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Comprehensive Study of Probability Theory

Research Project

Project/Area Number 01302008
Research Category

Grant-in-Aid for Co-operative Research (A)

Allocation TypeSingle-year Grants
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyushu University

Principal Investigator

KUNITA Hiroshi  Kyushu Univ. Fac. of Eng., Professor, 工学部, 教授 (30022552)

Co-Investigator(Kenkyū-buntansha) OGURA Yukio  Saga Univ. Fuc. of Sci. and Eng., Professor, 理工学部, 教授 (00037847)
SHIGA Tokuzo  Tokyo Institute of Technology Univ. Fac. of Sci. Professor, 理学部, 教授 (60025418)
ITO Yuuji  Keio Univ. Fac. of Sci. and Eng., Professor, 理工学部, 教授 (90112987)
KOTANI Shinichi  Tokyo Univ. Fac. of Sci., Professor, 理学部, 教授 (10025463)
WATANABE Sinzo  Kyoto Univ. Fac. of Sci., Professor, 理学部, 教授 (90025297)
土谷 正明  金沢大学, 教養部, 教授 (50016101)
田中 洋  慶応義塾大学, 理工学部, 教授 (70011468)
Project Period (FY) 1989 – 1990
Project Status Completed (Fiscal Year 1990)
Budget Amount *help
¥13,500,000 (Direct Cost: ¥13,500,000)
Fiscal Year 1990: ¥6,100,000 (Direct Cost: ¥6,100,000)
Fiscal Year 1989: ¥7,400,000 (Direct Cost: ¥7,400,000)
KeywordsStochastic analysis / Markov process / Limit theorems / Random environment / Gaussian process / Ergodic theory / Schrodinger operator / Stochastic control / 無限次元解析 / 抽象ウィ-ナ-空間 / 多粒子系マルコフ過程 / 測度値分枝過程 / 中心極限定理 / 加法過程 / マリアバン演算法 / 拡散過程 / ガウス型通信路
Research Abstract

We organized seven project teams in the first year and five ones in the second year and each team studied its own project. The belows are the main research results.
1. The research of the stochastic analysis : We analyzed the measures on the Wiener spaces. As results, we obtain the followings. (1) A probabilistic interpretation of the asymptotic behaviors of the fundamental solutions of a heat equation as t 0. (2) The structure of the Sobolev space in infinite demensional spaces. (3) The construction of the differential geometry on the infinite dimensional spaced such as De Rham decomposition theorem.
2. The research of Markov process : We studied the following problems for Marlkov processes with jumps. (1) The estimate of the fundamental solution. (2) The homogenization theorems. (3) Criteria for the recurrence (transience).
3. The research of the limit theorems : We studied the probability distributions related to limit theorems. In particular, we showed that the distribution of a self similar process with independent increment is L-distribution and an arbitrary L-distributio is realized by the above stochastic process.
4. The research of stochastic processes in random environments. We studied the asymptotic behaviors as t of random walks and diffusions in random environment.
5. The research of Gaussian measures and Gaussian processes. As a result, it turned out that the capacity of the channel with feedbacks is not greater than the twice of the capacity of the channel without feedback. 6. The research of probability theory and spectrum. We studied the following objects.
(1) The spectrum of the Schrodinger operators in a magnetic field. (2) The spectrum of the Laplacian in Sierpinsky's carpet and gaskets.
7. The research of the ergodic theorems. We studied the structures of Anozov flow, holocycle flow and other dynamical systems. Also, we studied the relation between the local limit theroems related to continued fraction expansions and prime number theorem.

Report

(3 results)
  • 1990 Annual Research Report   Final Research Report Summary
  • 1989 Annual Research Report
  • Research Products

    (47 results)

All Other

All Publications (47 results)

  • [Publications] A.Bensoussan,H.Nagai: "An ergodic control problem arising from the principal eigenfunctior of an elliptic operator" to appear in J.Math.Soc.Japan.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y.Ito: "Characterizations of ergodic dissipative flow" Preprint.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] G.Kallianpur I.Mitoma R.Wolpert: "Diffusion eqations in duals of nuclear spaces" Stochastics and Stochasoics Reports. 29. 285-329 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] H.Kunita: "Limit on random measures and stochastic difference equations related to mixing array of random variables" to appear in Proc.of Durham Symp.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y.NaKai U.Shiokawa: "A class of normal numbers" Japanese J.Math.16. 17-29 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y.Nakai U.Shiokawa: "A class of normal numbers II" in “Number Theory and Cryptography“,ed J.H.Loxton,London Math.Soc .,Lect.Note.Series. 154. 204-210 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] T. Nakata: "Urn models and Kompeitoh:an industrial application" European Journal of Engineering Education. 25ー3. 267-274 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y.Ogura M.Tomisaki: "Asymptotic behaviors of moments for oneーdimensional generalized diffusion opertors" Stochastic processes and their Applications. 245-273 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] T.Shiga: "A recurrent criterion for Markov processes of OrnsteinーUhlenbeck type" Probab.Th.Rel.Fields. 85. 425-447 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] S.Takanobu S.Watanabe: "Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations" to appear in Proc. Taniguchi Workshop at Sanda and Kyoto.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] M.Tomisaki: "Comparison theorems on Dirichlet norms and their applications" Forum Math.2. 277-295 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] S.Watanabe: "Short time asymptotic problems in Wiener functional integration theory.Applications to heat kernels and index theorems" LNM. 1444. 1-62 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Nobuyki Ikeda,Shinzo Watanaba: "Stochastic dfferential eqations and diffusion processes" North Holland , Kodansha, 555 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Hiroshi Kunita: "Stocastic flows stochastic differential equations" Cambridge University Press, 346 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y. Ito: "Characterizations of ergodic dissipative flow" Preprint.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] H. Kunita: "Limit on random measures and stochastic difference equations related to mixing array of random variables" Proc. of Durham Symp.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] Y. Ogura and M. Tomisaki: "Asymptotic behaviors of moments for one-dimensional generalized diffusion operators" Stochastic Processes and their Applications. 245-273 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] T. Shiga: "A recurrent criterion for Markov processes of Ornstein-Uhlenbeck type" Probab. Th. Rel. Fields. Vol. 85. 425-447 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] S. Takanobu and S. Watanabe: "Asympototic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations" proc. Taniguchi Workshop at Sanda and Kyoto.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] A.Bensoussan;H.Nagai: "An ergodic control problem arising from the principal eigenfunction of an elliptic operator" to appear in J.Math.Soc.Japan.

