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.     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

The exit distribution for iterated Brownian motion in cones

We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previously obtained.     

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.     

The local time of iterated Brownian motion

We define and study the local time process {L ^*(x,t);x¹,t0} of the iterated Brownian motion (IBM) {H(t):=W ₁(|W ₂ (t)|); t0}, whereW ₁(·) andW ₂(·) are independent Wiener processes.Research supported by Hungarian National Foundation for Scientific Research, Grant No. T 016384.Research supported by an NSERC Canada Grant at Carleton University, Ottawa.Research supported by a PSC CUNY Grant, No. 6-66364.     

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

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.     

Rates of convergence for increments of Brownian motion

The results of this paper concern rates of convergence for increments of Brownian motion. As a by-product we give some improvements of a result of Bolthausen dealing with Strassen's law of the iterated logarithm.     

N指标d维广义Brownian Sheet的像集及图集的Hausdorff测度

本文得到了Ｎ指标ｄ维广义Ｂｒｏｗｎｉａｎ Ｓｈｅｅｔ的像集及图集的Ｈａｕｓｄｏｒｆｆ测度的上、下界．     

Sharp integrability for Brownian motion in parabola-shaped regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains , 0<α<1. Together with some simple good-λ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for α=1/2 by Bañuelos et al. (Ann. Probab. 29 (2001) 882) and for general α by Li (Ann. Probab. 31 (2003) 1078). A sharp version of this result is also proved in higher dimensions.     

一类含参变量的Sierpinski垫片的Hausdorff测度

Sierpinski垫片是具有严格自相似性的经典分形集之一.本文给出了一类含参变量的Sierpinski垫片.通过它在x轴上的投影估计了这类Sierpinski垫片的Hausdorff测度的下界,然后精心构造了一个仿射变换,将参变量的范围由(0,π/3)的讨论转换到(π/3,π)的讨论,从而得到了这类Sierpinski垫片的Hausdorff测度的精确值.     

Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

On the connected components of the support of super Brownian motion and of its exit measure

Tribe proved in a previous paper that a typical point of the support of super Brownian motion considered at a fixed time is a.s. disconnected from the others when the space dimension is greater than or equal to 3. We give here a simpler proof of this result based on Le Gall's Brownian snake. This proof can then be adapted in order to obtain an analogous result for the support of the exit measure of the super Brownian motion from a smooth domain of ^d when d is greater than or equal to 4.     

Hitting probabilities and fractal dimensions of multiparameter multifractional Brownian motion

The main goal of this paper is to study the sample path properties for the harmonisable-type N-parameter multifractional Brownian motion, whose local regularities change as time evolves. We provide the upper and lower bounds on the hitting probabilities of an （N, d）-multifractional Brownian motion. Moreover, we determine the Hausdorff dimension of its inverse images, and the Hausdorff and packing dimensions of its level sets.     

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

Some Results on Fractional Brownian Sheets and Their Local Times

Let B0^H = {B0^H（t）,t ∈ R＋^N） be a real-valued fractional Brownian sheet. Define the （N,d）- Gaussian random field B^H by
B^H（t） = （B1^H（t）,...,Bd^H（t）） t ∈ R＋^N, where B1^H, ..., Bd^H are independent copies of B0^H. The existence and joint continuity of local times of B^H is proven in some given conditions in [22]. We then study further properties of the local times of B^H, such as the moments of increments of local times, the large increments and the maximum moduli of continuity of local times and as a result, we answer the questions posed in [22].