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Mathematical Analysis of Infinite Dimensional Stochastic Models

Research Project

Project/Area Number 09640246
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  Graduate School of Science and Technology, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (60025418)

Co-Investigator(Kenkyū-buntansha) TANIGUCHI Masaharu  Graduate School of Informatics and EngineeringScience, Tokyo Institute of Technn, 大学院・理工学研究科, 講師 (30260623)
TAKAOKA Koichiro  Faculty of Commerce, Hitotsubashi University, Lecturer, 商学部, 講師 (50272662)
NINOMIYA Hirokazu  Faculty of Science, Ryukoku University, Lecturer, 理学部, 講師 (90251610)
MORITA Takehiko  Graduate School of Science and Technology, Tokyo Institute of Technology, Associ, 大学院・理工学研究科, 助教授 (00192782)
UCHIYAMA Kohei  Graduate School of Science and Technology, Tokyo Institute of Technology, Profes, 大学院・理工学研究科, 教授 (00117566)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsinfinite dimensional diffusion / Flemig-Viot process / genetical model / unbounded selection / time-reversibility / stochastic partial differential equation / random walk in random environment / directed polymer model / 可逆定常分布 / 分数冪モーメント / ランダムポリマーモデル / ランダムウォーク / マルチンゲール / 相互作用のある拡散系 / 標本リアプノフ指数 / ブラウン運動
Research Abstract

Performing the reseach based on the project plan we obtained the following reseach results.
1. Fleming-Viot processes play an important role in population genetics, for which we obtained two significant results.
First, we considered the model with mutation and unbounded selectionas genetic factors. In this case it has not proved even the well-posedness of the diffusion processes, which we settled together with the uniqueness problem of the stationary distributions. This work was caried out jointly with S.N.Ethier (USA). Furthermore we solved the problem of diffusion approximation from discrete time Markov chain models.
Second, we solved a reversibility problem for the Fleming-Viot processes with mutation and selection, that is to characterize the mutation operator for the process to have a reversible distribution. This work was done with Z.H.Li (China) and L.Yau (USA). (Shiga)
2. We considered a suvival probability problem of random walker in temporarily and spatially varing random environ … More ment, and obtained a precise asymprotics of the suvival probability for small parameter rigion. To prove it we developed a detailed analysis of linear stochastic partial differential equations which are dual objects of the random walk model. This result appeared as ajoint work with T.Furuoya.
Directed polymer model is a closely related with this problem in mathematical context, and we get some significant results on asymptotical behaviorof the random partition function in low dimensional case, which is harder than higher dimensional case. (Shiga)
3. For a mechanical many particle system Uchiyama established the hydrodynamic limit and identified its hydrodynamic equation, that is a diffusion equation in this situation.
4. For a dynamical system in cofinite Fuchsian groups which can be regarded as a Markov system, Morita developed a perterbational analysis of the transfer operator and solved some ergodic problem that is related to number theory.
5. Motivated by mathematical finance Takaoka obtained a neccesary and suffucient condition for a continuous local martingale to be uniformly integrable. Less

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (49 results)

All Other

All Publications (49 results)

