Combination of stochastic differential equations and other theories

• Project/Area Number: 17K14204
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kyoto University (2019-2020); Okayama University (2017-2018)
• Principal Investigator: Kusuoka Seiichiro 京都大学, 理学研究科, 准教授 (20646814)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 確率微分方程式 / 確率解析 / 確率偏微分方程式 / ディリクレ形式 / 確率量子場モデル / マリアヴァン解析 / マルコフ過程 / 解析半群

Outline of Final Research Achievements

In this research I tried to develop the stochastic analysis and other theories by the combination of them. In particular, by applying the methods in stochastic differential equations to nonlinear partial differential equations, I succeeded to construct the stochastic processes and its stationary measures generated by singular stochastic partial differential equations which appear in the constructive quantum field theory. This result yields a new method for the construction of quantum fields, and is worthful both in stochastic analysis and quantum mechanics.

Academic Significance and Societal Importance of the Research Achievements

この研究における成果は特異確率偏微分方程式の解とその定常分布の構成である。特異確率偏微分方程式は近年盛んに研究されている話題であり、他の研究者は正則構造理論やパラ制御解析といった一般論を用いた研究を行っていた。一方、この研究ではこれらの一般論のアイデアだけを使った独自のアプローチを行うことにより、これらの理論の枠組みとは異なる形で方程式の解の構成を行った。この新しいアプローチの利点として明確な形で定常過程の構成ができることがあり、その成果の一部として量子場の構成に成功している。

Research Products

• [Int'l Joint Research] ボン大学(ドイツ)
• [Int'l Joint Research] ボン大学(ドイツ)
• [Int'l Joint Research] ボン大学(ドイツ)
• [Int'l Joint Research] ボン大学(ドイツ)
• [Journal Article] Stochastic quantization associated with the exp(\Phi)2-quantum field model driven by space-time white noise on the torus (2021)
  • Author(s): Masato Hoshino, Hiroshi Kawabi, Seiichiro Kusuoka
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 1 Pages: 339-375
  • DOI: 10.1007/s00028-020-00583-0
  • Peer Reviewed / Open Access
• [Journal Article] Recurrence of direct products of diffusion processes in random media having zero potentials (2020)
  • Author(s): Daehong Kim and Seiichiro Kusuoka
  • Journal Title: Electronic Journal of Probability Volume: 25 Issue: none Pages: 1-18
  • DOI: 10.1214/20-ejp540
  • Peer Reviewed / Open Access
• [Journal Article] The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model (2020)
  • Author(s): Albeverio Sergio、Kusuoka Seiichiro
  • Journal Title: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE Volume: 20 Pages: 1359-1427
  • DOI: 10.2422/2036-2145.201809_008
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Stochastic quantization associated with the $\exp(\Phi )_2$-quantum field model driven by space-time white noise on the torus in the full $L^1$-regime (2020)
  • Author(s): 楠岡誠一郎
  • Organizer: 確率解析とその周辺
• [Presentation] 回転不変な$\Phi ^4_3$測度とその流れの構成 (2020)
  • Author(s): 楠岡誠一郎
  • Organizer: 確率論シンポジウム
• [Presentation] Invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: One day workshop on stochastic analysis
  • Int'l Joint Research / Invited
• [Presentation] Invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: 量子場の数理とその周辺 2019
  • Invited
• [Presentation] Approach to the Quantum Field with Exponential Interactions by Singular SPDEs (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Analysis, Random Fields and Integrable Probability (The 12th Mathematical Society of Japan, Seasonal Institute)
  • Int'l Joint Research / Invited
• [Presentation] Approach to the quantum field with exponential interactions by singular SPDEs (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Japanese-German Open Conference on Stochastic Analysis 2019
  • Int'l Joint Research / Invited
• [Presentation] Approach to the quantum field with exponential interactions by singular SPDEs (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Nonlinear and stochastic partial differential equations
  • Int'l Joint Research / Invited
• [Presentation] Stochastic quantization associated with the $\exp( \Phi )_2$-quantum field model driven by space-time white noise on the torus (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: 確率解析とその周辺
• [Presentation] Invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2019)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Taiwan Mathematical Society Annual Meeting
  • Int'l Joint Research / Invited
• [Presentation] The invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2018)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Regularity Structures and Stochastic Systems
  • Int'l Joint Research / Invited
• [Presentation] The invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2018)
  • Author(s): Seiichiro Kusuoka
  • Organizer: 9th International Conference on Stochastic Analysis and its Applications
  • Int'l Joint Research / Invited
• [Presentation] The invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2018)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Gaussian Free Fields and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2018)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Quantum field theory, renormalisation and stochastic partial differential equations
  • Int'l Joint Research / Invited
• [Presentation] 3次元トーラス上のPhi4量子場モデルの不変測度とその流れ (2018)
  • Author(s): 楠岡誠一郎
  • Organizer: 慶應確率論ワークショップ
• [Presentation] Recurrence of the Brownian motion in multidimensional semi-selfsimilar random environments (2017)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Workshop on Jump Processes and Stochastic Analysis
  • Int'l Joint Research / Invited
• [Presentation] The invariant measure and the flow associated to the Φ43-quantum field model (2017)
  • Author(s): Seiichiro Kusuoka
  • Organizer: Japanese-German Open Conference on Stochastic Analysis 2017
  • Int'l Joint Research / Invited
• [Presentation] The invariant measure and the flow associated to the Φ43-quantum field model (2017)
  • Author(s): 楠岡誠一郎
  • Organizer: 第13回非線型の諸問題
  • Invited
• [Presentation] The invariant measure and the flow associated to the Φ43-quantum field model (2017)
  • Author(s): Seiichiro Kusuoka
  • Organizer: 確率解析とその周辺
  • Int'l Joint Research
• [Presentation] The invariant measure and flow associated to the Phi4-quantum field model on the three-dimensional torus (2017)
  • Author(s): Seiichiro Kusuoka
  • Organizer: 異分野基礎科学研究所(RIIS) 第2回研究報告会
  • Int'l Joint Research
• [Presentation] The invariant measure and flow associated to the Φ43-quantum field model (2017)
  • Author(s): 楠岡誠一郎
  • Organizer: 確率論シンポジウム
• [Presentation] 3次元トーラス上のΦ4量子場モデルの不変測度とその流れ (2017)
  • Author(s): 楠岡誠一郎
  • Organizer: Saga Workshop on Partial Differential Equations
  • Invited
• [Remarks] Seiichiro Kusuoka's webpage
  • URL: https://www.math.kyoto-u.ac.jp/~kusuoka/
• [Remarks] Seiichiro Kusuoka's webpage
  • URL: http://www.cc.okayama-u.ac.jp/~kusuoka/