Stochastic analysis and semi-classical problem in infinite dimensional spaces

无限维空间中的随机分析和半经典问题

• 批准号: 15540169
• 负责人: AIDA Shigeki
• 金额: $ 2.3万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540169/
• 关键词: Semi-classical limit; Stochastic analysis; Rough path analysis; Witten Laplacian; Log-Sobolev inequality; Path integral; Quantum_field theory; ループ空間; 径路積分表示; 推移確率; シュレーディンガー作用素; ラプラスの方法; 弱ポアンカレ不等式; ラブラスの方法

The results are as follows :(1)We determine the limit of the lowest eigenvalue of a Schr"odinger operator on a Wiener space under semi-classical limit. This is the case where the coefficient operator of the Dirichlet form is identity operator. Also we extend the result to the case where the coefficient operator is variable case and the case of a Schr"odinger operator on a path space over a compact Riemannian manifold in a recent preprint. The points are to use a unitary transformation by an approximate ground state function and rough path analysis.(2)We prove a weak Poincare inequality on a loop space over a compact Riemannian manifold by using the rough path analysis.(3)We prove a weak Poincare inequality on a loop space over a compact Riemannian manifold by using the rough path analysis.

主要结果如下：（1）在半经典极限下，确定了Wiener空间上Schr“odinger算子最低特征值的极限。这是狄利克雷形式的系数算子是恒等算子的情况。本文还将结果推广到了系数算子为可变算子的情形和Schr“odinger算子在紧致黎曼流形上的路径空间上的情形。其要点是通过近似基态函数和粗糙路径分析，使用么正变换。(2)We利用粗糙路分析方法证明了紧致黎曼流形上loop空间上的一个弱Poincare不等式. (3)We利用粗糙路分析方法证明了紧致黎曼流形上loop空间上的一个弱Poincare不等式.

项目成果

On a Certain Semiclassical Problem on Wiener Spaces

• DOI: 10.2977/prims/1145476107
• 发表时间: 2003-06
• 期刊: Publications of The Research Institute for Mathematical Sciences
• 影响因子: 1.2
• 作者: S. Aida

Small-time asymptotic estimates in local Dirichlet spaces

• DOI: 10.1214/ejp.v10-286
• 发表时间: 2005-10-07
• 期刊: ELECTRONIC JOURNAL OF PROBABILITY
• 影响因子: 1.4
• 作者: Ariyoshi, T;Hino, M
• 通讯作者: Hino, M

Weak Poincaré inequalities on domains defined by Brownian rough paths

• DOI: 10.1214/009117904000000478
• 发表时间: 2004-10
• 期刊: Annals of Probability
• 影响因子: 2.3
• 作者: S. Aida

Shigeki Aida: "Semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space"Journal of Functional Analysis. 203・2. 401-424 (2003)

会田茂树：“维纳空间上薛定谔算子的最低特征值的半经典极限”泛函分析杂志203・2（2003）。

Semiclassical limit of the lowest eigenvalue of a Schr"odinger operator on a Wiener space

维纳空间上薛定谔算子最低特征值的半经典极限

• 发表时间: 2003
• 期刊: Journal of Functional Analysis 203・2
• 作者: 芦屋隆一;萬代武史;守本晃;Kawamura Shinzo;Shigeki Aida
• 通讯作者: Shigeki Aida