  • [Publications] T.Shiga: "Exponential decay rate of survival probability of in a disastrous random environment." Probab.Th.Rel.Fields. 108. 417-439 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Furuoya and T.Shiga: "Sample Lyapunov exponent for a class of linear Markovian systems over Z^d." Osaka J.Math.35. 35-72 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Cox.A.Greven and T.Shiga: "Finite and infinite systems of interacting diffusions : Cluster formation and universality properties." Math.Nachrich.に掲載予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Wiener's test for the random walks with mean zero and finite variance." Ann.Probab.26. 368-376 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Green's functions for random walks on Z^N. Proc." London Math.Soc.77. 215-240 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Scaling limit for a mechanical system of interacting particles II," Commun.Math.Phys.196. 681-701 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Scaling limitis for large systems of in teracting particles, in Advances in non-linear partial differential equations and stochastics" Adv.Math.Appl.Sci.ed.by S.Kawasima and T.Yanagisawa. 48. (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Morita: "Markov systems and transfer operators associated with cofinite Fuchsian groups" Ergod.Th.& Dynam.Sys.17. 1147-1181 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Takaoka: "Some remarks on the uniform integrability of continuous martingales." Seminaire de probabilites XXXIII掲載予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Mizoguchi, H.Ninomiya, and E.Yanagida: "Diffusion-induced blowup in a nonlinear parabolic system" J.Dynamics and Differential Equations. 10-3. 619-638 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Iida, T.Muramatsu, H.Ninomiya, and E.Yanagida: "Diffusion-induced extinction of a superior species in a competition systems" Japan J.Industrial and Applied Mathematics. 15-2. 233-252 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Mizoguchi, H.Ninomiya, and E.Yanagida: "Critical exponent for the bipolar blowup in a seminlinear parabolic equation" J.Math.Anal.and Appl.218. 495-518 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi: "Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem" Quart.Appl.Math.掲載予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yuji Nomura: "Absence of diffusion near the bottom of the spectrum for a random Schroedinger operator on L^2(R^3)" J.Math.Kyoto Univ.37. 639-687 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Shiga: "Exponential decay rate of survival probability of in a disastrous random environment." Probab.Th.Rel.Fields. 108. 417-439 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Furuoya and T.Shiga: "Sample Lyapunov exponent for a class of linear Markovian systems over Z^d." Osaka J.Math. 35. 35-72 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Cox, A.Greven and T.Shiga: "Finite and infinite systems of interacting diffusions : Cluster formation and universality properties." Math.Nachrich.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Wiener's test for the random walks with mean zero and finite variance." Ann.Probab.26. 368-376 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Green's functions for random walks on Z^N." Proc.London Math.Soc.77. 215-240 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Scaling limit for a mechanical system of interacting particles II" Commun.Math.Phys.196. 681-701 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Uchiyama: "Scaling limits for large system of interacting particles" Adv.Math.Appl.Sci.48, ed.by S.Kawasima and T.Yanagisawa. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Morita: "Markov systems and transfer operators associated with cofinite Fuchsian groups" Ergod.Th.& Dynam.Sys. 17. 1147-1181 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Takaoka: "Some remarks on the uniform integrability of continuous martingales." Seminaire de Probabilites XXXIII.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Mizoguchi, H.Ninomiya, and E.Yanagida: "Diffusion-induced blowup in a nonlinear parabolic system" J.Dynamics and Differential Equations. 10-3. 619-638 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Iida, T.Muramatsu, H.Ninomiya, and E.Yanagida: "Diffusion-induced extinction of a superior species in a competition system" Japan J.Industrial and Applied Mathematics. 15-2. 233-252 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Mizoguchi, H.Ninomiya, and E.Yanagida: "Critical exponent for the bipolar blowup in a seminlinear parabolic equation" J.Math.Anal.and Appl.218. 495-518 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi: "Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem"" Quart.Appl.Math.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yuji Nomura: "Absence of diffusion near the bottom of the spectrum for a random Schroedinger operator on L^2(R^3)" J.Math.Kyoto Univ.Vol.37. 639-687 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Furuoya and T.Shiga: "Sample Lyapunov exponent for a class of linear Markovian systems over Z^d." Osaka J.Math.35. 35-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Cox, A.Greven and T.Shiga: "Finite and infinite systems of interacting diffusions: Cluster formation and universality properties." (Math.). Nachrich.に掲載予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Uchiyama: "Wiener's test for the random walks with mean zero and finite variance." Ann.Probab.26. 368-376 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Uchiyama: "Green's functions for random walks on Z^N" Proc.London Math.Soc.,. 77. 215-240 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Uchiyama: "Scaling limit for a mechanical system of interacting particles II," Commun.Math.Phys.196. 681-701 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Uchiyama: "Scaling limits for large systems of interacting particles, in Advances in non-linear partial differential equations and stochastics," Adv.Math.Appl.Sci.48, ed.by S.Kawasima and T.Yanagisawa. 48. (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Morita: "Markov systems and transfer operators associated with cofinite Fuchsian groups," Ergod.Th.& Dynam.Sys.17. 1147-1181 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Takaoka: "Some remarks on the uniform integrability of continuous martingales." (to appear in Seminaire de Probabilites XXXIII(1999)).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Mizoguchi, H.Ninomiya, and E.Yanagida: "Diffusion-induced blowup in a nonlinear parabolic system," J.Dynamics and Differential Equations. 10-3. 619-638 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Iida, T.Muramatsu, H.Ninomiya, and E.Yanagida: "Diffusion-induced blowup in a nonlinear parabolic system," Japan J.Industrial and Applied Mathematics. 15-2. 233-252 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Mizoguchi, H.Ninomiya, and E.Yanagida: "Critical exponent for the bipolar blowup in a seminlinear parabolic equation," J.Math.Anal.and Appl.218. 495-518 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Taniguchi: "Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem." Quart.Appl.Math.(掲載予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Shiga: "Exponential decay rate of survival probability of in a disatrous random environment." Probab.Th.Rel.Fields. 108. 417-439 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Furuoya and T.Shiga: "Sample Lyapunov exponent for a class of linear Markovian systems over Z^d." Osaka J.Math.34(掲載予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Cox, A.Greven and T.Shiga: "Finite infinite systems of interacting diffusions: Cluster formation and universality properties." Math.Nachrich.(掲載予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Morita: "Markov systems and transfer operators associated with cofinite Fuchsian groups," Ergod.Th.& Dynam.Sys.17. 1147-1181 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Takaoka: "On the martingales obtained by an extension due to Saisho,Tanemura and Yor of Pitman's theorem." Seminaire de Probabilites XXXI,LNM,Springer. 1655. 256-265 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Kusuoka and K.Takaoka: "On the Sparre Andersen transformation for multidimensional Brownian bridge," Jour.Math.Sci.Univ.of Tokyo. 4. 211-227 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Ninomiya: "Separatrices of competition-diffusion equations,in ″Reaction-Diffusion Equations and Their Applications and Computational Aspects″" edited by T.-T.Li et al. 118-125 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Iida and H.Ninomiya: "Nonlinear boundary value problems related to some shemical reactions on the interface in ″Reaction-Diffusion Equations and Their Applications and Computational Aspects″" edited T.-T.Li et al. 52-59 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Masaharu Taniguchi: "Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem" Quart.Appl.Math.(掲載予定).

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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