Multilateral research on infinite dimensional differential operators via stochastic analysis

基于随机分析的无限维微分算子多边研究

• 批准号: 23740107
• 负责人: KAWABI HIROSHI
• 金额: $ 2.75万
• 依托单位: Okayama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23740107/
• 关键词: 確率解析; 経路空間; 微分作用素; 一意性定理; 確率偏微分方程式; Rough Path理論; 推移確率; 漸近挙動; Gibbs測度; ラフパス解析

I mainly studied uniqueness problems of differential operators and corresponding stochastic dynamics on infinite dimensional spaces via stochastic analysis. In particular, I proved strong uniqueness of Dirichlet operators for Gibbs measures which appear in quantum field theory and constructed a unique solution to the corresponding stochastic PDE by using Dirichlet form theory. Besides, I studied precise asymptotic behavior (i.e., the method of stationary phase) of some oscillatory functional integrals by combining Rough Path theory with Malliavin calculus. Motivated by this study of the stationary phase method, I also obtained an explicit effect of non-symmetry on the long time asymptotic behavior for a class of non-symmetric random walks on the triangular lattice.

我主要通过随机分析研究无限维空间上微分算子的唯一性问题和相应的随机动力学问题。特别是，我证明了量子场论中出现的吉布斯测度的狄利克雷算子的强唯一性，并利用狄利克雷形式理论构造了相应随机偏微分方程的唯一解。此外，我还结合粗糙路径理论和马利亚文微积分研究了一些振荡函数积分的精确渐近行为（即平稳相方法）。受固定相方法研究的启发，我还获得了非对称性对三角晶格上一类非对称随机游走的长时间渐近行为的显式影响。

项目成果

Strong uniqueness of diffusions on a path space with interactions

具有相互作用的路径空间上的扩散具有很强的唯一性

• 发表时间: 2012
• 作者: 河備 浩司

Strong uniqueness of diffusions to Gibbs measures on a path space with exponential and polynomial interactions

具有指数和多项式相互作用的路径空间上吉布斯测度扩散的强唯一性

• 作者: Hiroshi Kawabi

Rockner, "Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions"

Rockner，“狄利克雷算子和随机动力学在具有指数相互作用的路径空间上的吉布斯测量具有很强的唯一性”

• 发表时间: 2012
• 作者: S. Albeverio; H. Kawabi;M
• 通讯作者: M

Stationary phase for rough differential equations

粗略微分方程的固定相

• 发表时间: 2014
• 作者: Hiroshi Kawabi

Uniqueness of Dirichlet forms related to stochastic quantization under exponential interaction in two-dimensional finite volume

二维有限体积指数相互作用下随机量子化狄利克雷形式的唯一性

• 作者: Hiroshi Kawabi