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Stochastic Analysis on Infinite Dimensional Spaces from a Geometric View

Research Project

Project/Area Number 20K03639
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 InstitutionKeio University

Principal Investigator

KAWABI Hiroshi  慶應義塾大学, 経済学部(日吉), 教授 (80432904)

Co-Investigator(Kenkyū-buntansha) 石渡 聡  山形大学, 理学部, 准教授 (70375393)
楠岡 誠一郎  京都大学, 理学研究科, 教授 (20646814)
星野 壮登  大阪大学, 大学院基礎工学研究科, 准教授 (20823206)
Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords確率論 / 確率解析 / 確率偏微分方程式 / ラフパス理論 / 離散幾何解析 / 大域解析学 / マリアヴァン解析 / ディリクレ形式 / Dirichlet形式 / 大域解析
Outline of Research at the Start

本研究課題では, 幾何学からの視点を組み合わせることで無限次元空間上の確率解析における新たな解析手段を提示する。大まかには以下の2つの研究に分類される。

(1) 構成的場の理論から派生した無限次元空間上の確率力学系に対する特異確率偏微分方程式および離散幾何解析を融合した研究

(2) ラフパス理論, マリアヴァン解析, 測度距離空間上の幾何解析などを援用した無限次元空間上のヘルマンダー型微分作用素の一意性問題, 関数不等式の研究

Outline of Final Research Achievements

(1) With Masato Hoshino and Seiichiro Kusuoka, I constructed a unique strong solution of singular stochastic partial differential equations which realize stochastic quantization of exp(Φ)_{2} quantum fields. We proposed a new method using peculiarity of the exponential model. The relation with Dirichlet form is also clarified.

(2) As a preparation for the discrete geometric study of quantum fields, I studied a discrete approximation of Schroedinger semigroup with drift on noncompact Riemannian manifolds with Satoshi Ishiwata. This study is regarded as a finite dimensional summation approximation of the Feynman-Kac functional integral on manifolds, and is interesting from the viewpoints of not only global analysis but also stochastic numerical analysis and manifold learning.

Academic Significance and Societal Importance of the Research Achievements

近年の特異確率偏微分方程式の理論の進展により, 無限次元空間上の確率解析と構成的場の量子論の数学的研究の融合が進んでいるが, 本研究で得られた成果は, その中でも中心的なexp(Φ)_{2}-モデルの数学解析における基礎定理である。また, リーマン多様体上のドリフト付きシュレーディンガー半群の離散近似の研究成果であるが, ファインマン-カッツ汎関数積分の有限次元和分近似を考えると言うことにも相当し, 今後, 離散幾何解析的視点からの量子場の解析, 確率数値解析, 更には多様体学習への波及も期待できる。

Report

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

    (31 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Open Access: 4 results) Presentation (20 results) (of which Int'l Joint Research: 10 results,  Invited: 14 results) Remarks (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Bonn大学/Bielefeld大学(ドイツ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Bonn大学/Bielefeld大学(ドイツ)

    • Related Report
      2022 Research-status Report
  • [Journal Article] A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold2024

    • Author(s)
      Ishiwata Satoshi、Kawabi Hiroshi
    • Journal Title

      Mathematische Annalen

      Volume: - Issue: 2 Pages: 1-37

    • DOI

      10.1007/s00208-024-02809-9

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Stochastic quantization associated with the exp(Φ)_{2}-quantum field model driven by space- time white noise on the torus in the full L_{1}-regime2022

    • Author(s)
      Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
    • Journal Title

      Probability Theory and Related Fields

      Volume: - Issue: 1-2 Pages: 391-447

    • DOI

      10.1007/s00440-022-01126-z

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Strong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus2022

    • Author(s)
      Sergio Albeverio, Hiroshi Kawabi, Stefan-Radu Mihalache, Michael Roeckner
    • Journal Title

      Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5)

      Volume: 22 Pages: 33-69

    • DOI

      10.2422/2036-2145.202105_106

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold2022

    • Author(s)
      Satoshi Ishiwata, Hiroshi Kawabi
    • Journal Title

      arXiv

      Volume: - Pages: 1-35

    • Related Report
      2022 Research-status Report
    • Open Access
  • [Journal Article] Stochastic quantization associated with the exp(Φ)_{2}-quantum field model and related topics (II)2021

    • Author(s)
      Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
    • Journal Title

      数理解析研究所講究録

      Volume: 2201 Pages: 76-86

    • Related Report
      2021 Research-status Report
    • Open Access
  • [Journal Article] Tightness of the solution to approximating equations of the stochastic quantization equation associated with the weighted exponential quantum field model on the two-dimensional torus2021

    • Author(s)
      Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
    • Journal Title