    • Related Report
      1990 Annual Research Report
  • [Publications] Y.Ito^^<^>: "Characterizations of ergodic dissipative flow" Preprint.

    • Related Report
      1990 Annual Research Report
  • [Publications] G.Kallianpur;I.Mitoma;R.Wolpert: "Diffusion equations in duals of nuclear spaces" Stochastics and Stochastics Reports. 29. 285-329 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] H.Kunita: "Limit on random measures and stochastic difference equations related to mixing array of random variables" to appear in Proc.of Durham Symp.

    • Related Report
      1990 Annual Research Report
  • [Publications] Y.Nakai;U.Shiokawa: "A class of normal numbers" Japanese J.Math.16. 17-29 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Y.Nakai;U.Shiokawa: "A class of normal numbers II" in “Number Theory and Cryptography",ed J.H.Loxton,London Math.Soc.,Lect.Note Series. 154. 204-210 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] T.Nakata: "Urn models and Kompeitoh:an industrial application" European Journal of Engineering Education. 25ー3. 267-274 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Y.Ogura;M.Tomisaki: "Asymptotic behaviors of moments for oneーdimensional generalized diffusion operators" Stochastic Processes and their Applications. 245-273 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] T.Shiga: "A recurrent criterion for Markov processes of OrnsteinーUhlenbeck type" Probab.Th.Rel.Fields. 85. 425-447 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] S.Takanobu;S.Watanabe: "Asympototic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations" to appear in Proc.Taniguchi Workshop at Sanda and Kyoto.

    • Related Report
      1990 Annual Research Report
  • [Publications] M.Tomisaki: "Comparison theorems on Dirichlet norms and their applications" Forum Math.2. 277-295 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] S.Watanabe: "Short time asymptotic problems in Wiener functional integration theory.Applications to heat kernels and index theorems" LNM. 1444. 1-62 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Nobuyuki Ikeda;Shinzo Watanabe: "Stochastic differential equations and diffusion processes" North Holland,Kodansha, 555 (1989)

    • Related Report
      1990 Annual Research Report
  • [Publications] Hiroshi Kunita: "Stochastic flows and stochastic differential equations" Cambridge University Press, 346 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Hiroshi Tanaka: "Time reversal of random walks in one-dimension" Tokyo Journal of Mathematics. 12. 159-174 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Hiroshi Tanaka: "A theorem of Pitman type for simple random walks on Z^d" Tokyo Journal of Mathematics. 12. 235-240 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Masaaki Tsuchiya: "Stochastic processes and semigroups associated with degenerate Levy's generating operators" Stochastics and stochastic reports. 26. 29-61 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Yukio Ogura: "One-dimensional bi-generalized diffusion processes" Journal of Mathematical Society of Japan. 41. 213-242 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kohei Uchiyama: "A tagged particle process in the Boltzman-Grad limit for the Broadwell model" Probability theory and related field. 82. 419-433 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kiyoshi Kawazu: "A one-dimensional birth and death process in random environment" Japan Journal of Applied Mathematics. 6. 97-109 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kiyoshi Kawazu: "Limit theorems for one-dimensional diffusions and random walks in random environments" Probability theory and related field. 80. 501-541 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Hiroshi Sato: "On the absolute continuity of infinite product measure and its application" Probability theory and related field. 81. 609-627 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Yasunari Higuchi: "A remark on the percolation for the 2D Esing model" Osaka Journal of Mathematics. 26. 207-224 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Setsuo Taniguchi: "Stochastic flows of diffemorphisms on an open set in R^d" Stochastics and stochstic reports. 28. 301-315 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Shinzo Watanabe: "A probabilistic proof of the Gause-Bonnet-Chern theorem for manifolds with boundary"

    • Related Report
      1989 Annual Research Report
  • [Publications] Hiroshi Kunita: "Limits of random measures by an array of independent random variables"

    • Related Report
      1989 Annual Research Report
  • [Publications] Nobuyuki Ikeda,Shinzo Watanabe: "Stochastic differential equations and diffusion processes" North Holland, 555 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Hiroshi Kunita: "Stochastic flows and stochastic differential equations" Cambridge University Press, 350 (1990)

    • Related Report
      1989 Annual Research Report

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Published: 1989-04-01   Modified: 2016-04-21  

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