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Stochastic Analysis and its applications to partial differential operators

Research Project

Project/Area Number 26400144
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionAoyama Gakuin University

Principal Investigator

Matsumoto Hiroyuki  青山学院大学, 理工学部, 教授 (00190538)

Co-Investigator(Renkei-kenkyūsha) Yuji Hamana  熊本大学, 理工学研究科, 教授 (00243923)
Research Collaborator Setsuo Taniguchi  九州大学, 基幹研究員・教授 (70155208)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords確率解析 / 拡散過程 / ベッセル過程 / 到達時刻 / ドリフト付きブラウン運動 / 管状近傍 / ブラウン運動 / コルモゴロフ作用素 / 特殊関数
Outline of Final Research Achievements

Following the former studies on Bessel processes in mind, I carried out studies on the probability distributions of the first hitting times to spheres of Brownian motions with constant drifts. By virtue of the skew-product representation of the Brownian motions and Stroock's representation of the Brownian motions on the spheres as the solutions of some stochastic differential equations, explicit forms of the distribution functions. As applications, we derived the asymptotic behavior of the tail probabilities and the volume of the corresponding Wiener sausage. Moreover, for the Bessel processes, the second term of the asymptotics of the tail probability has been reduced.
As other results, a book on the theory and applications of stochastic analysis was publushed. I studied Kolmogorov's diffusion process from the point of view of classical mechanics and, also investigate some important character of two-dimensional diffusion processes.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (9 results)

All 2017 2016 2015 2014

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 2 results,  Acknowledgement Compliant: 4 results) Presentation (2 results) Book (2 results)

  • [Journal Article] Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions2016

    • Author(s)
      Y. Hamana and H. Matsumoto
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 68 Issue: 4 Pages: 1615-1653

    • DOI

      10.2969/jmsj/06841615

    • NAID

      130005433281

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Hitting times to spheres of Brownian motions with and without drifts2016

    • Author(s)
      Y. Hamana and H. Matsumoto
    • Journal Title

      Proceedings of American Mathematical Society

      Volume: 144 Issue: 12 Pages: 5385-5396

    • DOI

      10.1090/proc/13136

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A formula for the expected volume of the Wiener sausage with constant drift2016

    • Author(s)
      Y. Hamana and H. Matsumoto
    • Journal Title

      Forum Mathematicum

      Volume: 29 Issue: 2 Pages: 369-382

    • DOI

      10.1515/forum-2016-0039

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The Kolmogorov operator and classical mechanics2015

    • Author(s)
      Nobuyuki Ikeda, Hiroyuki Matsumoto
    • Journal Title

      Lecture Notes in Mathematics

      Volume: 2137 Pages: 497-504

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Asymptotics of the probability distributions of the first hitting times of Bessel processes2014

    • Author(s)
      Y.Hamana, H.Matsumoto
    • Journal Title

      Electron. Commun. Probab.

      Volume: 19 Issue: none Pages: 1-5

    • DOI

      10.1214/ecp.v19-3215

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Precise asymptotic formulae for the first hitting times of Bessel processes2017

    • Author(s)
      濱名裕司,松本裕行
    • Organizer
      研究集会「マルコフ過程とその周辺」
    • Place of Presentation
      とくぎんトモニプラザ,徳島市
    • Year and Date
      2017-01-06
    • Related Report
      2016 Annual Research Report
  • [Presentation] Hitting times to spheres of Brownian motion with and without drift2015

    • Author(s)
      松本裕行
    • Organizer
      新潟確率論ワークショップ
    • Place of Presentation
      新潟大学
    • Year and Date
      2015-01-23
    • Related Report
      2014 Research-status Report
  • [Book] Stochastic Analysis2017

    • Author(s)
      H. Matsumoto and S. Taniguchi
    • Total Pages
      346
    • Publisher
      Cambridge University Press
    • Related Report
      2016 Annual Research Report
  • [Book] Stochastic Analysis2016

    • Author(s)
      Hiroyuki Matsumoto, Setsuo Taniguchi
    • Total Pages
      350
    • Publisher
      Cambridge University Press
    • Related Report
      2015 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2018-03-22  

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