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Stochastic Analysis and its application to differential operators

Research Project

Project/Area Number 12640117
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 InstitutionNAGOYA UNIVERSITY

Principal Investigator

MATUMOTO Hiroyuki  Nagoya University, Graduate School of Human Informatics., Professor, 大学院・人間情報学研究科, 教授 (00190538)

Co-Investigator(Kenkyū-buntansha) UEMURA Hideaki  Aichi Univ.of Education, Dept.Math., Ass.Prof., 教育学部, 助教授 (30203483)
IHARA Shunsuke  Nagoya Univ., Dept.Info.Sci., Professor, 情報文化学部, 教授 (00023200)
ITO Masayuki  Nagoya Univ., Dept.Info.Sci., Proferssor, 情報文化学部, 教授 (60022638)
UEKI Naomasa  Kyoto Univ., Graduate School of Human Eiviroments, Ass.Prof., 大学院・人間環境学研究科, 助教授 (80211069)
森本 宏  名古屋大学, 人間情報学研究科, 教授 (20115645)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,500,000 (Direct Cost: ¥1,500,000)
KeywordsBrownian motion / Wiener functionals / Mathematical finance / Hyperbolic spaces / Pitman's theorem / Levy's theorem / Laplacian / グリーン関数 / 双曲空間 / ウィナー汎関数 / ベッセル過程 / ブラウン運動 / メタプレクティック表現 / 対称空間
Research Abstract

If we consider a standard Brownian motion and its maximum process, the difference has a same probability law as reflecting Brownian motion and the difference between twice the maximun process and the original Brownian motion has the same law as a three dimensional Bessel process. These results are well known as Levy's and Pitman's theorems. The head investigator Matsumoto, in joint work with Yor, showed analogus or extensions of these theorems for an exponential Brownian functionals by using the classical Laplace method. The exponential Brownian functional appears also in the studies of mathematical finance and in stochastic analysis on hyperbolic spaces. Various aspects of this functional and their applications have been given under the support of this grant. Among others, some explicit expressions for the heat kernels of the semigroups generated by the Laplacians on non-compact symmetric spaces of rank one have been shown by using the results on the exponential Wiener functionals.
Other investigators gave many suggestions and comments on the works mentioned above and also worked on their own fields. Ihara studied information theory by applying the theory of large deviations. Uemura studied on the local time of multi-dimensional Brownian motions, which should be considered as generalized Wiener functionals. Ueki studied on the spectral properties of Schrodinger operators with random electro-magnetic fields.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (38 results)

All Other

All Publications (38 results)

