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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)对于路径空间上的傅里-拉普拉斯变换随机振荡积分（简称SOI），研究了其精确表达式、渐近性质及其在非线性偏微分方程中的应用。对于二次Wiener泛函的相函数，利用相关的Hilbert-Schmidt算子建立了Levy-Ito型指数表达式。此外，对于多项式系数随机线积分的相函数，给出了一个具体的渐近性质。通过展示高斯振幅函数的渐近特性在平稳点附近的集中，为路径空间上的平稳相位法打开了一扇大门。最后，我们发现了KdV层次的tau函数与与Ornstein-Uhlenbeck过程相关的SOI之间的双射关系，这是随机分析在非线性偏微分方程理论中的一个全新应用。(2)对路径空间的具体泛函进行了研究。本文计算了几何布朗运动的指数泛函的时间积分分布，并建立了其密度函数的递推公式。几何布朗运动在数学金融、随机环境中的扩散和双曲空间上的布朗运动的研究中起着关键作用。此外，还计算了Wishart过程分布的绝对连续性，给出了双曲空间上拉普拉斯形式的Selberg迹公式的一个概率证明。进一步，通过半稳定过程研究了p进上半平面上的示踪公式。(3)得到了一般空间上热核具有亚高斯估计的一个易于应用的判据。构造了基于复杂度的扩散过程，建立了关联热核的详细估计。