Research on differential operators on infinite dimensional spaces via stochastic analysis

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05300
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Keio University (2018-2021); Okayama University (2017)
• Principal Investigator: Kawabi Hiroshi 慶應義塾大学, 経済学部(日吉), 教授 (80432904)
• Co-Investigator(Kenkyū-buntansha): 楠岡 誠一郎 京都大学, 理学研究科, 准教授 (20646814)
• Project Period (FY): 2017-04-01 – 2022-03-31
• 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 / マリアヴァン解析 / Dirichlet形式

Outline of Final Research Achievements

I mainly studied uniqueness problems of differential operators and the corresponding stochastic dynamics on infinite dimensional spaces via stochastic analysis. In particular, I considered Dirichlet forms given by space-time quantum fields with interactions of exponential type, called exp(Φ)_{2}-measure, in Euclidean QFT, and proved strong uniqueness of Dirichlet operators defined through the Dirichlet forms. I also characterized the corresponding diffusion process as a unique strong solution to a singular stochastic partial differential equation. Besides, I proved two kinds of functional central limit theorems for non-symmetric random walks on nilpotent covering graphs by combining discrete geometric analysis with rough path theory.

Free Research Field

数物系科学

Academic Significance and Societal Importance of the Research Achievements

近年の特異な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。またベキ零被覆グラフ上のランダムウォークの中心極限定理に関する成果であるが, 確率微分方程式の解の離散近似とも解釈できるので, 確率数値解析への波及も期待できる。