  • [Publications] H.Matsumoto, S.Taniguchi: "Wiener functional of second order and their Levy measure"Electric Journal of Probability Theory. 7. 1-30 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Ihara, M.Kubo: "The asymptotics of string matching probabilities for Gaussian random Sequences"Nagoya Mathematical Journal. 166. 39-54 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Ueki: "Lifschitz tail of a Pauli Hamiltonian with an anomalous magnetic moment"Japanese Journal of Mathematics. 28. 261-286 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "Interpretation via Brownian motion of some independence properties Between GIG and gamma variables"Statistics and Probability Letters. 61. 253-259 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.matsumoto, Y.Ogura: "Markov or non-Markov property of cM-X processes"Journal of Mathematical Society of Japan. (印刷中). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts"Journal of Applied Probability. (印刷中). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki and YOR,Marc: "An analogue of Pitman's 2M-X theorem for exponential Brownian functional, 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
      2002 Final Research Report Summary
  • [Publications] DONATI-MARTIN,Catherine, MATSUMOTO,Hiroyuki, and YOR,Marc: "On positive and negative moments of the integrals of geometric Brownian motions"Stat.Prob.Lett.. 49. 45-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki and UEKI,Naomasa: "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
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki and YOR,Marc: "A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration"Osaka J.Math.. 38. 383-398 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki and YOR,Marc: "An analogue of Pitman's 2M-X theorem for exponential Brownian functional, Part II : the role of generalize dinverse Gaussian distributions"Nagoya Math.J.. 162. 65-86 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] ALILI,Larbi, MATSUMOTO,Hiroyuki and SHIRAISHI,Tomoyuki: "On a triplet of exponential Brownian functionals"Sem.Prob.XXXV, Lecture Notes in Math.1755. Springer, Berlin. 396-415 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki: "Closed form formulae for the heat kernels and the Green, functions for the Laplacians on the symmetric spaces of rank one"Bull.Sci.math.. 125. 553-581 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] DONATI-MARTIN,Catherine, MATSUMOTO,Hiroyuki and YOR,Marc: "Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motions"Publ.RIMS Kyoto Univ.. 37. 295-326 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO,Hiroyuki and TANIGUCHI,Setsuo: "Wiener functional of second order and their Levy measures"Electr.J.Prob.. 7, Paper no.14. 1-30 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] IHARA,Shunsuke: "Large deviation theorems for Gaussian processes and their applications in information theory"Acta Appl.Math.. 63. 165-174 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] UEMURA,Hideaki: "Plane wave decomposition of even-dimensional Brownian local times"J.Funct.Anal.. 173. 481-496 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] UEMURA,Hideaki: "Substitution to parametrized generalized Wiener functionals and pinned Brownian local times"Bull.Sci.math.. 125. 457-479 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] UEKI,Naomasa: "Simple examples of Lifschitz tails in Gaussian random magnetic fields"Ann.Henri Poincare. 1. 473-498 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] UEKI,Naomasa: "Estimates on the heat kernel of the Pauli Hamiltonian and its application to problems on hypoellipticity of the 〓-Laplacian"Math.Z.. 239. 69-97 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] NGUYEN,Laurent, MATSUMOTO,Hiroyuki and YOR,Marc, ed.by R.Buckdahn, H.Engelbert and M.Yor, Gordon and Breach: "Subordinates related to the exponential functionals of Brownian bridges and explicit formulae for the semigroup of hyperbolic Brownian motions, in Stochastic Processes and Related Topics"213-235 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Matsumoto, S.Taniguchi: "Wiener functional of second order and their Levy measure"Electric Journal of Probability Theory. 7. 1-30 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Ihara, M.Kubo: "The asymptotics of string matching probabilities for Gaussian random"Nagoya Mathematical Journal. 166. 39-54 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Ueki: "Lifschitz tail of a Pauli Hamiltonian with an anomalous magnetic moment"Japanese Journal of Mathematics. 28. 261-286 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Matsumoto, M.Yor: "Interpretation via Brownian motion of some independence properties Between GIG and gamma variables"Statistics and Probability Letters. 61. 253-259 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.matsumoto, Y.Ogura: "Markov or non-Markov property of cM-X processes"Journal of Mathematical Society of Japan. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Matsumoto, M.Yor: "On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts"Journal of Applied Probability. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroyuki Matsumoto: "A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtrations"Osaka J. Math.. 38. 383-398 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroyuki Matsumoto: "An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part II : the role of generalized inverse Gaussian distributive"Nagoya Math. J.. 162. 65-86 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Larbi Alili: "On a triplet of exponential Brownian functionals"Seminar de Probabilit'es. 35. 396-415 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Catherine Donati-Martin: "Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motions"Publ. RIMS Kyoto Univ.. 37. 295-326 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroyuki Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on symmetric spaces of rank one"Bull. de Sciences Math'ematiques. 125. 553-581 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideaki Uemura: "Substitution to parametrized generalized Wiener functionals and pinned Brownian local times"Bull. de. Sciences Math'ematiques. 125. 457-479 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroyuki Matsumoto: "An analogue of Pitman's 2M-X theorem for exponential wiener functionals, Part I :A time-inversion approach"Nagoya Math.J.. 159. 125-166 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroyuki Matsumoto: "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Enclidean space"J.Math.Soc.Japan. 52(2). 269-292 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Catherine Donati-Martin: "On positive and negative moments of the integral of geometric Brownian motions"Statistics and Probability Letters. 49. 45-52 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Naomasa Ueki: "Simple examples of Lifschitz tails in Gaussian random magnetic fields"Ann.H.Poincare. 1(3). 473-498 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hideaki Uemura: "Plane wave decomposition of even-dimensional Brownian local times"J.Func.Anal.. 173(2). 481-496 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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