Stochastic analysis on infinite dimensional spaces and its related differential operators

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400134
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Okayama University
• Principal Investigator: KAWABI HIROSHI 岡山大学, 自然科学研究科, 教授 (80432904)
• Co-Investigator(Renkei-kenkyūsha): HIROKAWA MASAO 広島大学, 大学院工学研究院, 教授 (70282788)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 確率解析 / 微分作用素 / 中心極限定理 / 確率偏微分方程式 / rough path理論 / 結晶格子 / ベキ零被覆グラフ / ランダムウォーク

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(\phi)_{2}-measure, in Euclidean QFT, and proved strong uniqueness of Dirichlet operators of the corresponding Dirichlet operator. I also discussed the related singular stochastic partial differential equation. Besides, I studied central limit theorems for non-symmetric random walks on both crystal lattices and nilpotent covering graphs from a viewpoint of discrete geometric analysis, and discussed its application to rough path theory. Finally, I proved that the space of Gaussian measures on an abstract Wiener space becomes a Hilbert-Riemannian manifold with non-positive sectional curvatures.

Free Research Field

数物系科学