A study of stochastic analysis - synthesizing and integrating

(1)As摘要研究了随机振荡积分在路径空间上的精确表示、渐近性态及其在非线性偏微分方程中的应用。在相函数为二次Wiener泛函的情况下，利用相应的Hilbert-Schmidt算子建立了Levy-Ito型指数表达式.在多项式系数随机线积分相函数的情形下，给出了一个具体的渐近性态。通过证明高斯振幅函数情形下渐近行为在驻点附近的集中性，打开了一扇通向路径空间上的驻相方法的大门。最后，我们发现了KdV族的tau函数与Ornstein-Uhlenbeck过程的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)We研究了路空间上的具体泛函。几何布朗运动在金融数学、随机环境中的扩散、双曲空间上的布朗运动等研究中起着重要的作用，本文计算了几何布朗运动的时间积分所得到的指数泛函的分布，并建立了其密度函数的递推公式.计算了Wishart过程分布的绝对连续性，并给出了双曲空间上Laplacian形式的Selberg迹公式的概率证明.利用半稳定过程研究了p-adic上半平面上的迹公式。(3)An得到了一般空间上热核具有次高斯估计的一个简单适用的判据。构造了关于复杂性的扩散过程，并给出了相关热核的详细估计。

(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.