      Advanced Studies in Pure Mathematics (ASPM)

      Volume: 87 Pages: 341-361

    • DOI

      10.2969/aspm/08710341

    • ISBN
      9784864970952
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stochastic quantization associated with the exp(\Phi)_2-quantum field model driven by space-time white noise on the torus in the full L_1-regime2020

    • Author(s)
      Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
    • Journal Title

      arXiv

      Volume: 2007.08171 Pages: 1-50

    • Related Report
      2020 Research-status Report
    • Open Access
  • [Presentation] Central limit theorems for non-symmetric random walks on nilpotent covering graphs2023

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Mathematics of Random Systems: Summer School 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic quantization associated with the exp(Φ)_{2}-quantum field model driven by space-time white noise2023

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      Ritsumeikan Workshop on Stochastic Analysis
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A discrete approximation of a diffusion with drift and killing on a Riemannian manifold2023

    • Author(s)
      Satoshi Ishiwata
    • Organizer
      Metrics and Measures
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction of a non-Gaussian and rotation-invariant Φ^{4}-measure and associated flow on R^{3} through stochastic quantization2023

    • Author(s)
      Seiichiro Kusuoka
    • Organizer
      2023 Spring Probability Workshop, Taiwan National University
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Random models on regularity-integrability structures2023

    • Author(s)
      Masato Hoshino
    • Organizer
      Ritsumeikan Workshop on Stochastic Analysis
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A graph discretized approximation of diffusion with drift and killing on a complete Riemannian manifold2023

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      立命館大学数理ファイナンスセミナー
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] リーマン多様体上の非対称拡散過程の離散近似2023

    • Author(s)
      石渡聡, 河備浩司
    • Organizer
      日本数学会年会
    • Related Report
      2022 Research-status Report
  • [Presentation] Stochastic quantization associated with the exp (Φ)_{2}-quantum field model driven by space-time white noise2022

    • Author(s)
      河備浩司, 星野壮登, 楠岡誠一郎
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] A graph discretized approximation of diffusions with drift and killing on a complete Riemannian manifold2022

    • Author(s)
      河備浩司
    • Organizer
      東京確率論セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] A graph discretized approximation of diffusions with drift and killing on a complete Riemannian manifold2022

    • Author(s)
      Hiroshi Kawabi
    • Organizer
      確率解析とその周辺 (Stochastic Analysis and Related Topics)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic quantization associated with the exp (Φ)_{2}-quantum field model2022

    • Author(s)
      Seiichiro Kusuoka
    • Organizer
      RIMS Symposium: Probability and Analysis on Random Structures and Related Topics
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic quantization associated with the exp (αΦ)_{2}-quantum field model2022

    • Author(s)
      Masato Hoshino
    • Organizer
      Open Japanese-German conference: Stochastic analysis and applications
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] リーマン多様体上の非対称拡散過程の離散近似2022

    • Author(s)
      石渡聡
    • Organizer
      確率論シンポジウム
    • Related Report
      2022 Research-status Report
  • [Presentation] 特異確率偏微分方程式に由来する確率微分方程式の問題2022

    • Author(s)
      楠岡 誠一郎
    • Organizer
      岡山確率解析ワークショップ2022
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] exp(Φ)_{2}-量子場モデルに付随する放物型確率偏微分方程式について2021

    • Author(s)
      星野 壮登
    • Organizer
      熊本大学応用解析セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Stochastic quantization associated with the exp(\Phi)_2-quantum field model on the torus in the full L_1-regime2021

    • Author(s)
      河備 浩司
    • Organizer
      東北確率論セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Geometric analysis on manifolds with ends2021

    • Author(s)
      石渡 聡
    • Organizer
      偏微分方程式姫路研究集会
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic quantization associated with the exp(\alpha\phi)_2-quantum field model2021

    • Author(s)
      星野 壮登
    • Organizer
      量子場の数理とその周辺
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Stochastic quantization associated with the exp(\alpha \phi)_2-quantum field model2020

    • Author(s)
      星野 壮登
    • Organizer
      2020年度確率論シンポジウム
    • Related Report
      2020 Research-status Report
  • [Presentation] Stochastic quantization associated with the exp (\Phi)_2 -quantum field model driven by space-time white noise on the torus in the full L_1-regime2020

    • Author(s)
      楠岡 誠一郎
    • Organizer
      2020年度確率解析シンポジウム
    • Related Report
      2020 Research-status Report
  • [Remarks] research map

    • URL

      https://researchmap.jp/hkawabi

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] 確率解析2023

    • Related Report
      2023 Annual Research Report

URL: 

Published: 2020-04-28   Modified: 2025-01-30  

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