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Potential theory for parabolic equations and related function spaces

Research Project

Project/Area Number 19K03523
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionChubu University (2021-2023)
Osaka City University (2019-2020)

Principal Investigator

Nishio Masaharu  中部大学, 理工学部, 教授 (90228156)

Co-Investigator(Kenkyū-buntansha) 下村 勝孝  茨城大学, 理工学研究科(理学野), 教授 (00201559)
竹内 敦司  東京女子大学, 現代教養学部, 教授 (30336755)
Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords熱方程式 / ベルグマン空間 / 再生核 / ジャンプ過程 / 多調和関数 / 多放物型関数 / 双対空間 / ブロッホ空間 / 放物型方程式 / 多放物型作用素 / 放物型ブロッホ空間 / 再生公式 / 多重放物型作用素 / 放物型ベルグマン空間 / ジャンプ型確率過程 / 関数空間 / 多重調和関数 / 多重放物型方程式 / ポテンシャル論 / マルチン境界
Outline of Research at the Start

本研究では、熱方程式やポアソン方程式を含む放物型方程式をポテンシャル論的手法および関数空間論的手法を用いて解析する。そして、主な研究対象は、放物型方程式の境界値問題およびその解の境界挙動、放物型方程式の解のなすベクトル空間および凸錐、そして放物型作用素を繰り返し作用させたときに生ずる現象の解析である。
キーワードをあげると、マルチン境界、ベルグマン空間、ハーディ空間、ディリクレ空間、ソボレフ空間、ブロッホ空間などである。

Outline of Final Research Achievements

In the present research, we studied the heat equation and more general linear parabolic equations by using potential theory and function space theory. We considered spaces of solutions of such parabolic equations on the upper half space, which are analogue of Bergman spaces and Bloch spaces. Hence, we established a reproducing property for functions in the space induced by two parabolic Bloch spaces with application to the dual space of bi-parabolic Bergman spaces.

Academic Significance and Societal Importance of the Research Achievements

一般的に言って、関数の性質を調べるとき、性質のよくわかった、あるいは良い性質を持った関数で近似することによって目的に到達しようと考えることは非常に有効である。今のところ我々の研究においては2回の繰り返しにとどまっているが、この回数をふやしていくことによって関数空間は大きくなり、関数を近似する目的に適うものが得られることが期待される。もう一つの視点は、分数べき放物型作用素とラプラス作用素との関連で、これはポテンシャル論的に興味深い問題である。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (30 results)

All 2024 2023 2022 2021 2020 2019 2018

All Journal Article (15 results) (of which Peer Reviewed: 12 results,  Open Access: 4 results) Presentation (14 results) (of which Int'l Joint Research: 4 results,  Invited: 4 results) Book (1 results)

  • [Journal Article] Wasserstein distance on solutions to stochastic differential equations with jumps2024

    • Author(s)
      A. Takeuchi
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 61

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Space-time boundedness and asymptotic behaviors of the densities of CME-subordinators2024

    • Author(s)
      M. Hayashi, A. Takeuchi and M. Yamazato
    • Journal Title

      Stochastic Processes and their Applications

      Volume: 167

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A remark on the dual spaces of bi-parabolic Bergman spaces2023

    • Author(s)
      Y. Hishikawa, M. Nishio, K. Shimomura and M. Yamada
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 55 Issue: 0 Pages: 1-16

    • DOI

      10.5036/mjiu.55.1

    • ISSN
      1343-3636, 1883-4353
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remark on asymptotic expansion of solutions to stochastic differential equations with jumps2023

    • Author(s)
      A. Takeuchi
    • Journal Title

      The Institute of Statistical Mathematics Cooperative Research Report

      Volume: 463

    • Related Report
      2022 Research-status Report
  • [Journal Article] Function spaces induced by two parabolic Bloch spaces2021

    • Author(s)
      Yosuke Hishikawa, Masaharu Nishio, Katsunori Shimomura, Masahiro Yamada
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 51 Issue: 3 Pages: 247-265

    • DOI

      10.32917/h2020031

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Rational function and time transformation of caloric morphism on semi-euclidean spaces2021

    • Author(s)
      Katsunori Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 53 Issue: 0 Pages: 35-42

    • DOI

      10.5036/mjiu.53.35

    • NAID

      130008136052

    • ISSN
      1343-3636, 1883-4353
    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Reproducing property for iterated parabolic operators of fractional order2021

    • Author(s)
      Masaharu Nishio, Katsunori Shimomura
    • Journal Title

      Mathematical Reports (Bucuresti)

      Volume: 23 Pages: 211-226

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Integration by Parts Formula on Solutions to Stochastic Differential Equations with Jumps on Riemannian Manifolds2021

    • Author(s)
      Hirotaka Kai, Atsushi Takeuchi
    • Journal Title

      Journal of Stochastic Analysis

      Volume: 2 Issue: 3 Pages: 1-10

    • DOI

      10.31390/josa.2.3.12

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Gradient formulas for jump processes on manifolds2021

