Functional analysis for paths of Markov processes via semi-Dirichlet forms

• Project/Area Number: 15K04941
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kansai University
• Principal Investigator: Toshihiro Uemura 関西大学, システム理工学部, 教授 (30285332)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: Dirichlet 形式 / マルコフ過程 / 加法汎関数 / semi-Dirichlet 形式 / 均質化問題 / Mosco 収束 / 正則部分空間 / 保存性 / 対称Dirichlet形式 / Mosco収束 / 再帰性 / 過渡性 / 純飛躍型マルコフ過程 / 容量不等式

Outline of Final Research Achievements

Global path properties, such as recurrence, transience and conservativeness, of Markov processes associated with Dirichlet forms are obtained in terms of the volume growth of balls with respect to the basic measure and the behaviors at the infinity of the coefficients of the infinitesimal generator associated with the processes. Moreover, we revealed the instability of such global path properties under Mosco’s convergence of Markov processes.

Academic Significance and Societal Importance of the Research Achievements

`連続的変化'の破綻が起きうる状況において，標本路が連続であるという拡散過程に限らず，不連続な状況を許容したジャンプ拡散過程，あるいは純飛躍型過程をモデル化した解析の研究が必要である現代において，当該研究はその先駆けともいえる研究である．また，拡散係数や，Levy係数などが滑らかでない場合においては，モデル化される確率過程の構成さえ自明ではない．Dirichlet 形式は，そのような場合においても，正則性と呼ばれれる条件さえ満たされれば確率過程が構成できるという優位性がある．

Research Products

• [Int'l Joint Research] TU Dresden(ドイツ)
• [Int'l Joint Research] TU Dresden/Institut fur Math. Stochastik(Germany)
• [Journal Article] Weak convergence of regular Dirichlet subspaces (2017)
  • Author(s): Li Liping，Toshihiro Uemura and Jiangang Ying
  • Journal Title: Osaka Journal of Mathematics Volume: 54
  • NAID: 120006358853
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the Conservativeness of Some Markov Processes (2017)
  • Author(s): Oshima Yoichi、Uemura Toshihiro
  • Journal Title: Potential Analysis Volume: 46 Issue: 4 Pages: 609-645
  • DOI: 10.1007/s11118-016-9596-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On instability of global path properties of symmetric Dirichlet forms under Mosco convergence (2016)
  • Author(s): Kohei Suzuki and Toshihiro Uemura
  • Journal Title: Osaka Journal of Mathematics Volume: 53 Pages: 567-590
  • NAID: 120005941454
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the recurrence of symmetric jump processes (2015)
  • Author(s): Hiroyuki Okura and Toshihiro Uemura
  • Journal Title: Forum Mathematicum Volume: 27 Pages: 3269-3300
• [Presentation] Homogenization of symmetric Dirichlet forms (2019)
  • Author(s): 上村稔大
  • Organizer: 研究集会「マルコフ過程とその周辺」 (まちなかキャンパス長岡）
  • Invited
• [Presentation] Homogenization of symmetric Levy processes on R^d (2019)
  • Author(s): 上村稔大
  • Organizer: 日本数学会2019年度年会「統計数学分科会」(東京工業大学）
• [Presentation] On convergence of symmetric Dirichlet forms (2018)
  • Author(s): Toshihiro Uemura
  • Organizer: TU Dresden Analysis & Stochastics Seminars 「Probability afternoon」
  • Int'l Joint Research
• [Presentation] On the conservativeness of some Markov processes (2017)
  • Author(s): Toshihiro Uemura
  • Organizer: Japanese-German Open Conference on Stochastic Analysis 2017
  • Int'l Joint Research / Invited
• [Presentation] Gamma-convergence of symmetric jump-type Dirichlet forms (2017)
  • Author(s): Toshihiro Uemura
  • Organizer: Workshop on Jump Processes and Stochastic Analysis 2017
  • Int'l Joint Research / Invited
• [Presentation] A note on convergence of Dirichlet forms (2017)
  • Author(s): Toshihiro Uemura
  • Organizer: Workshop on Interface between Commutative and Non-Commutative Stochastic Analysis
  • Place of Presentation: Hokkaido University, Japan
  • Invited
• [Presentation] On an optimal stopping problem and a variational inequality (2016)
  • Author(s): Toshihiro Uemura
  • Organizer: Workshop on Stochastic Analysis and related topics 2016
  • Place of Presentation: TU Dresden, Germany
  • Year and Date: 2016-05-09
  • Invited
• [Presentation] On a weak convergence of regular Dirichlet subspaces of H^1(I) (2016)
  • Author(s): Toshihiro Uemura
  • Organizer: Markov Processes and Related Fields
  • Place of Presentation: ヴェルクよこすか，神奈川県，日本
  • Year and Date: 2016-01-09
  • Invited
• [Presentation] A note on the Mosco convergence of symmetric Dirichlet forms (2015)
  • Author(s): Toshihiro Uemura
  • Organizer: Workshop on Stochastic Analysis and Related Topics
  • Place of Presentation: Dresden, Germany
  • Year and Date: 2015-11-05
  • Int'l Joint Research / Invited
• [Presentation] On the Mosco convergence of symmetric jump type Dirichlet forms (2015)
  • Author(s): Toshihiro Uemura
  • Organizer: The 11th Workshop on Markov Processes and Related Topics
  • Place of Presentation: Shanghai, China
  • Year and Date: 2015-06-27
  • Int'l Joint Research / Invited
• [Funded Workshop] 2nd Interdisciplinary & Research Alumni Symposium iJaDe2018 (Mathematics Session) (2018)
• [Funded Workshop] Dirichlet Forms and Their Geometry (2017)
  • Place of Presentation: Tohoku University, Japan
  • Year and Date: 2017-03-18
• [Funded Workshop] Workshop on Dirichlet forms and Stochastic Analysis (2017)
• [Funded Workshop] Workshop on Jump Processes and Stochastic Analysis 2017 (2017)
• [Funded Workshop] Workshop on Dirichlet Forms and Stochastic Analysis (2016)
  • Place of Presentation: Kansai University, Japan
• [Funded Workshop] Summer School on Dirichlet Forms and Stochastic Analysis at Kansai University 2015 (2015)
  • Place of Presentation: Kansai University, Suita, Osaka, Japan
  • Year and Date: 2015-08-23