A study of stochastic analysis - synthesizing and integrating

随机分析研究——综合与整合

基本信息

批准号：
14204010
负责人：
TANIGUCHI Setsuo
金额：
$ 20.47万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

(1)As for stochastic oscillatory integrals (SOI in short), which are Fourie-Laplace transforms on the path space, studies on exact expression, asymptotic behavior, and application to nonlinear partial differential equation were made. In the case of phase function of quadratic Wiener functional, a Levy-Ito type exponential expression was established with the associated Hilbert-Schmidt operator. Moreover, in the case of phase function of stochastic line integral with polynomial coefficients, a concrete asymptotic behavior was shown. A door to the stationary phase method on the path space was opened by showing the concentration around stationary points of the asymptotic behavior in the case of Gaussian amplitude function. Finally, we found out a bijective relation between the tau function of the KdV hierarchy and SOI' s associated with Ornstein-Uhlenbeck processes, which is a completely new application of stochastic analysis to nonlinear PDE theory. (2)We made studies on concrete functionals on path spaces. We computed the distribution of the exponential functional obtained as time-integral of geometric Brownian motion, which plays a key role in studies of mathematical finance, diffusions in random environments, and Brownian motion on hyperbolic space, and established the recursive formula for its density function. Moreover, the absolute continuity of the distribution of the Wishart process was computed, and a probabilistic proof of Selberg trace formulas for Laplacian on forms on hyperbolic spaces was given. Furthermore, the trace formula on p-adic upper half plane was studied via semi-stable processes. (3)An easily-applicable criterion for heat kernel on general space to have a sub-Gaussian estimate was achieved. Diffusion processes on complexity were constructed, and a detailed estimation of associated heat kernel was established.

（1）对于随机振荡积分（简称SOI），这是路径空间上的四型宽段转换，对精确表达，渐近行为的研究以及对非线性部分偏微分方程的应用。在二次Wiener函数的相位函数的情况下，与相关的Hilbert-Schmidt操作员建立了征ito类型的指数表达式。此外，在随机线与多项式系数积分的相位函数的情况下，显示了混凝土渐近行为。在高斯振幅功能的情况下，通过显示渐近行为固定点的固定点的浓度来打开路径空间上固定相法的门。最后，我们发现了KDV层次结构的Tau功能与SOI与Ornstein-Uhlenbeck过程相关的二光线关系，这是随机分析在非线性PDE理论中的全新应用。（2）我们对路径空间上的混凝土功能进行了研究。我们计算了作为几何布朗运动的时间综合性获得的指数功能的分布，后者在数学金融研究，随机环境中的扩散以及布朗在双曲空间上起着关键作用，并在双曲线空间上运动，并确定了其密度功能的递归公式。此外，计算了WishArt过程分布的绝对连续性，并给出了双曲线空间上selberg痕量公式的概率证明。此外，通过半稳定过程研究了padic上半平面上的痕量公式。（3）实现了一般空间上的热核的易于应用的标准，以实现高出估计的估计值。构建了关于复杂性的扩散过程，并建立了相关热内核的详细估计。