    • Author(s)
      Hirotaka Kai, Atsushi Takeuchi
    • Journal Title

      Electronic Journal of Probability

      Volume: 26 Issue: none Pages: 1-15

    • DOI

      10.1214/21-ejp660

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Remark on rates of convergence to extreme value distributions via the Stein equations2020

    • Author(s)
      Kusumoto Hideaki、Takeuchi Atsushi
    • Journal Title

      Extremes

      Volume: 23 Issue: 3 Pages: 411-423

    • DOI

      10.1007/s10687-020-00380-5

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Gradient formula for jump processes on Riemannian manifolds2020

    • Author(s)
      Takeuchi Atsushi
    • Journal Title

      The Institute of Statistical Mathematics Cooperative Research Report

      Volume: 434 Pages: 60-66

    • Related Report
      2020 Research-status Report
  • [Journal Article] Rates of convergence of extreme value distributions via the IBP formulas2020

    • Author(s)
      Takeuchi Atsushi
    • Journal Title

      The Institute of Statistical Mathematics Cooperative Research Report

      Volume: 433 Pages: 84-90

    • Related Report
      2020 Research-status Report
  • [Journal Article] Existence and non-existence of caloric morphisms with Bateman space-mapping for radial metrics2019

    • Author(s)
      K. Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 51 Pages: 1-12

    • NAID

      130007687024

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Integration by parts formulas for marked Hawkes processes2019

    • Author(s)
      Takeuchi Atsushi
    • Journal Title

      Statistics & Probability Letters

      Volume: 145 Pages: 229-237

    • DOI

      10.1016/j.spl.2018.10.003

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remark on pathwise uniqueness of stochastic differential equations driven by L?vy processes2018

    • Author(s)
      Takeuchi Atsushi、Tsukada Hiroshi
    • Journal Title

      Stochastic Analysis and Applications

      Volume: 37 Issue: 2 Pages: 155-170

    • DOI

      10.1080/07362994.2018.1541750

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] A remark on the dual spaces of bi-parabolic Bergman spaces2023

    • Author(s)
      下村勝孝, 菱川洋介, 西尾昌治, 山田雅博
    • Organizer
      日本数学会 2023 年度秋季総合分科会
    • Related Report
      2023 Annual Research Report
  • [Presentation] A remark on the dual spaces of bi-parabolic Bergman spaces2023

    • Author(s)
      下村勝孝
    • Organizer
      ポテンシャル論研究集会
    • Related Report
      2022 Research-status Report
  • [Presentation] Remark on asymptotic expansion of solutions to jump-type stochastic differential equations2023

    • Author(s)
      竹内敦司
    • Organizer
      研究集会「マルコフ過程とその周辺」
    • Related Report
      2022 Research-status Report
  • [Presentation] 多重放物型ベルグマン空間2022

    • Author(s)
      西尾昌治
    • Organizer
      第65回函数論シンポジウム
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Remark on asymptotic expansion of solutions to stochastic differential equations with jumps2022

    • Author(s)
      竹内敦司
    • Organizer
      統計数理研究所共同研究集会「無限分解可能過程に関連する諸問題」
    • Related Report
      2022 Research-status Report
  • [Presentation] 半Euclid空間上のcaloric morphismの時間変換と実有理関数2022

    • Author(s)
      下村勝孝
    • Organizer
      日本数学会2022年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] Jump-type stochastic differential equations on manifolds2022

    • Author(s)
      Atsushi Takeuchi
    • Organizer
      MFO-RIMS Tandem Workshop "Nonlocality in Analysis, Probability and Statistics"
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Function spaces induced by two parabolic Bloch spaces2021

    • Author(s)
      菱川洋介, 西尾昌治, 下村勝孝, 山田雅博
    • Organizer
      日本数学会2021年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] Wasserstein distance of solutions to stochastic differential equations with jumps2021

    • Author(s)
      竹内敦司
    • Organizer
      日本数学会2021年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] Weighted Bergman type spaces for the iterated parabolic operators of fractional order2019

    • Author(s)
      Masaharu Nishio
    • Organizer
      15th Romanian-Finich Analysis Seminar
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Reproducing properties and function spaces for parabolic operators of fractional order2019

    • Author(s)
      Masaharu Nishio
    • Organizer
      The Ninth Congress of Romanian Mathematicians
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Reproducing property for iterated paraboplic operators of fractional order2019

    • Author(s)
      下村勝孝、西尾昌治
    • Organizer
      日本数学界 2019 年度秋季総合分科会
    • Related Report
      2019 Research-status Report
  • [Presentation] Caloric morphism - 熱方程式の解を保つ変換2019

    • Author(s)
      下村勝孝
    • Organizer
      日本数学界 2019 年度秋季総合分科会, 特別講演
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Integration by parts formulas on marked Hawkes processes2019

    • Author(s)
      Atsushi Takeuchi
    • Organizer
      Workshop on Stochastic Analysis and Applications
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] Jump SDEs and the Study of Their Densities - A Self-Study Book -2019

    • Author(s)
      A. Kohatsu-Higa and A. Takeuchi
    • Total Pages
      355
    • Publisher
      Springer
    • ISBN
      9789813297418
    • Related Report
      2019 Research-status Report

URL: 

Published: 2019-04-18   Modified: 2025-01-30  

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