几类随机过程的Karhunen-Loeve展开及小球概率估计的研究

This project does the research on Karhunen-Loeve expansions for Gaussian processes. By the methods of stochastic Fubini theorem, Fredholm determinant theory and the boundary value theory of differential equations， we will study one dimensional complex Gaussian processes and non-tensered Gaussian processes. We will more focus on the Karhunen-Loeve expansions for the bivariate Brownian bridge and Gaussian processes in the p-adic field，Furthermore，new probable phenomenon and structure in the p-adic field will be discussed. As applications，large deviation， small deviation and Laplace transform for the L_2 norm of the processes will be given. There are only a few KL expansions for Gaussian processes known in the literature，and there are quite lots of Gaussian processes in theoretical and practical meaning，whose KL expansions are unknown and need to be solved by searching new useful methods. Through several special important Gaussian processes，new methods will be explored, and the set of KL expansions for Gaussian processes will be extended, further the associated results for general stochastic processes will be concentrated on. The project is an important theoretical significance and application prospects clearly.

本项目研究高斯过程的Karhunen-Loeve(KL)展开问题。通过运用随机Fubini定理，Fredholm 行列式理论，微分方程边值理论研究一维复杂高斯过程的 KL 展开问题及非张量高斯过程的KL展开问题。重点研究双变量布朗桥及p-adic数域上的高斯过程的KL展开方法，进而探索由非阿基米德数域出发所产生的新现象与新结构。作为应用对上述过程的L_2 范数给出大、小偏差及 Laplace 变换。从已有文献可知，目前具有精确KL展开的高斯过程的研究结果很少，还有很多有着理论和实际意义的高斯过程的KL展开工作急需有效办法解决。本项目将开拓新的研究方法，从几类具有特殊意义的重要的高斯过程入手，力求丰富已有KL展开的高斯过程的集合，并将研究推广到有实际背景的更一般的随机过程。本项目是高斯过程理论中具有重要理论意义和明确应用前景的研究课题。

在该项目中我们主要研究了以下与布朗运动有关的高斯过程的内容。我们研究了demeaned 平稳 Ornstein–Uhlenbeck 过程及其可加过程的 Karhunen–Loève（KL）和分布等式，分别证明了 L2 范数意义下的小球估计和 Laplace 变换。研究了可加 detrended 布朗运动及可加两边布朗运动的KL展开及Pythagorean型的分布等式，分别证明了在 L2 范数意义下的小球估计和 Laplace 变换。用Fredholm 积分方程理论研究了非张量高斯场，双变量布朗运动的 KL 展开问题。最后我们研究了可加 m 次 detrended 布朗运动的KL展开及Pythagorean型的分布等式，证明了在L2范数意义下的小球估计和 Laplace 变换。