Short time asymptotic problems in Wiener functional integration theory. Applications to heat kernels and index theorems
Short time asymptotic problems in Wiener functional integration theory. Applications to heat kernels and index theorems
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维纳泛函积分理论中的短时渐近问题。
DOI：
10.1007/bfb0083609
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
Shinzo Watanabe
中科院分区：
文献类型：
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作者：
Shinzo Watanabe
自1923年由N.Wiener提出Wiener测度空间以来，路径空间积分的严密理论在数学和数学物理中得到了许多有趣的应用。特别地，M.Kac建立了Feynman-Kac公式，并将其应用于薛定谔算子谱理论和位势理论中的几个问题。如果我们想把Kac的理论推广到弯曲的黎曼空间，我们需要利用Wiener空间上的一个重要的随机演算，即Ito的随机演算。事实上，黎曼流形上的布朗运动和随机活动标架等重要概念可以通过求解Ito的随机微分方程组来构造。我演讲的主要目的是讨论这种通过Wiener泛函积分获得热核迹(超迹)的短时渐近的概率方法。众所周知，拉普拉斯本征值的渐近性、指数定理、不动点公式、Morse函数的Morse不等式、向量场的Poincare-Hopf指数定理等分析、几何和数学物理中的许多重要问题本质上都与热核迹的估计问题有关。Wiener泛函积分方法首先用积分来表示热核
Since the Wiener measure space was introduced by N. Wiener in 1923, a rigorous theory of path space integrals has been developed with many interesting applications to mathematics and mathematical physics. Especially, the Feynman-Kac formula was established by M. Kac and it was applied to several problems in the spectral theory of Schrodinger operators and potential theory. If we want to extend Kac's theory to curved Riemannian spaces, we need to make use of an important stochastic calculus on the Wiener space, that is, Ito's stochastic calculus. Indeed, such important notions as Brownian motions and stochastic moving frames on Riemannian manifolds can be constructed by solving Ito's stochastic differential equations. The main purpose of my lecture is to discuss this probabilistic approach by the Wiener functional integration to obtain short time asymptotics of traces (supertraces) of heat kernels. It is wellknown that many important problems in analysis, geometry and mathematical physics, such as asymptotics of eigenvalues of the Laplacian, index theorems, fixed point formulas, Morse inequalities for Morse functions, Poincare-Hopf index theorem for vector fields and so on are essentially related to this problem of estimating traces of heat kernels. The method of Wiener functional integration consists of first representing the heat kernels by integrals of
反对称 Malliavin 微积分及其应用
DOI：
--
发表时间：
2007
期刊：
影响因子：
--
作者：
大江貴司;大中幸三郎;赤堀 次郎;M. Ikehata and T. Ohe;赤堀 次郎;K. Ohnaka;Jiro Akahori
通讯作者：
Jiro Akahori