Analysis of sample paths for stochastic processes

1. 我们得到了三维晶格上随机游走范围的一个几乎确定的不变性原理。这个结果是对中心极限定理等已知结果的改进，并且可以推导出各种极限定理作为该结果的必然结果。对于二维点阵上的随机游走范围，我们得到了迭代对数定律。出现的不是预期的对数项，而是对数项。这部作品将刊登在《Ann》杂志上。Probab.2。我们研究了当无序介质在欧几里德空间中时，热量如何从空间向介质传递的问题。应用Besov空间理论，构造了一个类似于介质上的扩散和介质外欧几里德空间上的布朗运动的穿透扩散过程。这项工作出现在J. Funct中。分析的。然后我们继续求解这个问题，得到了扩散过程的热核的短时间渐近性质。我们还得到了该过程的函数型大偏差。我们现在正在写一篇关于这些结果的论文。我们得到了无限维对称扩散过程中两个集合间转移概率的渐近性质。我们采用内禀距离而不是通常的集合间距离，我们的结果是相当普遍的。特别地，我们证明了riemann流形上环空间上的Orstein-Uhlenbeck过程的渐近性，这是一个开放问题。我们的结果对于欧氏空间上的简并对称扩散也是新的。这是与J. a .拉米雷斯合作的作品，现已提交出版。

1. We have obtained an almost sure invariance principle for the range of random walks on 3-dimensional lattice. This result is a refinement of the known results like, central limit theorem, and various limit theorem can be deduced as a corollary to this result. For the range of random walks on 2-dimensional lattice, we have obtained the law of iterated logarithm. Instead of the expected loglog term, logloglog term appears.This work will appear in Ann. Probab.2. We have studied a problem that when disordered media are in a Euclidean space, how does the heat transfers from the space to the media. Applying the theory of Besov spaces, we have constructed a penetrating diffusion process which behaves like the diffusion on the media and like Brownian motion on the Euclidean space outside the media. This work appears in J. Funct. Anal. We then continue to work this problem and obtain a short time asymptotic behavior of the heat kernel for the diffusion process. We also obtain a functional type large deviation for the process.We are now writing a paper on these results.3. We have obtained asymptotic behavior of the transition probabilities between two sets on infinite dimensional symmetric diffusion processes. We adopted intrinsic distance instead of the usual distance between sets and our result is fairly general. In particular, we have proved the asymptotic behavior for the Orstein-Uhlenbeck processes on loop spaces on Riemannian manifolds, which had been an open problem. Our result is new also for degenerated symmetric diffusions on Euclidean spaces. This is a joint work with J. A. Ramirez and is now submitted for publication.