布朗单极集的性质

本文研究了布朗单极集的性质，得到了布朗单极性的充分条件与必要条件。     

布朗单的极函数

本文研究了布朗单极函数的特征，得到了满足Ｌｉｐｓｃｈｉｔｚ条件的连续函数类与布朗单极函数类之间的关系。同时我们还得到了布朗单不动点的Ｈａｕｓｄｏｒｆｆ维数及Ｋｏｌｍｏｇｏｒｏｖ下熵指数的一个不等式。     

布朗单的钟泛函重对数律

本文研究了布朗单轨道的泛函极限问题,得到了布朗单的钟泛函重对数律.证明中的主要工具是布朗单的大偏差和小偏差.     

分数布朗单的幂函数随机积分逼近

本文研究了分数布朗单的逼近问题.利用Wiener积分,得到了分数布朗单的幂函数型随机积分逼近.     

布朗单增量的Strassen重对数律

利用布朗单与布朗单增量的大偏差,得到了布朗单与布朗单增量的泛函重对数律.     

N-d维广义Brownian单的像集一致Packing测度和一致Packing维数

我们获得了Ｎ－ｄ维广义Ｂｒｏｗｎｉａｎ单的像集的一致Ｐａｃｋｉｎｇ测度和一致Ｐａｃｋｉｎｇ维数。     

布朗单象集代数和的几个性质

本文研究了ｄ维布朗单Ｗ（ｔ）的象集代数和的几个性质。     

分数布朗单驱动的随机微分方程的传输不等式

本文研究了分数布朗单驱动的随机微分方程的问题,利用Girsanov变换,获得了该方程的解的分布在在连续轨道空间上关于一致度量满足T2传输不等式的结果,推广了现有文献中的结论.     

如何计算可实现Fuzzy矩阵的容度

本文给出一种［ｒ（Ａ）］ｎ２步内找到Ｆｕｚｚｙ矩阵Ｂ∈　Ｌｎ×ｒ（Ａ）,使Ａ＝Ｂ⊙ＢＴ成立,从而计算出给定可实现Ｆｕｚｚｙ矩阵Ａ的容度ｒ（Ａ）的算法．     

随机发展方程小扰动的容度大偏差原理

本文证明了随机发展方程小扰动的容度大偏差原理,所获结果改进和加强已知的结果     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

Strassen-type Laws of Iterated Logarithm for a Fractional Brownian Sheet

Abstract

We study Strassen-type laws of iterated logarithm for a fractional Brownian sheet including that for small time, which imply most of the former laws of the iterated logarithm and Strassen's laws for one-parameter and two-parameter Wiener processes.     

Large Time Asymptotics for Brownian Hitting Densities of Transient Concave Curves

With appropriate regularity assumptions on the increasing concave function x=(t)<0, the hitting time density p(t) for a transient curve x=(t) by a 1-dimensional Brownian motion is shown to satisfy Here r is the probability of eventually hitting the curve and (t)=t ^–1/2(t).     

Images of the Brownian sheet

An -parameter Brownian sheet in maps a non-random compact set in to the random compact set in . We prove two results on the image-set :

(1) It has positive -dimensional Lebesgue measure if and only if has positive -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999).

(2) If , then with probability one, we can find a finite number of points such that for any rotation matrix that leaves in , one of the 's is interior to . In particular, has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989).

This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).' Both ideas may be of independent interest.

We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).

     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.     

Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.