SG(2,2)上布朗运动的维数性质

本文研究分形集合ＳＧ（２,２）上布朗运动的维数性质,得到了ＳＧ（２,２）上布朗运动的样本图以及象集的Ｈａｕｓｄｏｒｆｆ维数与盒维数。     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

Kahane using brownian local times of intersection

We show that if a set E in the positive real line has Hausdorff dimension greater than d/2 m, then the m-fold algebraic sum of the image of E by d-dimensional Brownian motion has an interior point. This extends a result of Kahane. The proof uses techniques found in Rosen (1983) and Geman, Horowitz and Rosen. We then show that the results do not hold for random sets and demonstrate that the above condition on the Hausdorff dimension of E is not close to being necessary     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

SOME GEOMETRIC PROPERTIES OF BROWNIAN MOTION ON SIERPINSKI GASKET

Let {X(t),t≥0} be Brownian motion on Sierpinski gasket,The Hausdorff and packing dimensions of the image of a ompact set are studied,The uniform Hausdorff and packing dimensions of the inverse image are also discussed.     

Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.     

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

On sojourn of Brownian motion inside moving boundaries

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets, are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.     

Boundary trace of reflecting Brownian motions

We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains. Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are determined. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also given.Mathematics Subject Classification (2000):Primary 60G17, 60J60, Secondary 28A80, 30C35, 60G52, 60J50     

一类半鞅样本轨道的Fractal性质

本文用Ｈａｕｓｄｏｒｆｆ维数刻画了一类半这样本轨道的Ｆｒａｃｔａｌ性质，并由此研究了Ｂｒｏｗｎ运动与有界变差过程的碰撞问题．     

Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII

We penalise Brownian motion by a function of its one-sided supremum considered up to the last zero before t, respectively first zero after t, of that Brownian motion. This study presents some analogy with penalisation by the longest length of Brownian excursions, up to time t.     

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In a paper from 1954 Marstrand proved that if K⊂R² has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^1+α, α>0, for which the sum of their Hausdorff dimension is greater than 1.     

POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS

Let BH,K = {BH,K(t), t ∈ RN+ } be an (N, d)-bifractional Brownian sheet with Hurst indices H=(H1,···,HN) ∈(0,1)N and K=(K1,···,KN) ∈(0,1]N. The properties of the polar sets of BH,K are discussed. The sufficient conditions and necessary conditions for a compact set to be polar for BH,K are proved. The infimum of Hausdorff dimensions of its non-polar sets are obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.     

Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion

Burdzy and Khoshnevisan⁽⁹⁾ have shown that the Hausdorff dimension of the level sets of an iterated Brownian motion (IBM) is equal to 3/4. In this paper, the exact Hausdorff measure function and the packing measure of the levels set of IBM are given. Our approach relies on some accurate analysis on the local asymptotic of local times.     

非退化扩散过程的相交性与极函数

该文证明了在适当的条件下,任何两个独立的一维扩散过程相交的概率为1,相交的时间集的Hausdorff维数为;讨论了扩散过程的极函数,在适当的条件下,得到了类似于Brown运动一样的结果．     

CONVERGENCE RATE OF AN APPROXIMATION TO MULTIPLE INTEGRAL OF FBM

In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.     

Fractional Brownian Flows

We consider a stochastic flow on ? ⁿ driven by a fractional Brownian motion with Hurst parameter \(H\in(\frac{1}{2},1)\) and study a tangent flow and the growth of the Hausdorff measure of sub-manifolds of ? ⁿ as they evolve under the flow.The main result is a bound on the rate of (global) growth in terms of the (local) Hölder norm of the flow.     

广义分数布朗运动的若干性质

本文, 我们定义了一类新的分数布朗运动, 研究了它的局部非决定性和局部时的联合连续性, 最后给出了它的水平集的Hausdorff维数的上、下界.     

Bivariate occupation measure dimension of multidimensional processes

Bivariate occupation measure dimension is a new dimension for multidimensional random processes. This dimension is given by the asymptotic behavior of its bivariate occupation measure. Firstly, we compare this dimension with the Hausdorff dimension. Secondly, we study relations between these dimensions and the existence of local time or self-intersection local time of the process. Finally, we compute the local correlation dimension of multidimensional Gaussian and stable processes with local Hölder properties and show it has the same value that the Hausdorff dimension of its image have. By the way, we give a new a.s. convergence of the bivariate occupation measure of a multidimensional fractional Brownian or particular stable motion (and thus of a spatial Brownian or Lévy stable motion).     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.