic behavior of some symmetric diffusions on general state spaces"Potential Analysis. 16. 249-264 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hino, M.: "On Dirichlet spaces over convex sets in infinite dimensions, Finite and infinite dimensional analysis in honor of Leonard Gross(New Orleans, LA,2001)"Contemporary Mathematics, American Mathematical Society. 317. 143-156 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hino, M., Ramirez, J.A.: "Small-time Gaussian behavior of symmetric diffusion semigroups"Annals of Probability. 31. 1254-1296 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hino, M.: "Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space"Advanced Studies in Pure Mathematics. 41(to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hara, K., Ikeda, N.: "Quadratic Wiener Functionals and Dynamics on Grassmannian"Bull.Sci.math.. 125. 481-528 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.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)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.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.. 39. 897-930 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.-S.Yun, I.Shigekawa: "The existence of solutions for stochastic differential inclusion"Far East J.Math.Sci.. 7. 205-212 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.Shigekawa: "Vanishing theorem of the Hodge-Kodaira operator for differential forms on a convex domain of the Wiener space"Infin.Dimens.Anal.Quantum Probab.Relat.Top. 6. 53-63 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 高橋陽一郎: "漸近挙動入門"日本評論社. (2002)

    • 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 F redholm determinants I: fermion and boson point processes"J. Funct. Analysis. 205. 411-463 (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 F1 edhOlm determinants II: fermion shifts and their ergodic property"Annals of Probability. 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] T.Shirai, Y.Takahashi: "Random point fields associated with Fermion and Boson and other statistics"Advanced Studies in Pure Mathematics. 39(Math. Society of Japan/American Math. Society). 345-354 (2004)

    • 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.. 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] Yu.HIGUCHI, T.SHIRAI: "Weak Bloch property for discrete magnetic Schrodinger 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] Yu.HIGUCHI: "A remark on exponential growth and the spectrum of the Laplacian"Kodai Math. J.. 24. 42-47 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu.HIGUCHI: "Combinatorial curvature for planar graphs"J. Graph Therory. 38. 220-229 (2001)

    • 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"in Interdiscip. Inform. Sci.. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yu..HIGUCHI: "Boundary area growth and the spectrum of discrete Laplacian"Ann. Global Anal. Geom. 24. 201-230 (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: "Some spectral and geometric properties for infinite graphs"in The Proceedings for the 1st JAMS symposium, AMS Contemp. Math.. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.F ukushima, M..Hino: "On the space of BV functions and a related stochastic calculus in infinite dimensions"J. Funct. Anal.. 183. 245-268 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hino: "On short time asymptotic behavior of some symmetric diffusions on general state spaces"Potential Analysis. 16. 249-264 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hino: "On Dirichlet spaces over convex sets in infinite dimensions, Finite and infinite dimensional analysis in honor of Leonard Gross. (New Orleans, LA, 2001)"Contemporary Mathematics, CAmerican Mathematical Society. 317. 143-156 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hino, J.A.Ramirez: "Small-time Gaussian behavior of symmetric diffusion semigroups"Annals of Probability. 31. 1254-1296 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hino: "Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space"in Advanced Studies in Pure Mathematics. 41(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Hara, N.Ikeda: "Quadratic Wiener Functionals and Dynammics on Grassmannian"Bull.Sci.math. 125. 481-528 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.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)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.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. 39. 897-930 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.-S.Yun, I.Shigekawa: "The existence of solutions for stochastic differential inclusion"Far East J. Math. Sci. 7. 205-212 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.Shigekawa: "Vanishing theorem of the Hodge-Kodaira operator for differential forms on a convex domain of the Wiener space"Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6. 53-63 (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 Fredholm determinants I : fermion and boson point processes"J.Funct.Analysis. 205. 411-463 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with Fredholm determinants II : fermion shifts and their ergodic property"Annals of Probability. 31. 1533-1654 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with Fermion and Boson and other statistics"Advanced Studies in Pure Mathematics 39, Math.Society of Japan/American Math.Society. 345-354 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takashi Kumagai: "Diffusion processes on fractal fields and their large deviations"Probab.Theory Relat.Fields. 127. 305-352 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hino, M: "On short time asymptotic behavior of some symmetric diffusions on general state spaces"Potential Analysis. 16. 249-264 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Shigekawa: "Vanishing theorem of the Hodge-Kodaira operator for differential forms on a convex domain of the Wiener space"Infin.Dimens.Anal.Quantum Probab.Relat.Top. 6. 53-63 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with Fredholm determinants II : fermion shifts an their ergodic property"Annals of Probability. (発行予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Random point fields associated with Fermion and Boson and other statistics"Proceeding of MSJIRI 2002. (発行予定). (2003)

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

    • Related Report
      2002 Annual Research Report
  • [Publications] Takashi Kumagai: "Some remarks for stable-like jump processes on fractals"In : Trends in Math., Fractals in Graz 2001 (P. Grabner and W. Woess (eds.)). 185-196 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ichiro Shigekawa: "Littlewood-Paley inequality for a diffuision satisfying the logarithmic Sobolev inequality an for the Brownian motion on a Riemannian manifold with boundary"Osaka J. Math.. 39. 897-930 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masanori Hino: "On short time asymptotic behavior of some symmetric diffusions on general state spaces"Potential Anal.. 16. 249-264 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 高橋陽一郎: "漸近挙動入門"日本評論社. 145 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Kumagai: "Transition density asymptotics for some diffusion processes with multi-fractal structures,"Electronic Journal of Prob.. 6. 1-23 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Takahashi: "Function dynamics,"Japan J. Indust. Appl. Math.. 18. 405-423 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] I.Shigekawa: "The domain of generator and the intertwining property"In"Stochastics in Finite and Infinite Dimensions"(eds. T. Hida eaal). 401-410 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Hino: "On the space of BV functions and a related stochastic calculus in infinite dimensions"Journal of Funct. Anal.. 183. 245-268 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Shirai, Y.Takahashi: "Fermion process and Fredholm determinant"Proc. Second ISAAC Congress, Kluwer Academic Press. 1. 15-23 (2001)

    • Related Report
      2001 Annual Research Report

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

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