Stochastic analysis on loop space

循环空间的随机分析

• 批准号: 12640173
• 负责人: AIDA Shigeki
• 金额: $ 2.37万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640173/
• 关键词: Logarithmic Sobolev inequality; Schrodinger operator; Semiclassical limit; heat kernel; rough path; シュレーディンガー作用書; 準古典的近似; ラプラスの方法; スペクトルギャップ; 超縮小性

(1) We gave an estimate on the gap of spectrum of Schrodinger operators by using weak Poincare inequality. Also we gave an estimate on the distribution function of the ground state by the inequality.(2) Let H be the space of H^1-paths on a Euclidean space. Consider a Morse function on H which is a sum of the energy of the path and a smooth function on H which can be estended to a smooth function on the space of continuous paths. We defined a Witten Laplacian twisted by the Morse function on a Wiener space and proved that the first order behavior of the lowest eigenvalue under semiclassical limit is determined by the hessian of the Morse function.(3) Consider a continuous function F on the Cameron-Martin subspace of a classical Wiener space. Assume F can be extended to a continuous function F on the Wiener space. Then if the domain {F > 0} is a connected set, then weak Poincare inequalities hold on {F > 0}. We extend this result to the case where F is a continuous function of Brownian rough paths.(4) We proved very precise Gaussian estimates on heat kernels on Riemannian manifolds which possess poles under the assumptions that the curvature and the derivatives go to 0 sufficiently fast at infinity.

(1) We gave an estimate on the gap of spectrum of Schrodinger operators by using weak Poincare inequality. Also we gave an estimate on the distribution function of the ground state by the inequality.(2) Let H be the space of H^1-paths on a Euclidean space. Consider a Morse function on H which is a sum of the energy of the path and a smooth function on H which can be estended to a smooth function on the space of continuous paths. We defined a Witten Laplacian twisted by the Morse function on a Wiener space and proved that the first order behavior of the lowest eigenvalue under semiclassical limit is determined by the hessian of the Morse function.(3) Consider a continuous function F on the Cameron-Martin subspace of a classical Wiener space. Assume F can be extended to a continuous function F on the Wiener space. Then if the domain {F > 0} is a connected set, then weak Poincare inequalities hold on {F > 0}. We extend this result to the case where F is a continuous function of Brownian rough paths.(4) We proved very precise Gaussian estimates on heat kernels on Riemannian manifolds which possess poles under the assumptions that the curvature and the derivatives go to 0 sufficiently fast at infinity.

项目成果

S.Aida: "Precise Gaussian lower bounds on heat kernels"Stochastic in Finite and Infinite dimensions. 1-28 (2001)

S.Aida：“热核的精确高斯下界”有限和无限维度中的随机。

S.Aida: "An estimate of the gap of spectrum of Schrodinger operators which generate hyperhounded semi groups"Journal of Functional Analysis. 185. 474-526 (2001)

S.Aida：“生成超猎犬半群的薛定谔算子谱间隙的估计”泛函分析杂志。

S.Aida: "An estimate of the gag of spectrium of Sehrochinger operators which generate hyperbounded semigranps"Journal of Functional Analysis. 185. 474-526 (2001)

S.Aida：“对生成超界半粒的 Sehrochinger 算子谱的估计”函数分析杂志。

S.Aida: "Precise Gaussian estimater of heat kernols on asymptotically plat Riemannism menifolds with poles"Proceedings of the 1st Sino-German Conference on stochastic analysis. (to appear).

S.Aida：“带极点的渐近平黎曼流形上热核的精确高斯估计”第一届中德随机分析会议论文集。

S.Aida,T-S.Zhang: "On the short time asymptotics of transition probability of diffusion on path groups"Potential Analysic. (to appear).

S.Aida，T-S.Zhang：“路径群上扩散转移概率的短时渐近”势分析。